4. Imagine that production function uses two capital goods, physical and human c

Question

4. Imagine that production function uses two capital goods, physical and human capital: where Y, is the aggregate output, Kt is the aggregate physical capital stock, ht is the level of human capital per person and Lt is the number of persons. A represents the level of technology and is constant parameter representing the physical capital share. (a) Using the production function, write down an equation for output per capita yt as a function of capital per person, kt, and human capital per person, ht. (b) Firms rent physical capital at rate Rkt and human capital at rate Rht. Write down an expression of profits per capita assuming that the price of output is one. (c) Firms choose k and ht at every moment in order to maximize profits per capita. Find the first order conditions for kt and h, (d) In equilibrium the rates of return to physical and human capital must be identical so that Rkt- Rht. Using this relation and the first order conditions found in the previous question, write down an expression of the optimal amount of ht as a function of the optimal amount of

Explanation / Answer

a). Outper per capita as a function of kt = Yt/kt

= AKtalpha - 1(htLt)1 - alpha

Yt/ht =  AKtalphaht-alpha Lt1-alpha

= A(Kt/ht)alphaLt1-alpha

b). Profits = Price - Cost

1 - RktRhtAKtalpha-1(ht)-alphaLt1-alpha

c). First order condition for Kt = (alpha)AKtalpha-1(htLt)1-alpha

First order condition for ht = (1-alpha) A(Kt/ ht)alphaLt1-alpha

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