In this problem we will find the labor demand and labor supply for an economy, from there we will determine the equilibrium quantity of labor, and by plugging that into the production function we will find GDP. Consider Robin Crusoe who has utility function: U(Y,L) = Y - L^2/ 2 where Y is the output that Crusoe consumes, and L is labor hours Crusoe puts in. Suppose in this economy firms only use capital and labor (K and L) as inputs to production, and they can make things with a technology represented by the production function: Y = 0.75 ln(K) + 0.25 ln(L) where “ln” is the natural log.
a) Consider a firm that operates in a perfectly competitive market in which production is as described above, and write down its profit function. What is the objective of this firm? When trying to determine the demand for labor, the firm is trying to decide how many worker hours to employ in order to reach its objective. So what is their choice variable? Therefore, mathematically, what do you have to do in order to find the labor demand?
b) Go ahead and find the demand for labor. When you do, write it down in two ways: - as a function of the nominal wage w. - as a function of the real wage ω. Keep in mind that if y = f (x) then dy/dx = 1/x = ln(x)
c) Now find the labor supply for both the nominal wage and real wage.
d) Now that you have the Labor Demand and the Labor Supply, find the equilibrium: - nominal wage w*. - real wage ω*
e) Knowing this equilibrium wage, find now the equilibrium amount of labor in this economy: L* .
f) Lastly suppose that the price level P is given by the CPI and that it is 100. Suppose also that the equilibrium amount of capital is known, and that it is K* = 143. Find the equilibrium level of output of this economy, both in nominal terms and in real terms. What is the level of unemployment? g) now graph everything.