/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 (This follows Mankiw, D. Romer, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 5

(This follows Mankiw, D. Romer, and Weil, 1992.) Suppose output is given by \(Y(t)=K(t)^{\alpha} H(t)^{\beta}[A(t) L(t)]^{1-\alpha-\beta}, \alpha>0, \beta>0, \alpha+\beta<1 .\) Here \(L\) is the number of workers and \(H\) is their total amount of skills. The remainder of the notation is standard. The dynamics of the inputs are \(\dot{L}(t)=n L(t), \dot{A}(t)=g A(t), \dot{K}(t)=\) \(s_{k} Y(t)-\delta K(t), \dot{H}(t)=s_{h} Y(t)-\delta H(t),\) where \(0\) \(0 . L(0), A(0), K(0),\) and \(H(0)\) are given, and are all strictly positive. Finally, define \(y(t)=Y(t) /[A(t) L(t)], k(t)=K(t) /[A(t) L(t)],\) and \(h(t)=H(t) /[A(t) L(t)]\) (a) Derive an expression for \(y(t)\) in terms of \(k(t)\) and \(h(t)\) and the parameters of the model. (b) Derive an expression for \(\dot{k}(t)\) in terms of \(k(t)\) and \(h(t)\) and the parameters of the model. In \((k, h)\) space, sketch the set of points where \(\dot{k}=0\) (c) Derive an expression for \(\dot{h}(t)\) in terms of \(k(t)\) and \(h(t)\) and the parameters of the model. In \((k, h)\) space, sketch the set of points where \(\dot{h}=0\) (d) Does the economy converge to a balanced growth path? Why or why not? If so, what is the growth rate of output per worker on the balanced growth path? If not, in general terms what is the behavior of output per worker over time?

Short Answer

Expert verified
The economy converges to a balanced growth path; the growth rate of output per worker is \(g\).

Step by step solution

01

Express y(t) in Terms of k(t) and h(t)

Given that \(y(t) = \frac{Y(t)}{[A(t)L(t)]}\), substitute \(Y(t) = K(t)^\alpha H(t)^\beta [A(t)L(t)]^{1-\alpha-\beta}\). Then, we have:\[y(t) = \frac{K(t)^\alpha H(t)^\beta [A(t)L(t)]^{1-\alpha-\beta}}{[A(t)L(t)]}= K(t)^\alpha H(t)^\beta [A(t)L(t)]^{-\alpha-\beta}\]Since \(k(t) = \frac{K(t)}{[A(t)L(t)]}\) and \(h(t) = \frac{H(t)}{[A(t)L(t)]}\), express \(K(t)\) and \(H(t)\) in terms of \(k(t)\) and \(h(t)\), thus:\[y(t) = k(t)^\alpha h(t)^\beta\]This shows \(y(t)\) as a function of \(k(t)\) and \(h(t)\).
02

Derive Expression for \(\dot{k}(t)\)

The dynamic equation for \(k(t)\) is:\[\dot{k}(t) = \frac{\dot{K}(t)}{[A(t)L(t)]} - \frac{K(t)\dot{A}(t)L(t) + K(t)A(t)\dot{L}(t)}{[A(t)L(t)]^2}\]Substitute \(\dot{K}(t) = s_k Y(t) - \delta K(t)\), \(\dot{A}(t) = g A(t)\), and \(\dot{L}(t) = n L(t)\), leading to:\[\dot{k}(t) = s_k y(t) - (n+g+\delta)k(t)\]Express \(y(t)\) in terms of \(k(t)\) and \(h(t)\) from Step 1, giving:\[\dot{k}(t) = s_k k(t)^\alpha h(t)^\beta - (n+g+\delta)k(t)\]To find the points where \(\dot{k} = 0\), solve:\[s_k k(t)^\alpha h(t)^\beta = (n+g+\delta)k(t)\]Indicating the locus where \(\dot{k} = 0\).
03

Derive Expression for \(\dot{h}(t)\)

