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1 Analytics of the Solow Model In the Solow economy, people consume a good that firms produce with technology Y (which we assume to be constant) and f is a Cobb-Douglas production function Af (K, L), where A is TFP f(K, L) KL-a Here K is the stock of capital, which depreciates at rate δ E (0, 1) per period, and L is the labor force, which grows exogenously at rate n > 0. Here employment is always equal to the labor force L, because the L workers alive supply their labor services inelastically, whatever the wage rate. All L individuals alive and working at a point in time are identical, own equal shares of the aggregate capital stock, thus earn the same income, and save the same fraction s E (0,1) of total income, consuming the rest. So total savings are a fraction s of total gross output GDP) 1. Write the accumulation equation for the aggregate stock of capital K (t), namely Kt1) as a function of K (t), L (t) and parameters. 2. Write the accumulation equation for the stock of per capita capital k, namely k (t 1) as a function of k (t) and parameters. To do so, you have to find the production function in the intensive form function f (k) (K,1) for the Cobb-Douglas case (1) 3. Write the equation that determines a steady state k. 4. In a graph with per capita capital k (t) on the horizontal axis, plot the per capita savings function sAf (k (t)) and effective depreciation schedule (n + δ) k (t). Relate the graph to the change in per-capita capital k (t + 1 k (t). How many steady states k" can you see? Explain. 5. Compute steady state per capita income y and steady state per capita consumption c* as a function of parameters. 6. Suppose that that the economy is in steady state when, suddenly, the government implements a one-child policy, i order to slow population growth down. This policy reduces n to zero. Use a similar plot as in Question 4, to show the steady state both prior and after the new policy. Does the steady-state capital-labor ratio increase or decline? Also draw a graph, which traces out the evolution of capital per-capita (Hint: the graph should have "time" on the x-axis). Describe in words the resulting dynamics of per-capita output and consumption.

Answer 1

As a result of rising in steady state per capita capital, both consumption and output increases and reaches a new steady state eventually.

The accumulation equation: k(t + 1) = sY(t) + (1 - delta)K(t) = s[k (t)]^alpha [L(t)]^1 - alpha + (1 - delta) k(t) In per capita form: k(t + 1)/L(t + 1) = sY(t) + (1 - delta)k(t)/L(t + 1) or, k(t + 1) = sY(t) + (1 - delta)k(t)/(1 + n)L(t) therefore k(t + 1) = s/1 + n y (t) + (1 - delta)/1 + n k(t) Now y(t) = Y(t)/L(t) = [k(t)]^alpha [L(t)]^1 - alpha/[L(t)]^alpha[L(t)]^1 - alpha = [k(t)]^alpha Therefore k(t + 1) = s/1 + n[k(t)]^alpha + (1 - delta)/1 + n k(t) At the steady state k(t + 1) = k(t) = k* therefore k* = s/1 + n(k*)^alpha + (1 - delta)/1 + n k* or, [1 - 1 - delta/1 + n]k* = s/1 + n(k*)^alpha or, n + delta/1 + n k* = s/1 + n (k*)^alpha or, k*/(k*)^alpha = s/n + delta Therefore k* = (s/n + delta)^1/1 - alpha

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