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EconomicsDemonstrate that the consumer could actually save less – or could save more – if the return to savings (the price r) increases.

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) + δv(c2) where the discount factor 0 < δ < 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 (ω1, ω2) with ω1 much bigger than ω2. (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 intiial situation in which this consumer will save (i.e. c1 < ω1, and total savings is simply ω1 − c1 > 0.) The question we have is this: will the consumer surely increase savings if the interest rate increases?
The above question is an explaination of a whole question with parts abcde. I just need an answer to e:

e) (8 marks) 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 ω1 − 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. [Note: yes, I am essentially asking about the sign of a derivative of a function you don’t know, so you cannot just take the derivative. Yet, the question has a short and simple answer of maybe 4 or 5 lines. But how to get it?]


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