University of Calgary Department of Economics Econ 357 Assignment 2 Dr. Vojtassak Fall 2015 Instructions: You are encouraged to work with your...

Question 2:

On the planet Yummy there is just one commodity, cake, and two time periods. There are

two kinds of creatures, “old” and “young.” Old creatures have an income of I units of

cake in period 1 and no income in period 2. Young creatures have no income in period 1

and an income of I*units of cake in period 2. There are N1 old creatures and N2 young

creatures. The consumption bundles of interest to creatures are pairs (c1, c2), where c1 is

cake in period 1 and c2 is cake in period 2. All creatures, old and young, have identical

utility functions, representing preferences over cake in the two periods. This utility

function is U c1,c2 ( ) = c1

αc2

1−α where α is a number such that 0 ≤ α ≤ 1. Let the interest rate

is r.

a) Let the current cake is taken to be the numeraire, (that is, its price is set at 1),

write an expression for the present value of a consumption bundle (c1, c2).

b) What is the present value of income for old creatures? For young creatures?

c) The budget line for any creature is determined by the condition that the

present value of its consumption bundle equals the present value of its

income. Write down this budget equation for old creatures and for young

creatures.

d) What is an old creature’s demand for cake in period 1? In period 2?

e) What is a young creature’s demand for cake in period 1? In period 2?

f) If the interest rate is zero,

(i) how much cake would a young creature choose in period 1?

(ii) For what value of α would it choose the same amount in each period?

g) If α = .55, what would r have to be in order that young creatures would want

to consume the same amount in each period?

The total supply of cake in period 1 equals the total cake earnings of all old creatures,

since young creatures earn no cake in this period. There are N1 old creatures and each

earns I units of cake, so this total is N1I. Similarly, the total supply of cake in period 2

equals the total amount earned by young creatures. This amount is N2I.

h) At the equilibrium interest rate, the total demand of creatures for period-1

cake must equal total supply of period-1 cake, and similarly the demand for

period-2 cake must equal supply. If the interest rate is r, what is then the

demand for period-1 cake by each old creature? What is the demand for

period-1 cake by each young creature?

i) What is the total demand for period-1 cake at interest rate r

j) Write an equation that sets the demand for period-1 cake equal to the supply.

k) Write a general expression for the equilibrium value of r, given N1, N2, I, and

I*. Solve this equation for the special case when N1 = N2 and I = I* and α =

11/21.

l) In the special case at the end of the last section, show that the interest rate that

equalizes supply and demand for period-1 cake will also equalize supply and

demand for period-2 cake. (This illustrates Walras’s law.)