Using the analogous method as for \(k(t)\), the dynamic equation for \(h(t)\) is:\[\dot{h}(t) = \frac{\dot{H}(t)}{[A(t)L(t)]} - \frac{H(t)(\dot{A}(t)L(t) + A(t)\dot{L}(t))}{[A(t)L(t)]^2}\]Substitute \(\dot{H}(t) = s_h Y(t) - \delta H(t)\), \(\dot{A}(t) = g A(t)\), and \(\dot{L}(t) = n L(t)\), leading to:\[\dot{h}(t) = s_h y(t) - (n+g+\delta)h(t)\]Express \(y(t)\) in terms of \(k(t)\) and \(h(t)\) from Step 1, yielding:\[\dot{h}(t) = s_h k(t)^\alpha h(t)^\beta - (n+g+\delta)h(t)\]To determine the set of points where \(\dot{h} = 0\), solve:\[s_h k(t)^\alpha h(t)^\beta = (n+g+\delta)h(t)\]Representing the locus where \(\dot{h} = 0\).
04

Analyze Convergence to a Balanced Growth Path

For the economy to converge to a balanced growth path, both \(\dot{k}(t)\) and \(\dot{h}(t)\) must be zero, implying:\[s_k k(t)^\alpha h(t)^\beta = (n+g+\delta)k(t)\]\[s_h k(t)^\alpha h(t)^\beta = (n+g+\delta)h(t)\]Solving these conditions gives the steady-state values for \(k(t)\) and \(h(t)\) consistent with balanced growth. The growth rate of output per worker, \(y(t)\), will then equal the growth rate of \(A(t)\), which is \(g\), signifying balanced growth since technological progress drives long-term growth.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Balanced Growth Path
In the context of the Solow Growth Model, the balanced growth path refers to a situation where various economic variables grow at a constant rate, ensuring that the ratios between them remain stable over time. This concept is crucial because it identifies a trajectory along which an economy can sustainably develop without facing imbalances. The economy reaches this path when investment in capital and skills accumulates at a rate sufficient to offset depreciation and population growth, aided by technological advancements.
  • Both capital per effective worker, \(k(t)\), and skills per effective worker, \(h(t)\), grow at rates that allow their ratios to remain constant relative to output, \(y(t)\).
  • The growth rate of output per worker is driven by technological progress \(g\), as reflected in the economy's steady progression.
To achieve a balanced growth path, the conditions for \(\dot{k}(t) = 0\) and \(\dot{h}(t) = 0\) must be satisfied. This means the increments in capital and skills match the needs dictated by growth rates of labor and technology. These equilibrium conditions help determine the steady-state values of \(k(t)\) and \(h(t)\) that allow sustained economic growth.
Capital Accumulation
Capital accumulation in the Solow Growth Model represents the process of increasing the stock of capital resources, such as machinery and tools, within an economy. Capital is a vital component as it directly influences output and productivity.
  • It is governed by the equation \(\dot{K}(t) = s_k Y(t) - \delta K(t)\), where \(s_k\) indicates the portion of output invested in new capital.
  • Depreciation, \(\delta K(t)\), accounts for wear and tear on capital, diminishing its effectiveness over time.
The growth of capital stock relative to the effective number of workers, \(k(t) = \frac{K(t)}{[A(t)L(t)]}\), dictates how the economy can sustain and improve its productivity. Investment must exceed depreciation and account for the growth of labor and technological progress \((n+g)\) to ensure positive capital accumulation. A critical insight from this process is that increasing the saving rate \(s_k\) can lead to higher levels of \(k(t)\), albeit with diminishing returns due to the presence of \(\alpha < 1\).
Technological Progress
Technological progress is the engine of long-term economic growth in the Solow Growth Model. It enables improvements in efficiency, allowing the same set of labor and capital resources to produce more output over time. Unlike capital and labor, technology grows without direct bounds by physical or human limitations.
  • Described by \(\dot{A}(t) = g A(t)\), where \(g\) is the rate of technological growth, it reflects enhancements in methods of production or better technologies over time.
  • On a balanced growth path, \(g\) determines the sustainable growth rate of output per worker, maintaining progress even when capital and labor inputs stabilize.
This innovation-driven growth allows economies to expand beyond the constraints of merely accumulating more labor or capital. In real-world terms, technological progress could involve advancements in automation, improvements in information technology, or breakthroughs in production processes. Its critical role in the model underscores why economies invest in research and development to propel continual economic advancements.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(A different form of measurement error.) Suppose the true relationship between social infrastructure \((S I)\) and \(\log\) income per person \((y)\) is \(y_{i}=a+\) \(b S I_{i}+e_{i} .\) There are two components of social infrastructure, \(S I^{A}\) and \(S I^{B}\) (with \(S I_{i}=S I_{i}^{A}+S I_{i}^{B}\) ), and we only have data on one of the components, \(S I^{A}\) Both \(S I^{A}\) and \(S I^{B}\) are uncorrelated with \(e\). We are considering running an OLS regression of \(y\) on a constant and \(S I^{A}\) (a) Derive an expression of the form, \(y_{i}=a+b S I_{i}^{A}+\) other terms.

