"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { annotation: "annotation", blockquote: "blockquote", br: "br", em: "em", h2: "h2", h3: "h3", li: "li", math: "math", mfrac: "mfrac", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", msup: "msup", ol: "ol", p: "p", semantics: "semantics", span: "span", strong: "strong", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsx(_components.h2, { children: _jsx(_components.strong, { children: "Learning Outcomes" }) }), "\n", _jsx(_components.p, { children: "This article explains how neoclassical and endogenous growth models account for long-term economic growth and convergence patterns in a way that aligns with CFA Level 2 exam expectations, including:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Distinguishing the core assumptions of classical, neoclassical, and endogenous growth theories, with emphasis on how capital, labor, technology, and returns to scale drive long-run output per worker." }), "\n", _jsx(_components.li, { children: "Using the Cobb‑Douglas production function to derive steady‑state and sustainable growth rates, interpret capital deepening, and evaluate the impact of diminishing marginal product of capital." }), "\n", _jsx(_components.li, { children: "Applying growth accounting relations to decompose potential GDP growth into technology, capital, and labor contributions and to check the plausibility of long-run earnings and GDP forecasts." }), "\n", _jsx(_components.li, { children: "Explaining how endogenous growth models incorporate human capital, R&D, innovation incentives, and knowledge spillovers to generate permanently higher growth rates and justify policy intervention." }), "\n", _jsx(_components.li, { children: "Evaluating absolute, conditional, and club convergence hypotheses, linking them to cross‑country growth outcomes and institutional quality." }), "\n", _jsx(_components.li, { children: "Assessing policy implications under each framework, distinguishing clearly between temporary level effects and permanent growth rate effects, and relating these distinctions to investment and valuation analysis in typical CFA item sets." }), "\n", _jsx(_components.li, { children: "Comparing the qualitative predictions of neoclassical and endogenous models when analyzing shocks to savings, population growth, technology, and pro‑innovation policies." }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "CFA Level 2 Syllabus" }) }), "\n", _jsx(_components.p, { children: "For the CFA Level 2 exam, you are required to understand economic growth and cycles in the context of neoclassical and endogenous growth models, with a focus on the following syllabus points:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Compare classical, neoclassical, and endogenous growth theories." }), "\n", _jsx(_components.li, { children: "Interpret how steady-state (sustainable) growth rates are determined in neoclassical models." }), "\n", _jsx(_components.li, { children: "Apply the Cobb‑Douglas production function and growth accounting to forecast potential GDP growth." }), "\n", _jsx(_components.li, { children: "Explain the role of innovation, human capital, and government policies in endogenous growth models." }), "\n", _jsx(_components.li, { children: "Evaluate absolute, conditional, and club convergence hypotheses and their empirical support." }), "\n", _jsx(_components.li, { children: "Assess the expected effects of investment in technology and human capital on long-run growth." }), "\n", _jsx(_components.li, { children: "Discuss why governments may subsidize private investment in knowledge and innovation." }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Test Your Knowledge" }) }), "\n", _jsx(_components.p, { children: _jsx(_components.em, { children: "Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision." }) }), "\n", _jsx(_components.p, { children: "An analyst is comparing growth prospects for three economies—Alpha, Beta, and Gamma—using both neoclassical and endogenous growth frameworks." }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Alpha is a developed economy with high capital per worker, labor’s share of income of 60%, TFP growth of 1.5% per year, and labor force growth of 0.5% per year. Savings and investment rates are stable." }), "\n", _jsx(_components.li, { children: "Beta is an emerging economy with low capital per worker, labor’s share of 70%, TFP growth of 1.5% per year, and labor force growth of 2.0% per year. It has weak property rights and limited access to international capital." }), "\n", _jsx(_components.li, { children: "Gamma is similar to Beta in income per capita and demographics, but it has strong institutions, invests heavily in secondary and tertiary education, and recently introduced generous R&D tax credits for firms." }), "\n"] }), "\n", _jsx(_components.p, { children: "Use this information to answer Questions 1–4." }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.span, { children: "Using the neoclassical growth model, which economy is most likely to experience the highest sustainable growth rate of output per capita?" }), _jsxs(_components.ol, { type: "a", children: [_jsx(_components.li, { children: "Alpha" }), _jsx(_components.li, { children: "Beta" }), _jsx(_components.li, { children: "Gamma" }), _jsx(_components.li, { children: "All three will have the same sustainable per capita growth rate" })] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.span, { children: "Based on neoclassical theory, which statement best characterizes the effect of a higher savings rate in Beta (assuming institutions do not change)?" }), _jsxs(_components.ol, { type: "a", children: [_jsx(_components.li, { children: "It permanently increases Beta’s per capita growth rate by raising the investment share of GDP" }), _jsx(_components.li, { children: "It temporarily increases Beta’s growth rate until the economy reaches a higher steady-state level of output per worker" }), _jsx(_components.li, { children: "It has no impact on either the level or the growth rate of output per worker" }), _jsx(_components.li, { children: "It permanently increases both the level and growth rate of output per worker" })] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.span, { children: "According to endogenous growth theory, which feature of Gamma is most likely to justify a higher long-run growth rate than Beta?" }), _jsxs(_components.ol, { type: "a", children: [_jsx(_components.li, { children: "Larger labor force" }), _jsx(_components.li, { children: "Higher labor share of income" }), _jsx(_components.li, { children: "Stronger property rights and R&D incentives that raise the rate of innovation" }), _jsx(_components.li, { children: "Lower initial capital per worker" })] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.span, { children: "Observed over several decades, Gamma’s income per capita converges toward that of Alpha, while Beta continues to lag behind. Which convergence concept best explains this pattern?" }), _jsxs(_components.ol, { type: "a", children: [_jsx(_components.li, { children: "Absolute convergence" }), _jsx(_components.li, { children: "Conditional convergence driven purely by population growth differences" }), _jsx(_components.li, { children: "Club convergence