This problem combines equilibrium analysis with some of the things you learned in the chapter on intertemporal choice. It concerns the economics of saving and the life cycle on an imaginary planet where life is short and simple. In advanced courses in macroeconomics, you would study more-complicated versions of this model that build in more earthly realism. For the present, this simple model gives you a good idea of how the analysis must go.
On the planet Drongo 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) = ca1c12−a, where a is a number such that 0 ≤ a ≤ 1.
(a) If 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). ___________ Write down the present value of income for old creatures __________ and for young creatures __________ 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: ____________
(b) If the interest rate is r, write down an expression for an old creature’s demand for cake in period 1 __________ and in period 2 ___________ Write an expression for a young creature’s demand for cake in period 1 _____________ and in period 2 __________ (If its budget line is p1c1+p2c2 = W and its utility function is of the form proposed above, then a creature’s demand function for good 1 is c1 = aW/p and demand for good 2 is c2 = (1 − a)W/p.) If the interest rate is zero, how much cake would a young creature choose in period 1? ____________ For what value of a would it choose the same amount in each period if the interest rate is zero? ____________ If a = .55, what would r have to be in order that young creatures would want to consume the same amount in each period? __________
(c) 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 ______________
(d) 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, then the demand for period-1 cake by each old creature is ________ and the demand for period-1 cake by each young creature is __________ Since there are N1 old creatures and N2 young creatures, the total demand for period-1 cake at interest rate r is ____________
(e) Using the results of the last section, write an equation that sets the demand for period-1 cake equal to the supply. _________________ 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 a = 11/21. __________
(f) In the special case at the end of the last section, shows 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.)