5 (5 points) Use a diagram to show the effect of an increase in saving rate, s, in the Solow Model. In particular, please show both the original and new steady state quantity of capital per worker in the diagram, and explain dynamic adjustment of capital per worker from the initial steady state to the new steady state.

1 (5 points) Define st as output per effective worker, i.e., and k as capital per effiective worker, i.e, . Uise the constant return to scale production function to express the relationship between y and k.

2 (5 points) Derive the law of motion for capital per effective worker, express the relationship in terms of saving rate and depreciation rate and growth rate of labor-augmenting productivity. Compute the steady-state quantity of capital per effective worker and output per effective worker in terms of s, σ, andγs.

3 (5 points) Use a diagram to show the steady state quantity of capital per effective worker.

4 (5 points) What are the long-run growth rate of output per effective worker, and capital per effective worker? What are the long-run growth rate of output per worker, and capital per worker?

2 The labor search model (25 points) Assume the workers in the model are all in the labor force. That is, the workers are either employed or unemployed, with U denoting the fractions of workers who are unemployed, and (1 - U) the fraction who are employed. The jobs of the employed differ according to the wage that they are paid. Each period, there is a probability p that the unemployed worker can draw one job offer from the wage distribution, . The worker has the option to reject the offer, in which case, he or she receives unemployment compensation. Alternatively, the worker can accept the offer to work at w, in which case, he or she receives w each period. Let   denote the expected value for a worker who has offer with wage w in hand. is an increasing and concave function of wage, u. In addition, Vu denotes the value of the unemployed worker, which is assunned to equal b. In the model, there will be fows between the pool of employed workers and the pool of unemployed worker. Some employed workers will be separated from their jobs and become unemployed, while some nnemployed workers will receive job offers that are suficiently attractive to accept. 1 (5 points) Let the reservation wage be defined as the wage such that the worker is indifferent between accepting and rejecting an offer. Please write down the equation that pins down the reservation wage in the model. Use the reservation wage to characterize the decision regarding accepting the job offer or not.

2 (5 points) Please use the reservation wage to compute the fraction of workers that change from unemployment to employment in each period.

3 (5 points) Suppose s denotes the separation rate, i.e., the fraction of workers who will become randorly separated from their jobs every period. Please write down the fraction of workers that change from employment to unemployment in each period.

4 (5 points) In the equilibrium, the number of worker that flows from employment to unem- ployment will equal the number of worker that flow from unemployment to employment. Use your answer in (2) and (3) to compute the equilibrium unemployment rate.

5 (5 points) Can you use the one-sided search model to analyze the effects of an increase in the unemployment insurance beneft, b, on the reservation wage and unemployment rate.

3. Asset Pricing (25 points) Consider an endowment economy with two periods, t = 0, 1. There are agents with mass one, and an agent endows units of consumption goods at the beginning of period t = 0, 1. The consumption good is nondurable and will vanish after it is consumed. The agent's lifetime utility iswhere ct is the consumption at period t, andis the time discount factor. There is a bond market that opens at period O, and agents can purchase or issue one-period real bonds at the market. A one-period real bond issued at period O matures at period 1, and the issuer of the bond must pay 1 unit of consumption goods to the bond holder when the bond matures. The bond market is competitive, Let bo denote an agent's net purchases of the bond, and let qo denote the equilibrium price of the one-period real bond in terrns of period O consumption goods. Then the agent's period O and period 1 budget constraints are   Note that the agents are the only participants in the bond market, but there is no government or any other people outside the economy engaging in the market. (a) (10 points) Solve for the equilibrium price of the one-period real bond, qo.