based on institutional and policy similarity" }), _jsx(_components.li, { children: "No convergence, because initial income differences persist" })] }), "\n"] }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Introduction" }) }), "\n", _jsx(_components.p, { children: "Understanding why some economies grow rapidly while others stagnate is central to macroeconomic analysis and long-run investment decisions. Growth models provide structured ways to think about:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "How output per worker and living standards change over time" }), "\n", _jsx(_components.li, { children: "Whether poorer countries are likely to catch up to richer ones" }), "\n", _jsx(_components.li, { children: "Which policies can sustainably influence growth rates" }), "\n"] }), "\n", _jsx(_components.p, { children: "For Level 2, the key is not just memorizing definitions, but being able to:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Interpret growth forecasts built on these models" }), "\n", _jsx(_components.li, { children: "Analyze the likely impact of policy changes" }), "\n", _jsx(_components.li, { children: "Connect growth assumptions to equity and fixed-income valuation (e.g., forecasts of aggregate earnings and potential GDP)" }), "\n"] }), "\n", _jsx(_components.p, { children: "Two frameworks dominate the modern curriculum:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The ", _jsx(_components.strong, { children: "neoclassical growth model" }), ", which emphasizes diminishing returns to capital and exogenous technological progress."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Endogenous growth models" }), ", which bring innovation and human capital decisions inside the model and allow policy to permanently affect growth."] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: classical growth theory" }), _jsx(_components.br, {}), "\n", "Classical (Malthusian) growth theory argues that any increase in real GDP per capita above a subsistence level raises population growth, which eventually drives output per capita back to subsistence; long-run gains in living standards are not sustainable."] }), "\n"] }), "\n", _jsx(_components.p, { children: "Classical growth theory is historically important but is largely rejected empirically. Neoclassical and endogenous models, by contrast, provide the basis for current exam questions and for practical growth and valuation analysis." }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: neoclassical growth theory" }), _jsx(_components.br, {}), "\n", "Neoclassical growth theory focuses on the economy’s long-run steady-state (sustainable) growth rate, determined by exogenous technological progress, capital accumulation with diminishing marginal product, and labor force growth."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: convergence hypothesis" }), _jsx(_components.br, {}), "\n", "The convergence hypothesis is the idea that poorer economies will, under certain conditions, grow faster than richer ones and eventually catch up in terms of output per capita."] }), "\n"] }), "\n", _jsx(_components.p, { children: "The remaining sections examine how these models use the Cobb‑Douglas production function, how they treat technology, and what they imply for convergence and policy." }), "\n", _jsx(_components.h2, { children: "THE NEOCLASSICAL GROWTH MODEL" }), "\n", _jsx(_components.p, { children: "The neoclassical growth model explains long-run economic growth in terms of capital, labor, and an exogenous technology parameter. A convenient representation is the Cobb‑Douglas production function:" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "Y" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "T" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "K" }), _jsx(_components.mi, { children: "α" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "L" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Y = T K^{\\alpha} L^{1-\\alpha}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8641em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0037em" }, children: "α" }) }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8641em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0037em" }, children: "α" })] }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["where ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Y" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Y" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" })] }) })] }), " is output, ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), " is physical capital, ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "L" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " is labor, ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "T" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }), " is the level of technology, and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0 < \\alpha < 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6835em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), " is capital’s share of income."] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: Cobb-Douglas production function" }), _jsx(_components.br, {}), "\n", "A production function of the form ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "Y" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "T" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "K" }), _jsx(_components.mi, { children: "α" })] }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "L" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Y = T K^{\\alpha} L^{1-\\alpha}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6644em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0037em" }, children: "α" }) }) })] }) }) }) }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "L" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8141em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "1" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0037em" }, children: "α" })] }) })] }) }) }) }) })] })] })] })] }), ", with constant returns to scale and diminishing marginal products of each input; it expresses output as a function of capital, labor, and a technology factor."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: total factor productivity" }), _jsx(_components.br, {}), "\n", "Total factor productivity (TFP) is the technology term ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "T" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }), " in the production function; it captures the portion of output not explained by measured capital and labor inputs and reflects efficiency, innovation, and organizational know‑how."] