Question 1
This question has three subquestions (Question 1(a), Question 1(b), and Question 1(c)).
Consumption-Savings Question. Consider the two-period economy (with zero government spending and zero taxation), in which the representative consumer has no control over his real income (y1 in period 1 and y2 in period 2). The lifetime utility function of the representative consumer is:
u(c1, c2) = 2√c1 + 2√c2.
The lifetime budget constraint (in real terms) of the consumer is, as usual,
c1 + c2/ (1 + r) = y1 + y2/(1 + r) + (1 + r)a0.
Suppose the consumer begins period 1 with zero net assets (a0 = 0), and as per the notation in Chapters 3 and 4, r denotes the real interest rate.
For use below, it is convenient to define the gross real interest rate as R = 1 + r (as a point of terminology, "r" is the net real interest rate).
Thus, we have the lifetime Lagrangian formulation for the representative consumer's lifetime utility maximization problem:
L = 2√c1 + 2√c2 + λ[y1 + y2/(1 + r) + (1 + r)a1 - c1 - c2/(- 1 + r) ].
Question 1(a): Based on the Lagrangian, compute the first-order conditions with respect to c1 and c2. Then, use these first-order conditions to derive the consumption-savings optimality condition for the given utility function. NOTE: Your final expression of the consumption-savings optimality conditions should be presented in terms of the ratio c2/c1. Furthermore, in obtaining the representation of the consumption-savings optimality condition, you should express any (1+r) terms that appear as R instead (if you have not already done so).
Thus, the final form of the condition to present is: c2/c1 = ... in which the right hand side is for you to determine. Your final expression may NOT include any Lagrange multipliers in it. Clearly present the important steps and logic of your analysis.
Question 1(b): Starting from your expression in part a (that is, starting from the expression you obtained that has the form c2/c1 = ...), construct the natural logarithm of the expression.
Question 1(c): Now, recall from basic microeconomics and/or mathematical methods for economists that the elasticity of a variable x with respect to another variable y is defined as the percentage change in xinduced by a one-percent change in y. As you studied in basic microeconomics, elasticities are especially useful measures of the sensitivity of one variable to another because they do not depend on the units of measurement of either variable.
A convient method for computing an elasticity (which we will not prove here) is that the elasticity of one variable (say, x) with respect to another variable (say, y) is equal to the ftrst derivative of the natural log of x with respect to the natural log of y.
Starting from your solution in part b (that is, starting from the expression you obtained that has the form ln(c2/c1) = ...), compute the elasticity of the ratio c2/c1 with respect to the gross real interest rate R.
The resulting expression is the elasticity of consumption growth (between period one and period two) with respect to the (gross) real interest rate for the given utility function. Clearly present the important steps and logic of your analysis.