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MacroeconomicsHomework Set B1.Fall 2016Consider the production functionY = AKáLâwhere Y represents output, K and L represent the inputs of capital and labor, and A is totalfactor productivity. The coefficients á and â are positive.a.Show that this production function exhibits constant returns to scale if (and only if)á + â = 1.Note: A production functionY = F[K , L]exhibits constant returns to scale if for any positive constant z, we haveF[zK , zL] = z F[K , L]In particular, doubling the inputs will double the output under constant returns to scale.b.Consider an economy with perfectly competitive firms. Assume all the firms have theabove production function with á + â = 1. Show that the share of labor in output isconstant. Also, define and calculate the share of capital in output by assuming thathouseholds own the capital stock and rent it to firms each period.Note: The share of labor is wL / Y, where w denotes the real wage.c.Show that for the economy of Part b, the marginal product of labor (MPL) is proportionalto the average product of labor, and the constant of proportionality is the share of labor.Also, show that MPL can be expressed as a function of the capital-labor ratio.2.Consider a competitive firm with the Cobb-Douglas production function of Question 1.Set A = 100, K = 2,500, and á = â = 0.5.a.Derive the equation for the firmâs labor demand curve.b.Briefly explain how capital accumulation (an increase in K) and technological progress(an increase in A) would shift the labor demand curve.3.Consider an economy with a constant labor force L. Let Et and Ut denote the numbers ofemployed and unemployed individuals at time t, with Et + Ut = L. Assume that eachperiod, a fraction s of employed workers are separated from their jobs (for reasons relatedto frictional or structural unemployment) while a fraction f of unemployed workers findjobs.Note: This is a useful framework for modeling and analyzing the natural rate ofunemployment because we can actually measure the job finding rate f and the separationrate s.a.How would you model the dynamics of the unemployment rate over time starting from agiven unemployment rate at time t?b.What is the steady-state equilibrium of your dynamic model?c.Show that the steady-state equilibrium is stable in the following sense: if we start at alower (higher) unemployment rate at t, then the unemployment rate rises (falls) over time,converging to the steady-state value as time tends to infinity.