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["If we divide both sides by labor, we get output per worker ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "y" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" })] }) })] }), " as a function of capital per worker ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: 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_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7144em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.0037em" }, children: "α" }) }) })] }) }) }) }) })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "This relationship implies:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Increasing ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" })] }) })] }), " (capital per worker) raises ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "y" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "y" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "y" })] }) })] }), " (output per worker), but at a decreasing rate because ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0 < \\alpha < 1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6835em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5782em", verticalAlign: "-0.0391em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "<" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "."] }), "\n", _jsxs(_components.li, { children: ["Improvements in ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "T" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }), " (technology) shift the entire productivity curve upward at all levels of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "k" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "k" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03148em" }, children: "k" })] }) })] }), "."] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: capital deepening" }), _jsx(_components.br, {}), "\n", "Capital deepening is an increase in the amount of physical capital per worker (", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "K" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K/L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), "), which raises output per worker but is subject to diminishing marginal gains."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: marginal product of capital" }), _jsx(_components.br, {}), "\n", "The marginal product of capital (MPK) is the additional output generated by one more unit of capital, holding other inputs constant; in Cobb‑Douglas form, ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "M" }), _jsx(_components.mi, { children: "P" }), _jsx(_components.mi, { children: "K" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mi, { children: "Y" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "K" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "MPK = \\alpha Y/K" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "MP" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] })] })] }), "."] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["Firms invest in capital until the marginal product of capital equals its cost (the rental rate or required return). Because MPK falls as ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), " rises, even a very high savings rate cannot indefinitely raise the growth rate of output per worker in this model."] }), "\n", _jsx(_components.h3, { children: "Steady-state and sustainable growth in the neoclassical model" }), "\n", _jsxs(_components.p, { children: ["In neoclassical growth theory, the central concept is the ", _jsx(_components.strong, { children: "steady state" }), ", in which key ratios grow at constant rates:"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The capital-to-labor ratio ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "K" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K/L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " grows at the same rate as technology"] }), "\n", _jsx(_components.li, { children: "Output per worker grows at a constant rate" }), "\n", _jsx(_components.li, { children: "The output-to-capital ratio is constant" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: steady-state growth rate" }), _jsx(_components.br, {}), "\n", "The steady-state growth rate is the long-run equilibrium growth rate of output per worker when the capital-to-labor ratio, technology growth, and labor growth are aligned so that the economy grows on a balanced path."] }), "\n"] }), "\n", _jsx(_components.p, { children: "Using the Cobb‑Douglas framework, neoclassical theory derives explicit expressions for sustainable growth:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Let ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "θ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" })] }) })] }), " be the growth rate of technology (TFP)."] }), "\n", _jsxs(_components.li, { children: ["Let labor’s share of income be ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "(1 - \\alpha)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), "."] }), "\n", _jsxs(_components.li, { children: ["Let ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\Delta L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " be the long-run growth rate of the labor force."] }), "\n"] }), "\n", _jsx(_components.p, { children: "Then:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Sustainable growth of output per worker (or per capita)" }), " is:", "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "∗" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mi, { children: "θ" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "g^* = \\frac{\\theta}{1 - \\alpha}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9331em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7387em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.1408em", verticalAlign: "-0.7693em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7693em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Sustainable growth of total output" }), " is:", "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "G" }), _jsx(_components.mo, { children: "∗" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mi, { children: "θ" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" })] })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "G^* = \\frac{\\theta}{1 - \\alpha} + \\Delta L" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7387em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "G" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7387em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.1408em", verticalAlign: "-0.7693em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7693em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] })] })] }) }), "\n"] }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: sustainable growth rate" }), _jsx(_components.br, {}), "\n", "In the neoclassical context, the sustainable growth rate is the long-run steady-state growth rate of output (or output per capita) that can be maintained without changing capital’s share of income."] }), "\n"] }), "\n", _jsx(_components.p, { children: "Note what does not appear in these formulas: the growth rate of capital itself. Capital accumulation is important, but only in the transition to steady state; once there, long-run growth is pinned down by technology growth and labor’s share." }), "\n", _jsx(_components.h3, { children: "Growth accounting and capital deepening" }), "\n", _jsxs(_components.p, { children: ["From the Cobb‑Douglas production function, we can derive the ", _jsx(_components.strong, { children: "growth accounting relation" }), ":"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "Y" })] }), _jsx(_components.mi, { children: "Y" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mi, { children: "α" }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "K" })] }), _jsx(_components.mi, { children: "K" })] }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.mi, { children: "L" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{\\Delta Y}{Y} = \\frac{\\Delta T}{T} + \\alpha \\frac{\\Delta K}{K} + (1-\\alpha)\\frac{\\Delta L}{L}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0463em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3603em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0463em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3603em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0463em", verticalAlign: "-0.686em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3603em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0463em", verticalAlign: "-0.686em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3603em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "L" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }), "\n", _jsx(_components.p, { children: "This says:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The growth rate of potential GDP equals", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "TFP (technology) growth" }), "\n", _jsx(_components.li, { children: "plus capital growth weighted by capital’s income share" }), "\n", _jsx(_components.li, { children: "plus labor growth weighted by labor’s income share" }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "In practice, analysts:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Forecast long-term growth in ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "L" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " from demographic and investment trends"] }), "\n", _jsxs(_components.li, { children: ["Infer ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mi, { children: "T" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "T" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\Delta T/T" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }), " as the residual: realized GDP growth minus the contribution of capital and labor"] }), "\n"] }), "\n", _jsx(_components.p, { children: "This relation is often used in CFA exam questions to:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Decompose an economy’s growth into labor, capital deepening, and technological progress" }), "\n", _jsx(_components.li, { children: "Forecast potential GDP growth and relate it to long-run earnings growth and equity returns" }), "\n"] }), "\n", _jsx(_components.h3, { children: "Key neoclassical implications" }), "\n", _jsx(_components.p, { children: "Under neoclassical theory:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Diminishing returns to capital" }), " mean that capital deepening raises the level of output per worker but cannot permanently increase its growth rate."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "TFP growth is exogenous" }), ": the rate of technological progress is determined outside the model."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Savings and investment" }), ":", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "A higher savings rate leads to more capital per worker and a higher steady-state level of output per worker." }), "\n", _jsxs(_components.li, { children: ["Growth is temporarily faster during the transition but eventually reverts to ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "g^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8831em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), "."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Population (labor) growth" }), ":", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Raises the sustainable growth of total output ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "G" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "G^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6887em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "G" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), "."] }), "\n", _jsxs(_components.li, { children: ["But, all else equal, faster population growth reduces ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "K" }), _jsx(_components.mi, { mathvariant: "normal", children: "/" }), _jsx(_components.mi, { children: "L" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K/L" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "mord", children: "/" }), _jsx(_components.span, { className: "mord mathnormal", children: "L" })] }) })] }), " and therefore lowers steady-state income per worker."] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["These mechanics are very testable: item sets often ask you to determine whether a policy has a temporary ", _jsx(_components.strong, { children: "level effect" }), " or a permanent ", _jsx(_components.strong, { children: "growth rate effect" }), "."] }), "\n", _jsx(_components.h3, { children: _jsx(_components.strong, { children: "Worked Example 1.1" }) }), "\n", _jsxs(_components.p, { children: ["Suppose Country X has labor’s share of income of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mn, { children: "0.6" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "0.6" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0.6" })] }) })] }), " and long-run TFP growth of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "2\\%" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "2%" })] }) })] }), " per year. The labor force is expected to grow at ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "1\\%" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "1%" })] }) })] }), " per year. The government announces a policy that permanently raises the national savings rate, leading to a sustained increase in the investment share of GDP."] }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "What is Country X’s sustainable growth rate of output per capita under the neoclassical model?" }), "\n", _jsx(_components.li, { children: "Will the higher savings rate raise this sustainable growth rate?" }), "\n"] }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Answer:" }), _jsx(_components.br, {}), "\n", "The sustainable growth rate of output per capita is"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "∗" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mi, { children: "θ" }), _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "α" })] })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "2" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.mn, { children: "0.6" })] }), _jsx(_components.mo, { children: "≈" }), _jsx(_components.mn, { children: "3.33" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "g^* = \\frac{\\theta}{1 - \\alpha} = \\frac{2\\%}{0.6} \\approx 3.33\\%" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9331em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7387em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.1408em", verticalAlign: "-0.7693em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3714em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7693em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.113em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.427em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "0.6" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "2%" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≈" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "3.33%" })] })] })] }) }), "\n", _jsx(_components.p, { children: "The sustainable growth rate of total output is" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "G" }), _jsx(_components.mo, { children: "∗" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "3.33" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "4.33" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "G^* = 3.33\\% + 1\\% = 4.33\\%" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7387em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "G" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7387em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "3.33%" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "1%" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "4.33%" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["The increase in the savings rate raises capital per worker and pushes the economy to a higher ", _jsx(_components.strong, { children: "level" }), " of output per worker. During the transition, GDP growth is above ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "4.33" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "4.33\\%" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "4.33%" })] }) })] }), ". However, once the new steady state is reached, growth reverts to ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "3.33" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "3.33\\%" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "3.33%" })] }) })] }), " per capita (and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "4.33" }), _jsx(_components.mi, { mathvariant: "normal", children: "%" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "4.33\\%" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8056em", verticalAlign: "-0.0556em" } }), _jsx(_components.span, { className: "mord", children: "4.33%" })] }) })] }), " in total). So the policy does ", _jsx(_components.strong, { children: "not" }), " change the long-run sustainable growth rate; it only raises the level of output."] }), "\n"] }), "\n", _jsx(_components.p, { children: "This is the logic behind the common Level 2 result: under neoclassical theory, increasing the savings or investment rate generates a temporary growth effect but a permanent level effect." }), "\n", _jsx(_components.h2, { children: "ENDOGENOUS GROWTH THEORY" }), "\n", _jsx(_components.p, { children: "Endogenous growth theory challenges the neoclassical assumption that technology growth is independent of economic decisions. Instead, it brings innovation and human capital accumulation inside the model." }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: endogenous growth model" }), _jsx(_components.br, {}), "\n", "An endogenous growth model determines technological progress and productivity growth within the model itself, making them functions of decisions about R&D, education, and other investments that can be influenced by policy and market incentives."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: human capital" }), _jsx(_components.br, {}), "\n", "Human capital is the stock of skills, education, health, and knowledge embodied in workers that enhances their productivity."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: knowledge spillover" }), _jsx(_components.br, {}), "\n", "A knowledge spillover occurs when ideas or technologies developed by one firm or individual benefit others without full compensation, raising economy-wide productivity beyond the innovator’s private gain."] }), "\n"] }), "\n", _jsx(_components.p, { children: "Key features of endogenous growth models:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Returns to broad capital can be constant or increasing" }), _jsx(_components.br, {}), "\n", "Here “capital” often includes physical capital, human capital, and knowledge. Because ideas are non-rival (one firm’s use does not prevent others from using them), their social return can exceed their private return."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Technological progress is endogenous" }), _jsx(_components.br, {}), "\n", "TFP growth depends on purposeful decisions—such as spending on R&D, education, and infrastructure—and can thus be influenced by policy."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "No necessary steady-state growth rate" }), _jsx(_components.br, {}), "\n", "Because investment in knowledge can continuously raise productivity, increased investment can produce a ", _jsx(_components.strong, { children: "permanently higher growth rate" }), ", not just a one-time level shift."] }), "\n"] }), "\n", _jsx(_components.p, { children: "A simple version of an endogenous model might assume:" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "Y" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "A" }), _jsx(_components.mi, { children: "K" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Y = A K" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] })] })] }) }), "\n", _jsxs(_components.p, { children: ["with ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "A" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "A" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "A" })] }) })] }), " summarizing the stock of knowledge and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), " broadly defined to include physical and knowledge capital. If the effective return to ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), " is constant (not diminishing), then:"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["A higher savings rate permanently raises the growth rate of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] })] }), "\n", _jsxs(_components.li, { children: ["The growth rate of output ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "Y" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "Y" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.22222em" }, children: "Y" })] }) })] }), " increases permanently as well"] }), "\n"] }), "\n", _jsx(_components.p, { children: "This contrasts sharply with the neoclassical prediction." }), "\n", _jsx(_components.h3, { children: "Policy and externalities in endogenous growth" }), "\n", _jsxs(_components.p, { children: ["Because knowledge spillovers create ", _jsx(_components.strong, { children: "positive externalities" }), ", private agents may underinvest in R&D and education relative to the social optimum. This provides a rationale for growth-enhancing government policies, such as:"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Subsidies or tax credits for R&D" }), "\n", _jsx(_components.li, { children: "Public funding for basic research" }), "\n", _jsx(_components.li, { children: "Investment in education and health" }), "\n", _jsx(_components.li, { children: "Strong intellectual property protection (balanced against diffusion of ideas)" }), "\n"] }), "\n", _jsx(_components.p, { children: "Endogenous models therefore imply that:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Policies affecting human capital, innovation incentives, and knowledge diffusion can have ", _jsx(_components.strong, { children: "permanent effects" }), " on the growth rate."] }), "\n", _jsx(_components.li, { children: "Economies that successfully support innovation and human capital accumulation