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.

(b) (10 points) Now we change the model to make it three-period, so t = 0, 1,2. An agent endows units of consumption goods at the beginning of period t = 0, 1,2. The agent's lifetime utility becomes Besides the one-period real bond, an agent can also purchase or issue a two-period real bond at the credit market at period 0. The two-period real bond is a zero-coupon bond. That means the issuer of a two-period bond pays 1 unit of consumption goods to the bond holder at period 2, but the issuer does not pay any interest at period I to the bond holder. Solve for the equilibrium price of the two-period real bond at period 0.

(c) (5 points) Now we introduce one more good, the houses, into the economy. As in question (b), an agent endows units of consumption goods at the beginning of period t = 0, 1, 2. Moreover, an agent also owns e units of houses in the beginning of period 0. Houses are divisible, and houses provide living services to their holders: let h, denote an agent's holding of houses at the beginning of each period, then holding ht unit of houses generates ou(h.) units of utility to the agent. Where a ＞ O represents the relative importance of housing utility to consumption utility. We assume that houses are perfectly durable, and that means houses do not vanish or depreciate over time or after they provide the living service. Let an agent's consumption be ct, and let her house holdings at the beginning of a period be ha, then the agent's lifetime utility After agents receive the living service in each period, the housing market opens. The housing market allows agents to exchange between houses and consumption goods. Let pt denote the price of houses in terms of consumption goods at period t. Solve for po and p1.

(a) (5 points) Solve for the equilibrium consumption, c, working hours, n, and wage rate, w.

(b) (5 points) Now we extend the benchmark model to assume that housebolds are heterogeneous. Consider that half of the bouseholds are type 1, and half of the households are type 2, Type 1 and type 2 value consumptions differently, and this is captured by their difference in the parameter 7. Type I households' utility is and type 2 households' utility is

(c) (10 points) We further extend the model by introducing a government into the economy. The government charges a proportional tax on wage income. That is, suppose that a household's wage income is wn, then the government takes away tun from the household. The government transfers the tax revenue to all houscholds equally in a lump sum manner. How does the proportional tax on wage infuence the consumption equality? Explain your answer.

(d) (5 points) Now we go back to the benchmark economy with homogenous households; that is, now the parameter 7 is the same among households, and there is no government, tax, or transfer. However, we consider that firms are heterogeneous. That is, there are two types of firms. Type 1 firms' production function is  and A1 ＞ A2. Solve for the equilibrium wage rate and type 1 firms' and type 2 firrns' labor hiring in units of working hours.

1. Apply the Gram-Schmidt process to transform the vectors (1,1,1), (0,1,1),(1,0, 1) into orthonormal vectors with respect to the standard inner product.

2.  . Find an invertible matrix S that diagonalizes A, and compute AS. Apply the result, compute , where is defined by 

3.  Find a basis for the solution space of the following linear system, and find the dimension of the solution space. 

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