7) : Consider a consumer in a two period model. Denote consumption in period 1 by c1 and consumption in period 2 by c2. The consumer has preferences over consumption bundles (c1, c2) represented by the additively separable utility function U(c1, c2) = v(c1) + v(c2) where the discount factor 0 < < 1, and v() is some concave function.

We are interested in the consumer's saving behaviour and how savings depend on the interest rate. Let us assume that this consumer has an endowment vector of (1, 2) with 1 much bigger than 2. (So we are thinking of a person during his working age in period 1 and then retirement in period 2.) The consumer can trade period 1 consumption for period 2 consumption, and for each unit of period 1 consumption 'saved' will receive (1 + r) units of period 2 consumption. Here r is the interest rate (r > 0). [For clarity: We have a consumer with two goods, as always. We essentially assume that the 'price' for good 1 - which is period 1 consumption - is 1 + r, while the price for good 2 - which is period 2 consumption - is 1. ] Consider an intial situation in which this consumer will save (i.e. c1 < 1, and total savings is simply 1 c1 > 0.) The question we have is this: will the consumer surely increase savings if the interest rate increases? In order to get there we will use the following steps:

a) Set up the consumer's utility maximization problem.

b) Characterize the solution of the utility maximization problem. (as far as you can)

c) Provide a carefully labelled diagrammatic representation of the solution.

d) I claim that both goods here must be normal. Argue/demonstrate/prove that this is true.

e) We can denote the solution by the consumption demand functions c1(r, m) and c2(r, m). Both goods here are normal goods. (Note that r is sufficient to capture the price ratio (1 + r)/1 and we can treat m as being measured in units of good 2.) The savings function is simply 1 c1(r, m). Demonstrate that the consumer could actually save less - or could save more - if the return to savings (the price r) increases. We cannot tell.