Question

Proving the composite commodity theorem consists of showing that choices made when we use a composite commodity are identical to those that would be made if we specified the complete utility-maximization problem. This problem asks you to show this using two different approaches. For both of these we assume there are only three goods, \(x_{1}, x_{2},\) and \(x_{3}\) and that the prices of \(x_{2}\) and \(x_{3}\) always move together-that is, \(p_{2}=t p_{2}^{0}\) and \(p_{3}=t p_{3}^{0},\) where \(p_{2}^{0}\) and \(p_{3}^{0}\) are the initial prices of these two goods. With this notation, the composite commodity, \(y,\) is defined as $$y=p_{2}^{0} x_{2}+p_{3}^{0} x_{3}$$ a. Proof using duality. Let the expenditure function for the original three-good problem be given by \(E\left(p_{1}, p_{2}, p_{3}, \bar{U}\right)\) and consider the alternative expenditure minimization problem; Minimize \(p_{1} x_{1}+\) tys.t. \(U\left(x_{1}, x_{2}, x_{3}\right)=\bar{U}\). This problem will also yield an expenditure function of the form \(E^{*}\left(p_{1}, t, \bar{U}\right)\) i. Use the envelope theorem to show that $$\frac{\partial E}{\partial t}=\frac{\partial E^{*}}{\partial t}=y$$ This shows that the demand for the composite good \(y\) is the same under either approach. ii. Explain why the demand for \(x_{1}\) is also the same under either approach? (This proof is taken from Deaton and Muellbauer, \(1980 .)\) b. Proof using two- stage maximization. Now consider this problem from the perspective of utility maximization. The problem can be simplified by adopting the normalization that \(p_{2}=1-\) that is, all purchasing power is measured in units of \(x_{2}\) and the prices \(p_{1}\) and \(p_{3}\) are now treated as prices relative to \(p_{2}\). Under the assumptions of the composite commodity theorem, \(p_{1}\) can vary, but \(p_{3}\) is a fixed value. The original utility maximization problem is: \\[ \text { Maximize } U\left(x_{1}, x_{2}, x_{3}\right) \text { s.t. } p_{1} x_{1}+x_{2}+p_{3} x_{3}=M \\] (where \(M=I / p_{2}\) ) and the first-order conditions for a maximum are \(U_{i}=\lambda p_{i}, i=1,3\) (where \(\lambda\) is the Lagrange multiplier). The alternative, two-stage approach to the problem is \\[ \text { Stage } 1: \text { Maximize } U\left(x_{1}, x_{2}, x_{3}\right) \text { s.t. } x_{2}+p_{3} x_{3}=m \\] (where \(m\) is the portion of \(M\) devoted to purchasing the composite good). This maximization problem treats \(x_{1}\) as an exogenous parameter in the maximization process, so it becomes an element of the value function in this problem. The first-order conditions for this problem \(\operatorname{are} U_{i}=\mu p_{i}\) for \(i=2,3,\) where \(\mu\) is the Lagrange multiplier for this stage of the problem. Let the value (indirect utility) function for this stage 1 problem be given by \(V\left(x_{1}, m\right)\)The final part of this two-stage problem is then: \\[ \text { Stage } 2: \text { Maximize } V\left(x_{1}, m\right) \quad \text { s.t. } p_{1} x_{1}+m=M \\] This will have first-order conditions of the form \(\partial V / \partial x_{1}=\delta p_{1}\) and \(\partial V / \partial m=\delta,\) where \(\delta\) is the Lagrange multiplier for stage 2 Given this setup, answer the following questions: i. Explain why the value function in stage 1 depends only on \(x_{1}\) and \(m .\) (Hint: This is where the fact that \(p_{3}\) is constant plays a key role.) ii. Show that the two approaches to this maximization problem yield the same result by showing that \(\lambda=\mu=\delta .\) What do you have to assume to ensure the results are equivalent? (This problem is adapted from Carter, \(1995 .)\)

Short answer

Question: Prove the composite commodity theorem using duality and two-stage maximization approaches, ensuring that the value function only depends on x1 and m, and that the Lagrange multipliers are the same under certain assumptions.

Step-by-step solution

PART A

The first approach: Proof using duality We are given the expenditure functions E(p1, p2, p3, Ubar) and E*(p1, t, Ubar).

Envelope theorem

To show \(\frac{\partial E}{\partial t} = \frac{\partial E^*}{\partial t} = y\), we apply the envelope theorem as follows: 1. Evaluate the derivative of E(p1, p2, p3, Ubar) with respect to t. 2. Evaluate the derivative of E*(p1, t, Ubar) with respect to t. 3. Compare the results and prove that they are equal to y.

Demand for x1

To explain why the demand for x1 is the same under both approaches, we can show that in both cases, the maximization of utility depends on both x1 and the composite good y. Therefore, the choice of x1 will not be affected by the change in the definition of the expenditure function.

PART B

The second approach: Proof using two-stage maximization Now, we proceed with the two-stage maximization approach.

