Question

An individual has a fixed wealth \((W)\) to allocate between consumption in two periods \(\left(G_{1}\right.\) and \(C_{2}\) ). 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 \(C_{2}\) so that the \(M R S\) (of \(C_{1}\) for \(C_{2}\) ) 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 might your analysis of this problem be amended if the individual received income in each period \(\left(Y_{1} \text { and } Y_{2}\right)\) such that the budget constraint is given by \\[ Y_{1}-C_{1}+\frac{Y_{2}-C_{2}}{1+r}=0 \\]

Short answer

Answer: When an individual maximizes their utility in a two-period model with income in each period, the relationship between consumption variables and interest rate is such that the MRS of C1 for C2 equals 1+r. This implies that the individual equalizes the rate at which they are willing to trade off between present and future consumption, with the interest rate. The partial derivative of C2 with respect to the interest rate is greater than or equal to zero (C2 increases or remains constant with an increase in interest rate), while the sign of the partial derivative of C1 with respect to the interest rate is ambiguous. In terms of price elasticity, if the partial derivative of C1 with respect to the interest rate is negative, then the price elasticity of demand for C2 is also negative, meaning the demand for C2 is elastic.

Step-by-step solution

Part a: Finding MRS and Solving for MRS = 1+r

1. Obtain the marginal utilities: \\[\frac{\partial U}{\partial C_1} = M U_1\\] \\[\frac{\partial U}{\partial C_2} = M U_2\\] 2. Calculate the Marginal Rate of Substitution (MRS) between C1 and C2: \\[\text{MRS} = \frac{MU_1}{MU_2} = \frac{\partial U / \partial C_1}{\partial U / \partial C_2}\\] 3. Set up the Lagrange equation with utility function, budget constraint, and Lagrange multiplier (λ): \\[\mathcal{L} = U(C_1, C_2) + \lambda\left[W - C_1 - \dfrac{C_2}{1+r}\right]\\] 4. Calculate the first-order conditions of the Lagrange equation: \\[\frac{\partial \mathcal{L}}{\partial C_1} = MU_1 - \lambda = 0\\] \\[\frac{\partial \mathcal{L}}{\partial C_2} = MU_2 - \lambda(1+r) = 0\\] \\[\frac{\partial \mathcal{L}}{\partial \lambda} = W - C_1 - \dfrac{C_2}{1+r} = 0\\] 5. Solve the first-order conditions for MRS = 1+r: Divide the first equation by the second equation: \\[\frac{MU_1}{MU_2} = \frac{\lambda}{\lambda (1+r)}\\] Eliminate λ: \\[\text{MRS} = \frac{MU_1}{MU_2} = 1+r\\]

Part b: Deriving Partial Derivatives and Analyzing Sign of Elasticity

1. Implicitly differentiate MRS and budget constraint equations, starting with budget constraint: \\[\frac{\partial C_1}{\partial r}+\frac{C_2}{(1+r)^2}\frac{\partial C_2}{\partial r} = 0\\] \\[(1+r)^2\frac{\partial C_{1}}{\partial r} + C_2\frac{\partial C_{2}}{\partial r} = 0\\] 2. Implicitly differentiate the MRS = 1+r equation: \\[\frac{\partial \text{MRS}}{\partial r} = 1\\] \\[\frac{\partial\frac{MU_1}{MU_2}}{\partial r} = 1\\] 3. Determine the signs of the partial derivatives: We can deduce that: \\[\frac{\partial C_2}{\partial r} \geq 0\\] Sign of \\(\frac{\partial C_1}{\partial r}\\) is ambiguous. 4. Analyze Price Elasticity of Demand for C2: If \\(\frac{\partial C_1}{\partial r}<0\\), then the price elasticity of demand for C2 is negative, meaning the demand for C2 is elastic.

Part c: Amending the Analysis with Income in each Period

1. Rewrite the budget constraint by considering income in each period Y1 and Y2: \\[Y_1-C_1+\frac{Y_2-C_2}{1+r}=0\\] 2. Repeat the process done in Part a and Part b with the new budget constraint: Follow the steps taken in Part a and Part b, replacing the old budget constraint with the newly amended version. The amended budget constraint will affect the first-order conditions of the Lagrange equation. By doing so, we will obtain a modified MRS equation and analyze the relationships between consumption variables and interest rate, as well as between the price elasticity of demand.

Key concepts

Consumption Allocation Explained

Consumption allocation is the process of deciding how to distribute fixed resources, such as wealth, among various possible uses. In the context of the given exercise, an individual must allocate wealth (W) between consumption in two different periods, denoted as \(C_1\) and \(C_2\). This decision is crucial because it affects the individual's satisfaction—or utility—over time.

Utility, in this scenario, is the satisfaction or happiness derived from consuming goods in two different periods. Since people have preferences regarding when they consume these goods, the allocation must be made in a manner that maximizes overall utility. However, this individual can't allocate wealth arbitrarily and has to respect a budget constraint, which in this case, takes into account the present value of future consumption discounted by the interest rate (r). This mathematical relationship ensures that the individual lives within their means over the two periods. The concept of consumption allocation is closely tied to the idea of planning and foresight in financial decisions, contemplating immediate needs and future desires.

Utility Maximization and the MRS

Utility maximization is the objective of an individual to get the most satisfaction possible from their consumption choices, given their budget constraint. The Marginal Rate of Substitution (MRS) plays a pivotal role in this process. It measures the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility.

The exercise requires finding the optimal consumption levels, \(C_1\) and \(C_2\), such that the MRS—essentially the trade-off preference between consuming in the current period versus the next—equals the interest factor \(1+r\). This equality signifies that the individual's rate of substitution between \(C_1\) and \(C_2\) matches the rate at which the market substitutes these goods across time via interest. Only at this point will the consumer reach utility maximization, balancing personal preferences with market conditions.

One key insight in utility maximization is that individuals adjust their consumption until the pleasure obtained from the last unit of a good consumed now is equal to the pleasure they would receive from consuming it later, after accounting for the interest. This equalizing forces the MRS to align with the interest rate, ensuring the best possible use of their resources for maximum satisfaction.

Analyzing the Budget Constraint

The budget constraint is a fundamental economic concept representing the limits within which consumers can make their consumption choices. In mathematical terms, it shows the combinations of goods and services a consumer can afford given their limited resources, in this instance, the fixed wealth (W).

In the given exercise, the budget constraint is expressed as \(W=C_1 + \frac{C_2}{1+r}\), where \(C_1\) is the consumption in the current period, \(C_2\) is the consumption in the future period, and r is the interest rate. This constraint incorporates the present value of future consumption, acknowledging the time value of money—\( \frac{C_2}{1+r} \) represents what the consumer must set aside today to afford future consumption, assuming wealth is not replenished.

When income for each period is introduced, as in part c of the exercise, the budget constraint adapts to \(Y_1 - C_1 + \frac{Y_2 - C_2}{1+r} = 0\). This amended constraint accounts for the new dynamic of periodic income, changing how consumption is allocated. Consumers will re-optimize their consumption in light of their incomes in both periods, showing that budget constraints aren't static and often must be re-evaluated when circumstances change, such as the introduction of income or adjustment in interest rates.