Question

3. (Steady state in the Solow model) Consider two economies identical in everything except the production function. Economy 1 has a production function \(F(K, L)=K^{\alpha} L^{1-\alpha}\), economy 2 has a production function \(G(K, L)=\alpha K+(1-\alpha) L\). 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 \(F(K, L)=K^{\alpha} L^{1-\alpha}\)

c) Compute the growth rate of \(\mathrm{K}\) (aggregate capital) once \(\mathrm{k}\) (capital per worker) has reached its steady state. (Hint: start from the definition of \(k=\frac{K}{L}\) )

d) A demographical shift in population brings the growth rate of population down to \(n^{\prime}<n .\)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

Given that Economy 1 has a production function rightarrow F (k, L) = K^alpha L^1 - alpha Economy 2 has a production Function rightarrow G(K, L) = alpha k + (1 - alpha) (a) F(K, L) - k^alpha L^1 - alpha Then, y = y/L = k^alpha l^1 - alpha/L = (k)^alpha therefore (K) = KL G(K, L) = alpha k + (1 - alpha) L y = G(k, L)/L = alpha K + (1 - alpha) (b) at steady state, sy = (n + s) K n m = population growth rate for 1^st economy s(k)^alpha = (n + s)^K s/n + s = k^1- alpha therefore k^* = (s/n + s)^1/1 - alpha For 2^nd: - s(alpha k + 1 - alpha) = (n + s)k s alpha k + s(1 - alpha) = (n + s)K s(1 - alpha) = (n + s - s alpha)k Thus, k^k = s(1 - alpha)/n + s - s \r\n (C) Now, growth rate of k as k = K/L so, k = k.L