07北大高宏试题

Macroeconomics I:Midterm Exam

CCER,2007.10.21

Test time:3hours

1.Consider the following model economy discussed as in Chap.1.Sup-

pose goods are non-storable,and that all households have the following identical preference

∞

t=0

βt{ln(c t)+ln(1−l t)}1>β>0

and the budget constraint

b t−b t−1=w t l t+πt−

c t+r t−1b t−1,t=0,1,2,...

Suppose the production function takes the following form:

f t(l t)=(1+g)t l t,t=0,1,2,...

and g is the technological growth rate.

(1)(5points)Find equilibrium values of c t,l t,w t,r t.

If the utility function is replaced by

c1−γ1−γ+

(1−l)1−γ

1−γ

,γ>0,

(2)(10points)Find equilibrium values of c t,l t,w t,r t.

(3)(15points)Ifγincreases,what will happen to r t?Will your an-swer depend on whether g>0or g<0.Recall thatγalso implies the degree of impatience as discussed in the class,explain your result intutitively.

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2.Consider the cash-in-advance economy discussed in the classroom.

Suppose goods are non-storable,and that all households have the fol-lowing identical preference

∞

βt{αln(c t)+(1−α)ln(1−l t)}

t=0

the budget constraint

B t+1−B t+M t+1−M t=P t l t−P t c t+R t B t+T t+1,t=0,1,2,...

and the cash-in-advance constraint

P t c t=M t.

The government injects new money into the economy through lump-sum transfers T t,where T t+1=M t+1−M t=µM t.

(1)(15points)Ifα=0.5,β=.9,µ=0,what is the welfare cost of

inflation in terms of consumption relative to the OQM?

(3)(15points)Suppose l t=l for all t,bor supply isfixed ex-

ogenously.How is the welfare affected by changes inµ?In this case, could positive nominal interest rates R t be optimal?Why?

3.Consider the asset-pricing(no-money)economy discussed in Chap.3.

If the instantaneous utility function is ln(c t)andβ=0.9.Suppose the dividends are such that,for each t,d t=1or d t=.5with equal probability.

(1)(5points)Calculate the equilibrium asset prices q t numerically.

(Hint:you should derive two asset prices,one is when d t=1,and another is when d t=.5)

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(2)(10points)Calculate the equilibrium risk-free interest rates.(Hint: you should also derive two interest rates.)

(3)(10points)Calculate the equilibrium interest rates under risks. (Hint:you should derive four interest rates)

(4)(15points)Calculate the interest-rate premium.

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