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: In the two-period model, the optimal consumption satisfies the formula for the marginal rate of substitution (MRS) being equal to 1+r, as shown in the derivation of the utility maximization problem using the Lagrange method. The partial derivative of consumption in the second period with respect to the interest rate is non-negative, while the sign for the first period is ambiguous. If the consumption in the first period has a negative partial derivative, then the demand for consumption in the second period is price elastic. When including income in each period, the analysis remains unchanged as the partial derivatives with respect to the interest rate are not affected by income.

Step-by-step solution

Part a: Finding the optimal consumption

To find the optimal consumption, we can set up the Lagrange function (\(L\)) for the utility maximization problem with the budget constraint. $$ L = U(c_1, c_2) + \lambda [W - c_1 - \frac{c_2}{1+r}] $$ Take the partial derivatives of \(L\) with respect to \(c_1\), \(c_2\), and \(\lambda\): $$ \frac{\partial L}{\partial c_1} = \frac{\partial U}{\partial c_1} - \lambda = 0 $$ $$ \frac{\partial L}{\partial c_2} = \frac{\partial U}{\partial c_2} - \lambda(1+r) = 0 $$ $$ \frac{\partial L}{\partial \lambda} = W - c_1 - \frac{c_2}{1+r} = 0 $$ From the first two equations, we can find the marginal rate of substitution (MRS) of \(c_1\) for \(c_2\) as: $$ \frac{\frac{\partial U}{\partial c_1}}{\frac{\partial U}{\partial c_2}} = 1+r $$ This condition ensures that the individual allocates consumption between the two periods in a way that maximizes utility given the budget constraint.

Part b: Analyzing the partial derivatives

In order to study the sign of \(\partial c_1/\partial r\) and \(\partial c_2/\partial r\), we differentiate the budget constraint with respect to \(r\): $$ \frac{\partial W}{\partial r} - \frac{\partial c_1}{\partial r} + \frac{c_2}{(1+r)^2}\frac{\partial c_2}{\partial r} = 0 $$ Since wealth \(W\) doesn't depend on the interest rate, its derivative is 0. Now, we have: $$ \frac{\partial c_1}{\partial r} - \frac{c_2}{(1+r)^2}\frac{\partial c_2}{\partial r} = 0 $$ From this equation, we can conclude that: $$ \frac{\partial c_2}{\partial r} \geq 0 $$ However, the sign of \(\partial c_1 / \partial r\) is ambiguous. If it is negative, then the price elasticity of demand for \(c_2\) is: $$ \frac{\partial c_2}{c_2} \geq \frac{-\partial c_1}{c_1} $$ If \(\partial c_1 / \partial r\) is negative, it indicates that the demand for consumption in the second period \(c_2\) is price elastic.

Part c: Amending the budget constraint with income

If the budget constraint includes income received in each period \((y_1, y_2)\), then it can be written as: $$ y_1 - c_1 + \frac{y_2 - c_2}{1+r} = 0 $$ Now, differentiate the new budget constraint with respect to \(r\): $$ \frac{\partial y_1}{\partial r} - \frac{\partial c_1}{\partial r} + \frac{y_2 - c_2}{(1+r)^2}\frac{\partial c_2}{\partial r} = 0 $$ Since incomes \(y_1\) and \(y_2\) don't depend on the interest rate, their derivatives are 0. The equation becomes: $$ \frac{\partial c_1}{\partial r} - \frac{y_2 - c_2}{(1+r)^2}\frac{\partial c_2}{\partial r} = 0 $$ We can now see that: $$ \frac{\partial c_2}{\partial r} \geq 0 $$ However, the sign of \(\partial c_1/\partial r\) remains ambiguous, and the analysis of price elasticity of demand for \(c_2\) from part (b) remains unchanged.

Key concepts

Marginal Rate of Substitution

Understanding the Marginal Rate of Substitution (MRS) is key when studying the behavior of consumers, particularly when making inter-temporal consumption choices. The MRS represents the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of satisfaction.

The MRS of consumption in one period for consumption in another can be thought of as a trade-off rate. For instance, in the exercise provided, the MRS of consumption in the first period for the second period is the amount of future consumption a consumer is willing to sacrifice for additional current consumption, keeping utility constant. In mathematical terms, this is expressed as the ratio of partial derivatives of the utility function with respect to each good:\[\frac{\frac{\partial U}{\partial c_1}}{\frac{\partial U}{\partial c_2}}\].

For utility maximization, consumers adjust their consumption until the MRS is equal to the price ratio of the two goods, which is influenced by the interest rate, depicted as \(1+r\) in this scenario. Therefore, the optimal consumption choice occurs where the individual's MRS of \(c_1\) for \(c_2\) is equal to \(1+r\), which reflects the time value of money.

Utility Maximization

The goal of every consumer is to achieve the highest satisfaction from their available resources. Utility maximization refers to the process consumers undergo to determine the most satisfying allocation of their goods and services within their budget constraints. This is often determined by the utility function \(U(c_1, c_2)\), which calculates the total satisfaction received from different combinations of goods \(c_1\) and \(c_2\).

By solving the utility maximization problem, as depicted in the Lagrangian method presented in the exercise, we can find the optimal levels of consumption for each period that provide the greatest utility. This problem takes into account both the individual's preferences, as described by their utility function, and their financial limitations, described by the budget constraint.

Budget Constraint

A budget constraint represents a major limitation that consumers face; it details the combination of goods and services they can afford with their fixed wealth or income. It's the trade-off between various baskets of goods that can be purchased with their available resources. In the context of the exercise, the budget constraint is defined by the equation:\[W=c_1 + \frac{c_2}{1+r}\].

This equation shows how the wealth \(W\) can be allocated between present consumption \(c_1\) and future consumption \(c_2\), factoring in the interest rate \(r\). The constraint also underlines the financial reality that any consumption today will reduce the consumer's ability to consume in the future, or vice versa, highlighting the inter-temporal trade-off.

Price Elasticity of Demand

Price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. A beautiful illustration of this concept is present in the exercise when it discusses the reaction of consumption to changes in the interest rate.

If the quantity demanded changes significantly with a small change in price, the demand is considered 'elastic', whereas if the quantity demanded doesn't change much with a significant change in price, the demand is termed 'inelastic'. In terms of inter-temporal choices, if \(\partial c_1 / \partial r\) is negative, it indicates that an increase in interest rates leads to a fall in current consumption, suggesting a relatively elastic demand for future consumption \(c_2\). This elasticity concept is crucial in understanding consumer behavior and making economic predictions.