Problem 1: Departing from Cobb-Douglas
Consider the production model where the aggregate production function is:
F(K;L) = �A KL
K +L with �A > 0; (1)
where K denotes capital and L denotes labor.
(i) (1 point) Does this production function exhibit constant returns to scale? (Give a proof).
(ii) (2 points) All markets are perfectly competitive. Derive the expressions for the wage (w) and the rental price of capital (r) as a function of K and L.
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(iii) (2 points) Express the labor share of income as a function of K and L.
(iv) (2 points) Express per capita GDP, y � Y=L, as a function of k � K=L. Draw a graphical representation of this relationship (with k on the horizontal axis and y on the vertical axis).
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(v) (1 point) Suppose that capital is accumulated as in the Solow growth model: a fraction �s of total income is invested into capital and the depreciation rate of the capital stock is �d. Write down the law of motion of the capital stock.
(vi) (2 points) Represent graphically the determination of the steady-state capital stock, K�.
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(vii) (2 points) Give the closed-form expressions for the steady-state values for k and y (denoted k� and y�, respectively). Under which condition does a steady state with positive capital stock exist?
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(viii) (2 points) Show analytically and graphically the effects of an increase of �s on k� and y�.
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(ix) (BONUS: 2 points) Assume �d < �A. Find the closed-form solution for steady-state per capita consumption, c�, as a function of �s, �A, and �d. Plot c� as a function of �s. What is the golden-rule value for �s?
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Problem 2: Public infrastructures and income differences
Consider an economy with the following aggregate production function, F(K;L) = AK1=3L2=3. We suppose that total factor productivity depends on the pro- vision of public goods as captured by per capita government purchases, g = G=L. We assume the following functional form: A = �Ag2=3. Hence,
F(K;L) = �Ag2=3K1=3L2=3:
(i) (2 points) Express per capita income, y � Y=L, as a function of �A, g, and k � K=L.
(ii) (2 points) Suppose that government purchases are �nanced with a proportional tax on income, g = ty where t is the tax rate. Express y as a function of �A, t and k.
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(iii) (2 points) Suppose all countries have the same tax rate and factor productivity. We adopt the normalization �A3t2 = 1. Use the model to predict per capita income for the following countries:
country observed k predicted y observed y U.S. 1.000 1.000 1.000 U.K. 0.671 0.828 India 0.061 0.084 China 0.147 0.172
Textbook prediction 1 0.876 0.394 0.528
Does this model outperform the textbook model to predict per capita income?
