Question

An individual has a fixed wealth \((W)\) to allocate between consumption in two periods \((Q\) and \(\mathrm{C}_{2}\) ). The individual's utility function is given by \\[ T J(C \quad C | \\] and the budget constraint is \\[ w=c+\frac{c_{*}}{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 \(C,\) and \(C_{2}\) so that the \(M R S\) (of \(Q\) for \(C_{2}\) ) is equal to \(1+r\) b. Show that \(d C / d r>0\) but that the sign of \(d Q / d r\) is ambiguous. If \(d C_{x} / d r\) is negative, what can you conclude about the price elasticity of demand for \(\mathrm{C}_{2} ?\) c. How might your analysis of this problem be amended if the individual received income in each period \(\left(Y_{f} \text { and } Y_{2}\right)\) such that the budget constraint is given by

Short answer

Question: Show that in order to maximize utility given the budget constraint, the individual should choose C₁ and C₂ such that the MRS (of C₁ for C₂) is equal to 1+r. Also, discuss the change in analysis for the case where the individual receives income in each period. Answer: To maximize utility, the individual should choose C₁ and C₂ such that the Marginal Rate of Substitution (MRS) of C₁ for C₂ is equal to 1+r. This would require calculating the marginal utility of C₁ and C₂, MRS = (T'(C₁)) / (J'(C₂)), and setting this equal to 1+r. If the individual receives income in each period (Y₁ and Y₂), the budget constraint would be amended to C₁ + (C₂ / (1+r)) = Y₁ + (Y₂ / (1+r)). In this case, the same steps would be followed to find the effects on utility maximization for C₁ and C₂ consumption using the new budget constraint.

Step-by-step solution

Rewrite the budget constraint

First, rewrite the budget constraint in terms of C₂: \\[C_2 = (W - C_1)(1+r)\\]

Calculate the Marginal Utility of C₁ and C₂

Differentiate the utility function with respect to C₁ and C₂: \\[\frac{\partial U}{\partial C_1} = T'(C_1)\\] \\[\frac{\partial U}{\partial C_2} = J'(C_2)\\]

Calculate the Marginal Rate of Substitution (MRS)

Calculate the Marginal Rate of Substitution (MRS) of C₁ for C₂: \\[MRS = \frac{T'(C_1)}{J'(C_2)}\\]

Set up the condition for utility maximization

For utility maximization, the MRS should be equal to 1+r: \\[MRS = 1+r\\] It implies: \\[\frac{T'(C_1)}{J'(C_2)} = 1+r\\]

Differentiate the budget constraint with respect to r

By differentiating the modified budget constraint with respect to r, we can find the rate of change of C₁ and C₂ concerning r: \\[\frac{dC_1}{dr} = 0\\] \\[\frac{dC_2}{dr} = W - C_1 > 0\\]

Analyze the sign of dC₁/dr and dC₂/dr

Since \\[\frac{dC_1}{dr} = 0\\], C₁ is independent of r. For dC₂/dr, we observe that: \\[\frac{dC_2}{dr} = W - C_1 > 0\\] This implies that an increase in interest rate, r, will lead to an increase in C₂, while C₁ remains unchanged. The sign of dC₂/dr is positive, but the relationship depends on the shape of the utility function.

Determine the price elasticity of demand for C₂

If dC₂/dr is negative, the price elasticity of demand for C₂ can be concluded as: \\[\frac{dC_2}{dQ} < 0\\] This means the demand for C₂ is price elastic, implying that an increase in interest rate (r) will lead to a more significant percentage decrease in the consumption of C₂.

Amend the analysis for income in each period

If the individual receives income in each period (Y₁ and Y₂), the budget constraint should reflect the earned incomes: \\[C_1 + \frac{C_2}{1+r} = Y_1 + \frac{Y_2}{1+r}\\] To analyze the problem with this criterion, we need to include income in each period in the budget constraint and follow the same steps as above to find the effects on utility maximization for C₁ and C₂ consumption using the new budget constraint.

Key concepts

Marginal Rate of Substitution (MRS)

Understanding the Marginal Rate of Substitution (MRS) is pivotal when dissecting inter-temporal consumption choice. In simple terms, MRS represents the rate at which a consumer is willing to give up some amount of one good (in this case, consumption in one period) in exchange for another good (consumption in another period), while maintaining the same level of overall utility.

The MRS is derived from the consumer's indifference curve, which is a graph showing various combinations of two goods that give the consumer equal satisfaction and utility. Mathematically, it's calculated by taking the ratio of the marginal utilities of the two goods. In the context of our textbook problem, we are looking at the consumption in two periods, with the utility function involving \( C_1 \) and \( C_2 \). The formula for MRS, as shown in the solution, is \( \frac{T'(C_1)}{J'(C_2)} \).

For a consumer to achieve utility maximization, their MRS should align with the opportunity cost of making one choice over another. This opportunity cost is influenced by the prevailing interest rate; hence the MRS needs to equal \( 1+r \), the factor by which consumption is effectively 'priced' between the two periods. If this condition is satisfied, we can say the consumer is making a utility-maximizing decision regarding their consumption over time.

Budget Constraint

The concept of a budget constraint is a fundamental one in economics, often visualized as a straight line on a graph representing the combinations of different goods or services that a consumer can purchase given their income or wealth. It essentially defines the spending limit.

In the context of inter-temporal consumption, the budget constraint comes into play by illustrating the trade-off between current and future consumption. The constraint takes into account not only the wealth available but also the interest rate, which affects the future value of money. In our exercise, the original budget constraint was \( W = C_1 + \frac{C_2}{1+r} \), which states that the total wealth \( W \) is distributed between the present consumption \( C_1 \) and the discounted value of future consumption \( C_2 \). This brings time into the consumption choices, where the interest rate \( r \) acts as the price of current consumption in terms of future consumption.

When income is received in each period (\( Y_1 \) and \( Y_2 \)), this changes the budget constraint to reflect the additional money available to spend in each period, resulting in a new equation \( C_1 + \frac{C_2}{1+r} = Y_1 + \frac{Y_2}{1+r} \). This modification can lead to different consumption choices as the individual now has more options on how to allocate consumption over the two periods.

Utility Maximization

Utility Maximization is the goal of consumers to achieve the highest satisfaction possible within their budget. In other words, it is about making the best choices with limited resources. The principle of utility maximization is what drives a consumer's decision-making and underlies their consumption patterns.

In our problem, the individual seeks to maximize utility derived from consuming goods in two different periods. This is achieved when the Marginal Rate of Substitution (MRS) of consumption in the first period for the second period is equal to the one-period interest rate plus one (\( MRS = 1+r \)). This implies that the consumer is indifferent to consuming one more unit now versus the additional units that could be consumed in the future, factoring in the interest rate.

The various steps outlined in the solution take us through the process of how to calculate the change in consumption with respect to changes in the interest rate and evaluate the conditions for utility maximization. As the solution suggests, when the interest rate increases, so does future consumption \( C_2 \), assuming the utility function remains the same. However, whether present consumption \( C_1 \) increases or decreases can be ambiguous and dependent on the specifics of the individuals' utility function and their preference between current and future consumption.