Question 2. Consider a situation with 1 consumption good, and 2 dates,t 1 andt 2

Question

Question 2. Consider a situation with 1 consumption good, and 2 dates,t 1 andt 2. Thus, technically, there are two goods, the consumption good in date 1 and that in date 2. There is only one agent (so this is a demand question) who has the following utility on the consumption bundles given by (zIF2) where zt denotes the consumption of that agent in date t = 1.2: For everyR , the preferences of our finance agent is represented by u ; R R defined by u(z) = In zit 1nz2, where € (0.1) (and is generally called the discount factor Suppose the agents initial endowments are given by e = (ei, e2> 0. A pro e vector is p = P1,P2) 0, moreover, because that the budget set and the demand is homogenous of degree zero in prices, we may normalize these prices to (1,p2). Thus,1 of the good in period 1 corresponds to 1/p2 of the good in period 2. Hence, given p = (1,P2), the interest rate is nothing but P2 = . Therefore, non-negative interest rates correspond to p 1. a. (5pts.) Are these preferences weakly convex? Are they strictly convex? Justify your result. b. (15pts.) Identify the demand as a correspondence of p0. Discuss the optimal behavior of this agent to non-negative interest rates r>0.

Explanation / Answer

a. The utility function of the consumer shows that the preferences of the consumer are strictly convex in the above case. The logarithmic utility function is downward sloping and convex to origin. Thus, it is strictly convex.

b. The demand function in the above case = Discount factor/ 1+ discount factor * Income / Price of good 2.Corresponding to non negative interest rates in the economy, the optimal behavior will remain the same as the optimum bundle demanded will remain same.

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