Problems 1. In the Solow model, suppose that the depreciation rate increases. Determine the effect of this on the quantity of capital per worker and on output per working the steady state. Explain the economic intuition behind your results. 2. Derive an expression for the steady state capital-output ratio in the Solow model in levels and per-capita terms. 3. In the Solow model, consider the following calibration that gives something close to the U.S. economy. Suppose the production technology is zK\"N 1'", that the depreciation rate is .10, and the population growth rate is .02, and z = 1. Suppose initially that a is .36. (a) What savings rate generates a capital-output ratio of 3 (which is roughly the value in the U.S.)? (b) Imagining that w = MPN, which is what we had in Chapter 4. With this Cobb- Douglass production function, 1 e is called the labor share of income because one can show that (1 HJY = wN, i.e. (1 a.) fraction of total revenue goes to pay labor costs. Prove that this equality}.' holds (for any {1,3 vain-l). (c) What is the per worker production hmction? (d) What is the stead},r state real wage? Hint: First determine the steady state capi- tal per person, then determine output per person, then use output per person to determine a: from part (b). (?) + SRPG 2021 A LRPC INFLATION RATE (Percent) UNEMPLOYMENT RATE (Percent) Based on your answers to the previous questions, use the black line (plus symbol) to draw the short-run Phillips curve for this economy in 2021 (SRPC2021. The short-run Phillips curve is line: o At Natural Real GDP At the natural rate of unemployment o Representing the tradeoff between unemployment and inflation Now consider the long-run effects of this policy. Suppose, in particular, that following Implementation of the policy, the aggregate demand curve remains at ADB. Designate the long-run equilibrium that would follow such a policy as outcome C. Going back to the first graph, place the grey point (star symbol) at outcome C. Because output at point C is Natural Real GDP, the unemployment rate associated with outcome C is the natural rate of unemployment. Finally, use the green line (triangle symbol) to draw the long-run Phillips curve (LRPC) on the second graph. This line is line: At Natural Real GDP At the natural rate of unemployment Representing the tradeoff between unemployment and inflation5. Consider a Solow growth model with population growth rate n > 0. Assume the technology growth rate is 0. Let k*(s) be the steady-state capital-labor ratio given savings rate s. (a) Show that the steady-state capital-labor ratio k* (s) is increasing in the savings rate s. (b) Express the steady-state per-capita consumption c* as a function of savings rate s. (c) Using the function above, express the condition for the savings rate s* that maximizes the steady-state per-capita consumption (i.e., what condition(s) would s* have to satisfy?). (Such a savings rate is called the golden-rule savings rate.) (d) Assume Cobb Douglas production function F(K, L) = KoLl-a where 0
