Question

3. (Steady state in the Solow model) Consider two economies identical in everything except the production function. Economy 1 has a production function F(K, L)KoL1-a, economy 2 has a production function G(K, L)-aK(1-a)L. For both economies capital grows according to (1). a) Write output per worker as a function of capital per worker for both economies. b) Compute the steady state value of capital per worker for both these economies or, if it does not exist, show graphically that it does not exist and list the neoclassical assumptions that are not satisfied in that case. For the following questions only consider the production function F(K, L-K'Li c) Compute the growth rate of K (aggregate capital) once k (capital per worker) has reached its steady state. (Hint: start from the definition of k d) A demographical shift in population brings the growth rate of population down to n' < n. How does the new steady state level of capital per worker compare to the one before the shift? How does the growth rate of aggregate capital (once the new steady state is reached) compare to the one before the shift?

Answer 1

a). For the first economy, given the production function$F(K,L)=K^{\alpha }L^{1-\alpha }$. Taking the intensive form of this production function i.e. dividing by L we get $y=k^{\alpha }$. where y=Y/L and k=K/L. For the second economy, given the production function $G(K,L)=\alpha K &plus; (1-\alpha )L$ ,dividing by L we get $y=1-\alpha (1-k)$ where y=Y/L and k=K/L. Thus, output per worker is a function of capital per worker for both economies.

b.) The law of motion for capital is given by $\Delta k=i-(\delta &plus;n)k$ where $\delta$ represents depreciation of capital and n represents the rate of population growth.

Now for Solow model, i=sy since it has assumed a closed economy.

Substituting into the above equation we get $\Delta k=s.k^{\alpha } -(\delta &plus;n)k$ . Assuming change in stock of capital at steady state =0 and solving for k we arrive at the steady state value of capital per worker, $k^{\ast }=(s/n&plus;\delta )^{1/1-^{\alpha }}$ .

This is true for the first economy.

For the second economy, the production function depicts constant marginal productivity of capital. This means diminishing marginal productivity of capital does not hold and hence more than one neoclassical assumptions are not satisfied.The assumtiption of concavity of production function and hence the Inada conditions are violated in this case. Hence we may conclude that steady state value of capital per worker does not exist.