can sustain higher long-term growth than otherwise identical economies." }), "\n"] }), "\n", _jsx(_components.h3, { children: _jsx(_components.strong, { children: "Worked Example 1.2" }) }), "\n", _jsx(_components.p, { children: "Country Y is an emerging market that devotes a small share of GDP to R&D and has weak university-industry links. The government introduces generous tax credits for private R&D and expands university funding targeted at science and engineering. Assume these policies significantly increase the economy’s rate of innovation." }), "\n", _jsx(_components.p, { children: "According to endogenous growth theory, what is the expected long-run effect on Country Y’s growth path?" }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Answer:" }), _jsx(_components.br, {}), "\n", "In an endogenous growth framework, higher R&D and human capital investment increase the rate at which new ideas are created and diffused. Because these investments raise TFP growth ", _jsx(_components.strong, { children: "within" }), " the model, the long-run ", _jsx(_components.strong, { children: "growth rate" }), " of output per worker can increase permanently, not just its level. Knowledge spillovers mean that the social return exceeds the private return, so the policy can generate a sustained increase in the growth path of GDP per capita."] }), "\n"] }), "\n", _jsx(_components.h3, { children: _jsx(_components.strong, { children: "Exam Warning" }) }), "\n", _jsxs(_components.p, { children: ["Neoclassical and endogenous growth models both recognize that technological progress is critical. The difference is ", _jsx(_components.strong, { children: "where it comes from" }), " and ", _jsx(_components.strong, { children: "what policy can do" }), ":"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Neoclassical: Technology growth is exogenous; increasing physical capital alone cannot sustain faster growth once the steady state is reached." }), "\n", _jsxs(_components.li, { children: ["Endogenous: Technology growth responds to incentives; policies that support R&D, education, and knowledge diffusion can raise the ", _jsx(_components.strong, { children: "long-run" }), " growth rate."] }), "\n"] }), "\n", _jsx(_components.p, { children: "In item sets, be careful not to assume that higher investment automatically implies permanent higher growth; you must identify which model is being used (or implied) in the vignette." }), "\n", _jsx(_components.h2, { children: "CONVERGENCE HYPOTHESES" }), "\n", _jsx(_components.p, { children: "A central policy question is whether poor countries will eventually catch up to rich countries—that is, whether productivity and living standards tend to converge over time." }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: absolute convergence" }), _jsx(_components.br, {}), "\n", "Absolute convergence is the hypothesis that poorer countries will eventually reach the same level of real income per capita as richer countries, regardless of differences in savings rates, population growth, or institutions, assuming similar access to technology."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: conditional convergence" }), _jsx(_components.br, {}), "\n", "Conditional convergence is the hypothesis that economies converge to similar levels of income per capita only if they share structural characteristics such as savings rates, population growth, human capital, and institutional quality."] }), "\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: club convergence" }), _jsx(_components.br, {}), "\n", "Club convergence is the hypothesis that only countries with sufficiently similar structural characteristics and institutions (a “club”) converge toward each other’s income levels; countries outside the club may not catch up."] }), "\n"] }), "\n", _jsx(_components.h3, { children: "Convergence in the neoclassical model" }), "\n", _jsxs(_components.p, { children: ["The neoclassical model assumes that all countries have access to the same technology ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "T" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }), " and share a common Cobb‑Douglas functional form. Under these conditions:"] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["With the same savings rate, population growth, and production function, ", _jsx(_components.strong, { children: "poorer countries should grow faster" }), ", because they are farther from their steady-state capital per worker."] }), "\n", _jsxs(_components.li, { children: ["This implies ", _jsx(_components.strong, { children: "conditional convergence" }), ": convergence of living standards conditional on having the same structural parameters."] }), "\n"] }), "\n", _jsx(_components.p, { children: "However:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Equal growth rates do not imply equal levels of income per capita. Differences in savings, population growth, or institutions can lead to different steady-state levels." }), "\n", _jsxs(_components.li, { children: ["The model does ", _jsx(_components.strong, { children: "not" }), " imply absolute convergence across all countries, because structural parameters differ widely."] }), "\n"] }), "\n", _jsx(_components.p, { children: "Empirically, evidence is more supportive of conditional and club convergence than of absolute convergence:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Among groups of countries with similar policies and institutions (e.g., OECD economies), poorer members often grow faster and catch up." }), "\n", _jsx(_components.li, { children: "Cross-country differences in property rights, education systems, macro stability, and openness help explain why many low-income countries do not converge." }), "\n"] }), "\n", _jsx(_components.h3, { children: "Club convergence and institutions" }), "\n", _jsx(_components.p, { children: "Club convergence directs attention to the preconditions for growth:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Sound property rights and rule of law" }), "\n", _jsx(_components.li, { children: "Adequate financial markets and intermediation" }), "\n", _jsx(_components.li, { children: "Investments in human capital (education, healthcare)" }), "\n", _jsx(_components.li, { children: "Tax and regulatory systems that do not unduly deter entrepreneurship" }), "\n", _jsx(_components.li, { children: "Openness to trade and capital flows" }), "\n"] }), "\n", _jsx(_components.p, { children: "Countries that meet these conditions can form a “club” with similar long-run growth prospects. Poorer members of the club tend to grow faster than richer ones until they converge to similar income per capita. Countries that lack these features may remain outside the club and fail to catch up." }), "\n", _jsx(_components.h3, { children: _jsx(_components.strong, { children: "Worked Example 1.3" }) }), "\n", _jsx(_components.p, { children: "Several low-income countries in Region A undertake reforms to strengthen property rights, improve banking regulation, and expand primary and secondary education. Over the next 20 years, these countries experience rapid growth and converge toward the income per capita of the region’s richer economies. Neighboring countries that do not reform continue to grow