Problem 1: Two-Period Consumption-Savings 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(c_{1},c_{2})= lnc_{1} +lnc_{2}
The lifetime budget constraint (in real terms) of the consumer is, as usual,
c_{1} + c_{2}/(1+r)=y_{1} +Y_{2}/(1+r)+(1+r)a_{0}
Suppose the consumer begins period 1 with zero net assets (a_{0} = 0), and 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).
a. Set up a lifetime Lagrangian formulation for the representative consumer’s lifetime utility maximization problem. Define any new notation you introduce.
b. Based on the Lagrangian from part a, 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 condition should be presented in terms of the ratio c_{2}/c_{1}. 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 c_{2}/c_{1}
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.
Starting from
c. Starting from your expression in part b (that is, starting from the expression you obtained that has the form c_{2}/c_{1}=... ), construct the natural logarithm of the expression. In doing so, recall the following results regarding algebraic manipulation of natural logs: for any two x > 0 and y > 0, ln(xy) = ln x + ln y, and ln(x^{y})= y ln x. Your final expression here should be of the form ln c_{2}/c_{1}
Clearly present the important steps and logic of your analysis.
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 x induced 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 convenient 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 first derivative of the natural log of x with respect to the natural log of y. (Read this statement very carefully.)
d. Starting your expression in part c (that is, starting from the expression you obtained that has the form ln c_{2}/c_{1}=....) compute the elasticity of the ratio c_{2}/c_{1} 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.