Question

The general CES utility function is given by $$U(X, Y)=\frac{X^{\delta}}{\delta}+\frac{Y^{\delta}}{\delta}$$ a. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion $$\frac{X}{Y}=\left(\frac{P_{X}}{P_{Y}}\right)^{\frac{1}{\delta-1}}$$ b. Show that the result in part (a) implies that individuals will allocate their funds equally between \(X\) and \(Y\) for the Cobb-Douglas case \((\delta=0),\) as we have shown before in several problems. c. How does the ratio \(P_{x} X / P_{y} Y\) depend on the value of \(\delta ?\) Explain your results intuitively. (For further details on this function, see Extension E4.3.)

Short answer

Explain the effects of δ on the degree of substitutability between goods X and Y.

Step-by-step solution

Write down the given utility function

The CES utility function is given by: $$U(X,Y)=\frac{X^{\delta}}{\delta}+\frac{Y^{\delta}}{\delta}$$

Set up the budget constraint

The budget constraint for this problem is: $$P_X X + P_Y Y = I$$ Where \(P_X\) and \(P_Y\) are the prices of goods X and Y, respectively, and I is the consumer's income.

Form the Lagrange function for the constrained optimization problem

To find the first-order conditions, we need to create a Lagrangian function for this constrained optimization problem: $$\mathcal{L}(X,Y,\lambda) = \frac{X^{\delta}}{\delta}+\frac{Y^{\delta}}{\delta} - \lambda (P_X X + P_Y Y - I)$$

Calculate first-order conditions

Now, take the partial derivatives of the Lagrange function with respect to X, Y, and λ: $$\frac{\partial \mathcal{L}}{\partial X} = X^{\delta-1} - \lambda P_X = 0$$ $$\frac{\partial \mathcal{L}}{\partial Y} = Y^{\delta-1} - \lambda P_Y = 0$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = P_X X + P_Y Y - I = 0$$

Solve the first-order conditions to find the optimal consumption of goods X and Y

From the first two equations, we can write: $$\frac{X^{\delta-1}}{Y^{\delta-1}}=\frac{P_Y}{P_X}$$ Now, isolate the \(X/Y\) ratio: $$\frac{X}{Y}=\left(\frac{P_X}{P_Y}\right)^{\frac{1}{\delta-1}}$$

Show the result for the Cobb-Douglas case (δ=0)

In the case of a Cobb-Douglas utility function, where δ=0, the first-order condition becomes: $$\frac{X}{Y}=\left(\frac{P_X}{P_Y}\right)^{\frac{1}{-1}}=\frac{P_Y}{P_X}$$ Multiply both sides by \(P_X P_Y\): $$P_Y X = P_X Y$$ Hence, in the Cobb-Douglas case, individuals will allocate their funds equally between X and Y, as shown before in previous problems.

Analyze the ratio \(P_{x} X / P_{y} Y\) depending on the value of δ

Using the optimal proportion of goods X and Y obtained in Step 5: $$\frac{X}{Y}=\left(\frac{P_X}{P_Y}\right)^{\frac{1}{\delta-1}}$$ Multiply both sides by \(P_Y\): $$\frac{P_x X}{P_y Y}=\frac{P_X^{\frac{\delta}{\delta-1}}}{P_Y^{\frac{\delta}{\delta-1}}}$$ The ratio \(P_{x} X / P_{y} Y\) depends on the value of δ, and if δ is greater than 1, the ratio will be greater than 1, meaning the consumer will spend more on good X than good Y. If δ is less than 1, the ratio will be less than 1, meaning the consumer will spend more on good Y than good X. When δ is equal to 0 (Cobb-Douglas case), the ratio will be equal to 1, indicating equal spending on both goods. The value of δ determines the degree of substitutability between goods X and Y. A higher δ indicates that goods X and Y are easier to substitute for one another, and the consumer will allocate their funds more sensitive to prices changes. A lower δ means the consumer will find it more difficult to substitute good X for good Y, resulting in the funds allocated being less sensitive to price changes.

Key concepts

Lagrangian optimization

Lagrangian optimization is a powerful mathematical technique used to find the optimal solution of a problem subject to constraints. It is particularly useful in economics, where consumers seek to maximize utility while staying within a budget. A Lagrangian function combines the original function and the constraint into one expression, making it easier to solve.
In the context of utility maximization, the Lagrangian function allows us to incorporate the budget constraint directly into the optimization process. For a given utility function, such as the CES utility function, we can use the Lagrangian approach to find the optimal consumption of different goods.
Here, the Lagrangian function \[ \mathcal{L}(X,Y,\lambda) = \frac{X^{\delta}}{\delta} + \frac{Y^{\delta}}{\delta} - \lambda (P_X X + P_Y Y - I) \]helps determine how a consumer can optimally allocate their income \(I\) across goods \(X\) and \(Y\). By solving the partial derivatives of this function, we obtain the first-order conditions that are crucial for finding the maximum utility.

Cobb-Douglas utility function

The Cobb-Douglas utility function is a specific form of utility function that assumes a particular type of consumer preference. It is characterized by constant expenditure shares on goods, regardless of changes in income or prices.
In mathematical terms, a Cobb-Douglas utility function for two goods would typically be expressed as \[ U(X, Y) = X^{a} Y^{b}, \]where \(a\) and \(b\) are positive constants indicating the elasticity of substitution between the goods. When \(\delta = 0\), the CES function behaves like a Cobb-Douglas function.
With this utility function, the first-order conditions show that consumers will allocate their spending equally between goods \(X\) and \(Y\). This result emerges because the elasticity of substitution is constant, and consumers proportionally adjust their spending based on relative price changes, maintaining equal marginal rate of substitution.

First-order conditions

First-order conditions are critical components in optimization problems. They are derived from the Lagrangian function and provide the necessary conditions for a maximum or minimum.
To identify the optimal point, we take partial derivatives of the Lagrangian with respect to each variable. For the CES utility problem, the first-order conditions are given by:

• \(\frac{\partial \mathcal{L}}{\partial X} = X^{\delta-1} - \lambda P_X = 0\)
• \(\frac{\partial \mathcal{L}}{\partial Y} = Y^{\delta-1} - \lambda P_Y = 0\)
• \(\frac{\partial \mathcal{L}}{\partial \lambda} = P_X X + P_Y Y - I = 0\)

These equations determine the balance between marginal utility of each good and its price, adjusted by the Lagrange multiplier \(\lambda\), representing the marginal utility of income.
These conditions can be solved to reveal important relationships, such as the ratio \(\frac{X}{Y}\), representing the optimal consumption mix. Understanding these conditions is essential as they guide us in how a consumer should best allocate their resources to maximize satisfaction under budget constraints.