Question

An individual has a fixed wealth ( \(W\) ) to allocate between consumption in two periods \(\left(c_{1} \text { and } c_{2}\right) .\) The individual's utility function is given by \\[ U\left(c_{1}, c_{2}\right) \\] and the budget constraint is \\[ W=c_{1}+\frac{c_{2}}{1+r} \\] where \(r\) is the one-period interest rate. a. Show that, in order to maximize utility given this budget constraint, the individual should choose \(c_{1}\) and \(c_{2}\) such that the \(M R S\left(\text { of } c_{1} \text { for } c_{2}\right)\) is equal to \(1+r\) b. Show that \(\partial c_{2} / \partial r \geq 0\) but that the sign of \(\partial c_{1} / \partial r\) is ambiguous. If \(\partial c_{1} / \partial r\) is negative, what can you conclude about the price elasticity of demand for \(c_{2} ?\) c. How would your conclusions from part (b) be amended if the individual received income in each period ( \(y_{1}\) and \(y_{2}\) ) such that the budget constraint is given by \\[ y_{1}-c_{1}+\frac{y_{2}-c_{2}}{1+r}=0 ? \\]

Short answer

Answer: The partial derivatives of consumption in period 1 and period 2 with respect to the interest rate would change with respect to the individual's preferences and response to interest rate changes under the new setup with income in both periods. Consumption levels would adjust to the change in income, but the definitive results would rely on the individual's preferences towards consumption in each period and their reactions to interest rate changes.

Step-by-step solution

a. Deriving the condition for utility maximization

To maximize the utility function, the consumer will choose the consumption levels \(c_{1}\) and \(c_{2}\) such that the MRS of \(c_{1}\) for \(c_{2}\) is equal to the relative price of consuming \(c_{1}\) and \(c_{2}\). The MRS is found by taking the ratio of the marginal utilities: \\[ MRS=\frac{\partial U\left(c_{1},c_{2}\right) / \partial c_{1}}{\partial U\left(c_{1},c_{2}\right) / \partial c_{2}} \\] Now, we can write down the condition for utility maximization: \\[ MRS = 1+r \\]

b. Analyzing the influence of the interest rate on consumption

To find the effect of interest rate on consumption, we need to compute the partial derivatives of \(c_{1}\) and \(c_{2}\) with respect to \(r\). \\[ \frac{\partial c_{1}}{\partial r} \\] and \\[\frac{\partial c_{2}}{\partial r} \\] We are given that the partial derivative of \(c_{2}\) with respect to \(r\) is non-negative, which indicates that an increase in interest rate leads to an increase or no change in the consumption in period 2. However, the sign of the partial derivative of \(c_{1}\) with respect to \(r\) is ambiguous. If the partial derivative is negative, it means that an increase in the interest rate leads to a decrease in the consumption in period 1. In this case, we can conclude that the price elasticity of demand for \(c_{2}\) is elastic.

c. Amending conclusions with income in each period

Now, we need to analyze how the partial derivatives change if there is income in each period with a revised budget constraint: \\[ y_{1}-c_{1}+\frac{y_{2}-c_{2}}{1+r}=0 \\] Again, we find the partial derivatives of \(c_{1}\) and \(c_{2}\) with respect to \(r\): \\[ \frac{\partial c_{1}}{\partial r} \\] and \\[\frac{\partial c_{2}}{\partial r} \\] The relationship between these partial derivatives with the initial ones can be determined by analyzing how an additional income in each period affects the consumption in the corresponding periods. Given the new income streams, it is likely that the consumption levels would adjust to the change in income. However, the conclusive results will depend on the individual's preference for consumption in each period and their responses to the interest rate changes under this new setup.