Extra Practice Problems # 1
Jorge F. Chavez∗
November 8, 2012
Question 1: Constant v. Increasing Returns
This is a very general question intended to highlight difference between CRS and IRS. Suppose
you run firm with production function Y = aL. You buy labor at wage w, and sell your output on
the open market at a price of p. You are a price taker in all markets so without loss of generality
we can normalize prices at p = 1. Briefly argue your response to the following questions:
(a) For a given wage rate w, find the1 :
i. Profit-maximizing output Yt .
ii. Profit-maximizing labor demand Lt .
iii. Total profit
(b) Now suppose that labor markets are competitive. Find the:
i. Market-clearing wage
ii. Firm profits at that wage
(c) Now we allow for increasing returns to scale. The production function is Y = aL2 . For a
given wage rate w, find the:
i. Profit-maximizing output Yt .
ii. Profit-maximizing labor demand Lt .
iii. Total profits.
Will there be a market-clearing wage?
∗
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e-mail:j.chavez-cotrado@warwick.ac.uk
Hint: “Infinity” is a potentially valid answer.
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Question 2: Country Comparisons with the Solow Model
Suppose that there are two countries M and N. Both countries are well-described by the Solow
model and have identical CRS technology:
Yti = Ai (Kti )α (Lit )1−α
The countries have the same level of technological progress (Ai = Aj , ∀i ̸= j) but have different
capital to labor ratios, and thus different per capita income. Define the gross return on capital as
rt +1−δ (in other words, the rental value for the current period, plus the amount of undepreciated
capital left at the end of the period. Define the net return on capital as rt − δ, or the gross
return minus one. Let δ = 0, and α = 1/3. (Note: no specific assumption regarding population
growth is required)
(a) If country N has 10 times country M’s per capita income, what is the ratio of per-capita
capital in country N to that in country M?
(b) If country N has a 5 percent net return on capital, (and 10 times the per capita income of
country M), what is the net return on capital in country M?
(c) Suppose you were born in country N. Under these circumstances with some savings to invest,
would you invest it in country N or or send it to country M?
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