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.