Consider the following model with physical and human capital: \\[ \begin{array}{l} Y(t)=\left[\left(1-a_{K}\right) K(t)\right]^{\alpha}\left[\left(1-a_{H}\right) H(t)\right]^{1-\alpha}, \quad 0 < \alpha < 1, \quad 0 < a_{K} < 1, \quad 0 < a_{l l} < 1 \\ \dot{K}(t)=s Y(t)-\delta_{K} K(t) \\ \dot{H}(t)=B\left[a_{K} K(t)\right] \gamma\left[a_{H} H(t)\right]^{\phi}[A(t) L(t)]^{1-\gamma-\phi}-\delta_{H} H(t), \gamma > 0, \phi > 0, \gamma+\phi < 1 \\\ \dot{L}(t)=n L(t) \\ \dot{A}(t)=g A(t) \end{array} \\] where \(a_{K}\) and \(a_{H}\) are the fractions of the stocks of physical and human capital used in the education sector. This model assumes that human capital is produced in its own sector with its own production function. Bodies \((L)\) are useful only as something to be educated, not as an input into the production of final goods. Similarly, knowledge \((A)\) is useful only as something that can be conveyed to students, not as a direct input to goods production. (a) Define \(k=K /(A L)\) and \(h=H /(A L)\). Derive equations for \(\dot{k}\) and \(\dot{h}\) (b) Find an equation describing the set of combinations of \(h\) and \(k\) such that \(\dot{k}=0 .\) Sketch in \((h, k)\) space. Do the same for \(h=0\) (c) Does this economy have a balanced growth path? If so, is it unique? Is it stable? What are the growth rates of output per person, physical capital per person, and human capital per person on the balanced growth path? (d) Suppose the economy is initially on a balanced growth path, and that there is a permanent increase in \(s .\) How does this change affect the path of output per person over time?

Endogenizing the choice of \(E\) (This follows Bils and Klenow, 2000 .) Suppose that the wage of a worker with education \(E\) at time \(t\) is \(b e^{g t} e^{\phi E}\). Consider a worker born at time 0 who will be in school for the first \(E\) years of life and will work for the remaining \(T-E\) years. Assume that the interest rate is constant and equal to \(\bar{r}\) (a) What is the present discounted value of the worker's lifetime earnings as a function of \(E, T, b, \bar{r}, \phi,\) and \(g ?\) (b) Find the first-order condition for the value of \(E\) that maximizes the expression you found in part (a). Let \(E^{*}\) denote this value of \(E .\) (Assume an interior solution.) (c) Describe how each of the following developments affects \(E^{*}\) (i) A rise in \(T\) (ii) A rise in \(\bar{r}\) (iii) A rise in \(g\).

Briefly explain whether each of the following statements concerning a crosscountry regression of income per person on a measure of social infrastructure is true or false: (a) "If the regression is estimated by ordinary least squares, it shows the effect of social infrastructure on output per person." (b) "If the regression is estimated by instrumental variables using variables that are not affected by social infrastructure as instruments, it shows the effect of social infrastructure on output per person." (c) "If the regression is estimated by ordinary least squares and has a high \(R^{2},\) this means that there are no important influences on output per person that are omitted from the regression; thus in this case, the coefficient estimate from the regression is likely to be close to the true effect of social infrastructure on output per person."

Suppose the production function is \(Y=K^{\alpha}\left(e^{\phi E} L\right)^{1-\alpha}, 0<\alpha<1 . E\) is the amount of education workers receive; the rest of the notation is standard. Assume that there is perfect capital mobility. In particular, \(K\) always adjusts so that the marginal product of capital equals the world rate of return, \(r^{*}\) (a) Find an expression for the marginal product of capital as a function of \(K\) \(E, L,\) and the parameters of the production function. (b) Use the equation you derived in \((a)\) to find \(K\) as a function of \(r^{*}, E, L,\) and the parameters of the production function. (c) Use your answer in ( \(b\) ) to find an expression for \(d(\ln Y) / d E\), incorporating the effect of \(E\) on \(Y\) via \(K\) (d) Explain intuitively how capital mobility affects the impact of the change in \(E\) on output.
See all solutions

Recommended explanations on Economics Textbooks

Taxation Economics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.