Stage 1 Value function

To explain why the value function in stage 1 depends only on x1 and m, we argue that in this stage, the consumer maximizes utility only with respect to the composite good y = p2*x2 + p3*x3. And since p3 is constant, the value function will only depend on x1 and the choice of total expenditure m towards the purchase of the composite good.

Equivalence of Lagrange multipliers

To show that the two approaches to this maximization problem yield the same result by showing that λ = μ = δ, we do the following: 1. Compare the first-order conditions from the original maximization problem with the first-order conditions from stage 1, and show that since both conditions imply maximization of utility, λ = μ. 2. Compare the first-order conditions from stage 2 maximization problem with the conditions in stage 1, and show that since both conditions imply maximization of the value function, μ = δ. Finally, if we can show that λ = μ = δ, we would have demonstrated that both approaches lead to the same result.

Assumptions

To ensure the results are equivalent, we need to assume that the utility function is continuous, increasing, and concave, satisfying the second-order conditions for utility maximization problems. And the budget constraint is convex.

Key concepts

Expenditure Function Minimization

Understanding the expenditure function minimization is essential for grasping the composite commodity theorem. This approach involves finding the least costly way to achieve a specific level of utility given the prices of goods and the consumer's income. For instance, if you want to maintain a specific level of happiness or satisfaction, you would seek to do so at the minimum possible cost.

In our exercise, the expenditure function is denoted as \(E(p_1, p_2, p_3, \bar{U})\), representing the minimum expense needed to reach utility level \(\bar{U}\), with \(p_1\), \(p_2\), and \(p_3\) as the prices for goods \(x_1\), \(x_2\), and \(x_3\) respectively. The problem introduces a composite good \(y\) and shows that treating \(x_2\) and \(x_3\) as a single good doesn't alter the demand for \(x_1\), thereby simplifying the analysis. This is an application of the theory that leads to the simplification of complex economic models, focusing analysis on the composite good \(y\) and \(x_1\).

Utility Maximization

Utility maximization is the cornerstone of consumer choice theory, highlighting that individuals aim to maximize their satisfaction within their budget constraints. A consumer will allocate their spending across various goods to maximize their happiness or utility, symbolized as \( U(x_1, x_2, x_3) \).

In the given exercise, the utility maximization problem is presented in two formats: the initial problem and the two-stage maximization. The goal is to show that regardless of the method, the consumer's demand for \(x_1\) remains consistent. This demonstrates that the consumers' choices are the same, whether they consider goods separately or lump some together as a composite good.

Envelope Theorem

The envelope theorem is a useful tool in economics that facilitates the analysis of how a function's optimal value changes as a parameter changes. Specifically, it tells us how the minimum expenditure required to achieve a certain utility level changes as the price of a good changes.

Applying the envelope theorem to our expenditure minimization problem allows us to see how the total expense to reach a particular utility \( \bar{U} \) fluctuates with respect to price changes. This strategy reveals that the derivative of the expenditure function in relation to price \( t \) is equal to the demand for the composite good \( y \), proving a key aspect of the composite commodity theorem and confirming that the expenditures on this composite good is unaffected by how we aggregate goods \( x_2 \) and \( x_3 \) into \( y \).

Lagrange Multipliers

The method of Lagrange multipliers is the mathematical strategy used in optimization problems subject to constraints. It helps in finding the local maxima or minima of a function subject to equality constraints.

In our exercise, Lagrange multipliers help us identify the optimal quantities of goods \( x_1 \), \( x_2 \) and \( x_3 \) that a consumer will purchase to maximize their utility subject to their budget constraint. Whether using the standard utility maximization approach or the two-stage process, the Lagrange multiplier \( \lambda \) will equalize the marginal utility per dollar spent on each good. This demonstrates the consistency of consumer choice across different methods of problem structure and reinforces the validity of the composite commodity theorem.

First-order Conditions

First-order conditions are essentially the 'critical points' in optimization problems. They represent the requirements for a solution to potentially be an optimal point - either a maximum or minimum - of a function with respect to its variables.

In the exercise, the first-order conditions are derived from setting the derivatives of the utility function with respect to each good equal to the corresponding Lagrange multiplier times the price of the good. This is indicated as \( U_i = \lambda p_i \) for original utility maximization and \( U_i = \mu p_i \) for stage 1 of the two-stage approach. By showing this equivalence, we confirm that the selection of \( x_1 \) is consistent across different structuring of the maximization problem, serving as a fundamental step in proving the composite commodity theorem.

Two-Stage Maximization Approach

The two-stage maximization approach is a technique that simplifies complex decision-making by breaking it down into more manageable parts. Initially, it separates a consumer's choices into two distinct steps, which is quite useful in scenarios involving composite commodities.

In the first stage, the consumer maximizes their utility concerning a composite good, treating other goods as fixed parameters. The second stage then involves optimizing the remaining choices given the results of the first stage. This approach proves that a consumer's decisions about non-composite goods, like \( x_1 \) in our problem, remain unchanged even when we simplify the model by considering composite goods. It adds an intuitive layer to understanding how consumers allocate their budget across various goods, and is particularly important in supporting the composite commodity theorem.