Consider a static (one-period), closed economy with one representative consumer, one representative firm, and a government. The level of capital K and government expenditures G in the economy are both fixed exogenously. The government levies a lump sum tax T in order to fund its purchases, and the government budget must balance. Suppose the price of consumption is normalized to one (p = 1). The consumer has 24 hours of time available (h = 24), which she can use only for labor or leisure. She receives labor income, profits from the firm (π), and pays lump sum taxes. The consumer’s utility function is u(c, l) = α ln(c)+(1−α) ln(l), and the firm’s production function is Y = zK^θN^1−θ .

(a) Formally define a competitive equilibrium in this economy just as we did in the video lectures. Make sure you include the formal definitions of the optimization problems for both consumer and firm, and the correct equations for government budget balance and market clearing.

(b) Now we’re going to plug in parameter values for this economy. Suppose z = 12, θ = 1/2 , K = 36 and w = 10. Solve the firm’s optimization problem and call your optimal labor choice N∗ . What would be the firm’s profit at the optimal choice of N∗ ?

(c) Now assume that the parameter on the consumer’s utility function is α = 1/2 . Suppose the consumer gets wage w = 10, pays taxes T = 25 and receives the profits from the firm that you found in part b above. Solve the consumer’s optimization problem with these values, and call the optimal choices c ∗ and l ∗.

(d) Set up the market clearing equations for the output and labor markets. Determine whether or not these markets clear, i.e. whether or not supply equals demand in each. [Note that G = 25 since T = 25 and the government budget must balance.]

(e) This economy is not in equilibrium, and in order to achieve equilibrium the wage rate would need adjust. Would the wage would need to increase or decrease to achieve equilibrium? Why? Your explanation should include a description of how both consumer and firm behavior would change, and why those changes would move the economy towards equilibrium.