slowly and remain far behind." }), "\n", _jsx(_components.p, { children: "Which convergence concept best explains this outcome?" }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Answer:" }), _jsx(_components.br, {}), "\n", "This pattern is best explained by ", _jsx(_components.strong, { children: "club convergence" }), ". Countries that adopted reforms achieved institutional and policy conditions similar to those of richer countries, effectively “joining the club” and converging toward their income levels. Countries that did not reform remained outside the club and did not converge."] }), "\n"] }), "\n", _jsx(_components.p, { children: "For CFA analysis, recognizing club convergence can inform assessments of whether a given emerging market is likely to catch up, which in turn affects assumptions about long-run earnings growth and valuation multiples." }), "\n", _jsx(_components.h2, { children: "POLICY IMPLICATIONS AND GROWTH" }), "\n", _jsx(_components.p, { children: "The neoclassical and endogenous growth frameworks lead to different views on what government policy can achieve." }), "\n", _jsx(_components.h3, { children: "Policy in the neoclassical model" }), "\n", _jsx(_components.p, { children: "Under neoclassical theory:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Policies that raise the savings and investment rate" }), " (e.g., tax incentives for investment)", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Increase capital per worker and the level of output per worker" }), "\n", _jsx(_components.li, { children: "Temporarily raise growth above the steady-state rate" }), "\n", _jsxs(_components.li, { children: ["Do ", _jsx(_components.strong, { children: "not" }), " change the long-run sustainable growth rate ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "g^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8831em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), " unless they affect technology growth ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "θ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" })] }) })] })] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Policies that affect population growth" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Higher population growth raises total output growth ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "G" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "G^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6887em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "G" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), " but can lower steady-state income per capita by diluting capital per worker."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Policies that influence technology (TFP)" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Even though technology is modeled as exogenous, in practice analysts will treat investments that plausibly raise ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "θ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" })] }) })] }), " (e.g., better education, openness to trade, stronger institutions) as the main drivers of permanent growth improvements."] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "Neoclassical analysis is particularly useful for:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Separating ", _jsx(_components.strong, { children: "temporary growth effects" }), " (transitions to a new steady state) from ", _jsx(_components.strong, { children: "permanent growth effects" }), " (changes in ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "θ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" })] }) })] }), ")."] }), "\n", _jsx(_components.li, { children: "Understanding why capital-deepening strategies alone cannot sustain high growth indefinitely." }), "\n"] }), "\n", _jsx(_components.h3, { children: "Policy in endogenous growth models" }), "\n", _jsx(_components.p, { children: "Endogenous growth models give a much larger role to policy because TFP growth is responsive to incentives:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "R&D subsidies and tax credits" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Offset underinvestment due to knowledge spillovers" }), "\n", _jsx(_components.li, { children: "Increase the rate of innovation and TFP growth" }), "\n", _jsx(_components.li, { children: "Can permanently raise the growth rate of output per worker" }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Education and human capital policies" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Public education spending, especially on secondary and tertiary education, raises human capital and the capacity to innovate and adopt technologies." }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Institutions and intellectual property rights" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Strong but balanced IP protection encourages innovation while allowing diffusion of knowledge." }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "Because of constant or increasing returns to broad capital (physical plus human plus knowledge), increased investment in these areas can keep marginal returns from falling, allowing high growth to be sustained." }), "\n", _jsx(_components.h3, { children: _jsx(_components.strong, { children: "Worked Example 1.4" }) }), "\n", _jsx(_components.p, { children: "An analyst is evaluating the growth impact of two policies in a middle-income country:" }), "\n", _jsxs(_components.ol, { children: ["\n", _jsx(_components.li, { children: "A one-time public investment program to expand highways and ports over the next five years." }), "\n", _jsx(_components.li, { children: "A permanent increase in public and private spending on R&D, backed by long-term tax incentives." }), "\n"] }), "\n", _jsx(_components.p, { children: "How would neoclassical and endogenous growth theories differ in their assessment of these policies’ long-run effects on the growth rate of GDP per capita?" }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Answer:" }), _jsx(_components.br, {})] }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Neoclassical view:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["The infrastructure program (Policy 1) raises the capital stock and may improve efficiency, leading to a higher level of output per worker and temporarily faster growth. Once the new capital stock is in place and the economy returns to its steady state, growth reverts to the prior ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "g^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8831em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), "."] }), "\n", _jsxs(_components.li, { children: ["The R&D initiative (Policy 2) could be interpreted as raising the rate of technology growth ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "θ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "θ" })] }) })] }), ". If sustained, this can increase ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "g" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "g^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8831em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "g" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), " permanently. However, this effect is not modeled explicitly; technology remains exogenous."] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Endogenous view:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Policy 1 still mainly has a level effect (higher ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), "), unless it significantly changes incentives for innovation."] }), "\n", _jsxs(_components.li, { children: ["Policy 2 directly increases the rate of innovation and human capital accumulation. Because TFP growth is endogenous, the model predicts a ", _jsx(_components.strong, { children: "permanent" }), " increase in the growth rate of GDP per capita, assuming the R&D incentives persist."] }), "\n"] }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.p, { children: ["On the exam, when a vignette describes persistent R&D and human capital policies, and asks about permanent growth effects, it is usually signaling an ", _jsx(_components.strong, { children: "endogenous" }), " growth framework."] }), "\n", _jsx(_components.h3, { children: "Link to equity returns and valuation" }), "\n", _jsx(_components.p, { children: "Growth models are relevant for valuation because:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Long-run equity returns are tied to the long-run sustainable growth rate of aggregate earnings, which is constrained by ", _jsx(_components.strong, { children: "potential GDP growth" }), "."] }), "\n", _jsx(_components.li, { children: "Using the growth accounting relation, an analyst can evaluate whether a forecast of, say, 6–7% real earnings growth is plausible given demographic trends, expected capital accumulation, and realistic TFP growth." }), "\n"] }), "\n", _jsx(_components.p, { children: "If a country’s potential GDP growth is estimated at 3–4%, forecasts of long-run earnings or dividend growth significantly above that may be unrealistic, suggesting over-optimistic equity valuations." }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Summary" }) }), "\n", _jsx(_components.p, { children: "Neoclassical and endogenous growth theories provide two lenses for analyzing long-run economic growth:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Neoclassical theory, grounded in the Cobb‑Douglas production function, emphasizes diminishing returns to capital and treats technological progress as exogenous. It predicts that:", "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Capital deepening raises the ", _jsx(_components.strong, { children: "level" }), " of output per worker but not its long-run growth rate."] }), "\n", _jsx(_components.li, { children: "Sustainable growth per capita is determined by technology growth and labor’s income share." }), "\n", _jsx(_components.li, { children: "Less developed countries can grow faster and converge conditionally, provided they share similar structural features and institutions." }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["Endogenous growth theory brings technological progress inside the model. It highlights:", "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "The roles of human capital, R&D, and knowledge spillovers." }), "\n", _jsxs(_components.li, { children: ["Constant or increasing returns to broad capital, allowing policies to have ", _jsx(_components.strong, { children: "permanent" }), " growth effects."] }), "\n", _jsx(_components.li, { children: "Strong incentives for innovation and education as key drivers of sustained high growth." }), "\n"] }), "\n"] }), "\n"] }), "\n", _jsx(_components.p, { children: "Convergence outcomes depend critically on institutions and policies. Absolute convergence is not supported; instead, conditional and club convergence better fit observed data. For investors and CFA candidates, these models guide the assessment of:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "The plausibility of long-run growth assumptions in valuation models" }), "\n", _jsx(_components.li, { children: "The likely impact of policy changes on potential GDP and corporate earnings" }), "\n", _jsx(_components.li, { children: "Which countries are more likely to catch up in living standards and investment opportunities" }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Key Point Checklist" }) }), "\n", _jsx(_components.p, { children: _jsx(_components.em, { children: "This article has covered the following key knowledge points:" }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Distinction between classical, neoclassical, and endogenous growth theories and their core assumptions." }), "\n", _jsx(_components.li, { children: "Structure and interpretation of the Cobb‑Douglas production function and total factor productivity." }), "\n", _jsx(_components.li, { children: "Definition of capital deepening and its role in raising the level of output per worker under diminishing returns." }), "\n", _jsx(_components.li, { children: "Derivation and use of the growth accounting relation to forecast potential GDP growth." }), "\n", _jsx(_components.li, { children: "Neoclassical expressions for sustainable growth of output per capita and total output." }), "\n", _jsx(_components.li, { children: "Why higher savings and investment have only temporary growth effects in the neoclassical model." }), "\n", _jsx(_components.li, { children: "How endogenous growth models treat technology, human capital, and R&D as endogenous drivers of permanent growth changes." }), "\n", _jsx(_components.li, { children: "The economic rationale for government subsidies to education and R&D due to knowledge spillovers and externalities." }), "\n", _jsx(_components.li, { children: "Definitions and implications of absolute, conditional, and club convergence hypotheses." }), "\n", _jsx(_components.li, { children: "How institutional quality and policy choices determine whether poorer countries catch up or remain outside the convergence “club”." }), "\n", _jsx(_components.li, { children: "Application of growth models to evaluate policy proposals and long-run equity return assumptions in CFA exam scenarios." }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Key Terms and Concepts" }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "classical growth theory" }), "\n", _jsx(_components.li, { children: "neoclassical growth theory" }), "\n", _jsx(_components.li, { children: "convergence hypothesis" }), "\n", _jsx(_components.li, { children: "Cobb-Douglas production function" }), "\n", _jsx(_components.li, { children: "total factor productivity" }), "\n", _jsx(_components.li, { children: "capital deepening" }), "\n", _jsx(_components.li, { children: "marginal product of capital" }), "\n", _jsx(_components.li, { children: "steady-state growth rate" }), "\n", _jsx(_components.li, { children: "sustainable growth rate" }), "\n", _jsx(_components.li, { children: "endogenous growth model" }), "\n", _jsx(_components.li, { children: "human capital" }), "\n", _jsx(_components.li, { children: "knowledge spillover" }), "\n", _jsx(_components.li, { children: "absolute convergence" }), "\n", _jsx(_components.li, { children: "conditional convergence" }), "\n", _jsx(_components.li, { children: "club convergence" }), "\n"] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent };

Economic growth and cycles - Neoclassical and endogenous growth models