博弈论试题

Problem set for week 7

1. In the partial equilibrium intertemporal model, assume the utility function as in the handout, and assume that the consumer lives only two periods, hence the intertemporal budget constraint is

()()111t t t t C Y r Y C ++=++−

i) Derive the first order conditions for the Lagrangian optimisation problem for a

consumer who chooses optimal consumption in the two periods, subject to the intertemporal budget constraint, and show that they imply the Euler Equation for optimal consumption is

111t t C r C θ

++=+ answer:

()()

()()()11111111111

1max ln ln 1subject to

1or

01hence Lagrangean is

1max ln ln 11first order conditions

101101101henc t t t t t t t t t t t t t t t t t t t t t t C C C Y r Y C C Y Y C r

C Y C C Y C r C C r

C Y Y C r

θ

λθλλθ++++−++−+−+++−+−++=++−−−=+⎡⎤+

+−⎢⎥++⎣⎦

−=−=++−=+1e

111111which rearranges to give the answer.

t t C r C θ+=++

iii) Assume that

r = θ. Use this to show that, for sufficiently small r, ()1112

t t t t C C Y Y ++=≈+ and interpret your answer (hint, use the Euler equation and substitute into the budget constraint)

Answer:

()()

()()1111=hence from budget constraint

121121where the bracketed term is the present value of lifetime income.

1Hence for sufficiently small ,2

t t t t t t

t t r C C C

C Y r Y C r C Y r Y r Y C Y r r r C θ++++=⇒==++−+=+++⎛⎞⎛⎞=+⎜⎟⎜⎟++⎝⎠⎝

⎠≈()1This is a simple example of consumption smoothing.

t t Y Y ++

2 Now assume that the consumer’s intertemporal problem is that of the representative

consumer in a real business cycle model, and therefore that their income is GDP. Assume that labour supply and the current capital stock is fixed, and the consumer treats their income as exogenous, but that it is in fact generated by the production function y=zk αN , where k is capital per worker. Initial income per worker is 100.

a) Suppose that a transitory productivity shift hits the economy, raising y to 110. By what percentage has z increased?

[10%.]

b) Suppose that z=A 1-α as in lecture notes. For the same shift in output, will A have risen by more or less than z ? Explain the difference.

[If we have a 10% increase in output, then we must have (A 1/A 0)1-α = 1.1. If

α=1/3,then we must have A 1/A 0= 1.11/(1-α)=1.11/(1-1/3)=1.13/2=1.154≈1+3/2*(0.1): the rise in A is equivalent to a roughly 15% rise in “effective” labour supply, but labour is not the only factor of production, so, given initially fixed capital, output rises by less.]

c) Show that this shift must increase the marginal product of capital, and hence the return earned by the representative consumer. Explain the economic mechanism.

[From notes we have MPK t = α(A t /k t )1-α Given fixed labour supply and fixed capital stock, k t is constant, hence MPK t will rise. Economic rationale: a higher level of A t is like increasing the supply of labour for a given amount of capital. Given

complementarity in production, this raises both the absolute and marginal product of capital. Since r t =R t –d , the return to the consumer will also rise.]

d) Assume that in the next period A will revert to its original value.

i) Given the same preferences as above, will the consumer increase consumption in the current period by more or less than the increase in income? (you do not need to give precise answers)

ii) What does this imply for the level of k in the next period? What will be the impact on the marginal product of capital and hence r ? (again, you do not need to give precise answers).