I. Assume that the aggregate production function is Yt = F(Kt-1,Nt) =Kt-1 αNt 1-α where Kt-1 is total capital stock at the beginning of t carried over from t-1; Nt is total physical labor; and 0 <α < 1. The above production function is a constant return to scale production function jointly with respect to capital (= Kt-1) and labor (Nt), and therefore, the per capita output:yt = (Yt/Nt) is a function of kt-1=(Kt-1/Nt-1) and n (population or labor growth rate). That is , yt = (Yt/Nt) = ???????????????? = ???????????????? ???????????????? = ????????( ????????????????−1 (1+????????) )

Assume population growth, n = 0. The representative agent can hold money, nominal risk free debt, and capital for asset, and receives per capita real money transfer = τt at the beginning of each period t.

Suppose the representative agent chooses the time path of ct (consumption), kt (capital), bt (real per capita nominal bonds), and mt (real per capita money holdings)for all t>=0 to maximize lifetime discounted utility given by

(1) Σt=0 βt u (ct, mt) subject to the following per capita budget constraint

(2) f(kt-1) + (1-δ)kt-1 + [(1+it-1) bt-1] /(1+πt) + τt + mt-1/(1+πt) = ct + kt + bt + mt for all t >=0

where β =individual subjective discount rate, it-1= 1 period nominal interest rate at time t-1, δ=a constant capital depreciation rate, and πt= inflation rate from t-1 to t.

1-a) The first order conditions of the optimization problems imply the following conditions:

(3) ( , ) c t mt u c = β ( , ) c t+1 mt+1 u c ( f ‘(kt) +1− δ ) for all t >=0

(4) ( , ) c t mt u c = β ( , ) c t+1 mt+1 u c (1 ) (1 ) + +1 + t t i π for all t >=0

(5) ( , ) c t mt u c = ( , ) m t mt u c + β (1 ) ( ( , )) 1 1 1 + + + + t c t mt u c π for all t >=0

Furthermore, using equation (4) and (5), the following equation can be derived:

(6) t t c t t m t t i i u c m u c m + = ( ( , )) 1 ( ( , ))

I-1) a) Explain what the right hand side of each of the first order conditions (equation (3)-(5)) represents (12pts);

and b) Explain why equations (3)-(5) are necessary conditions for the intertemporal utility maximization (3 pts).

I-2) Steady state level of k, m, y, c, and π are defined as the k* , m* , y* , c* , and π*, where Δk* = Δm* = Δy* = Δc* = 0. From the condition that Δm* =0, we can show that the steady state level of inflation π* = θ, where θ= the growth rate of nominal money supply.

Suppose the utility function is given as follows: (????????????????, ????????????????) = ???????????????? 1 2???????????????? 1 2 . Furthermore, the values forα, β, δ, and θ are given as follows: α=1/3, β=0.95, δ =0.05, and θ=0.03. What is the value of steady state k*, c*, i*(nominal interest rate), and m*? Show your work. (16pts)

I-3) a) Explain neutrality of money, super-neutrality of money, and Fisher Effect (9pts); b) Do these effects hold in this model? Why or Why not? Explain.(9pts)

I-4) The above model assumes perfect information (no uncertainty). Suppose we extend the above model to a stochastic setting where both real return on capital (????????????????+1 ???????? = ????????′ (????????????????) − ????????) and real return on nominal bond (????????????????+1 ???????? = ???????????????? − ????????????????+1) are all uncertain, and assume that the representative agent maximizes the expected discounted lifetime utility. Furthermore, let us assume that there exists one period real bond, which offers risk free real interest rate, f tr . You can think of one period real bond as inflation indexed bond, whose nominal interest rate is indexed to the rate of realized inflation rate, and therefore guarantees risk free real interest rate exante.

a) Explain Consumption CAPM (CCAPM). (If you are going to use any equation, make sure to explain what the equation means.) (8 pts)

b) Does Consumption CAPM imply that the expected real returns from capital (????????????????+1 ???????? ) ????????nd nominal bonds (????????????????+1 ???????? = ???????????????? − ????????????????+1) be the same as the risk free real return from real bond ( f tr )? Why or why not? Explain. (5 pts) II. Consider the following equation (equation (7)) for the gov’t budget constraint: (7) ( ) + −1 t−1 = t + t − t−1 e t t g r b t b b + + + + − −1 −1 (1 ) 1 t e t t e t t r b π π π t t t h h +π − − 1 1 ,where 1 (1 ) (1 ) 1 1 − + + = − − t t t i r π = ex-post real interest rate; 1 (1 ) (1 ) 1 1 − + + = − − e t e t t i r π = ex-ante real interest rate; = e π t expected inflation and π t = realized inflation; bt, gt, tt, ht are real gov’t debt, real gov’t spending, real non-seigniorage tax revenue, and real monetary base (or high powered money).

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