Question

The CES utility function we have used in this chapter is given by \\[U(x, y)=\frac{x^{8}}{\delta}+\frac{y^{8}}{\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)^{1 /(8-1)}.\\] 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.) d. Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.

Short answer

Short Answer: In this exercise, we derived the proportion of goods x and y that consumers consume depending on their given prices \(p_x\) and \(p_y\), and analyzed how the ratio of expenditures on goods x and y depends on the value of δ. We found that for the Cobb-Douglas case, consumers split their funds equally between x and y. In general, the ratio \(p_x x / p_y y\) depends on δ through the exponent 6/7, capturing the degree to which the consumer views goods x and y as substitutable or complementary. Moreover, we derived the indirect utility and expenditure functions and verified their homogeneity properties.

Step-by-step solution

Part a: Derive the First-Order Conditions

First, we need to set up the Lagrangian for the maximization problem. The consumer has the budget constraint, which can be written as: \\[p_x x + p_y y = m.\\] The Lagrangian is defined as, \\[L(x,y,\lambda) = \frac{x^8}{\delta} + \frac{y^8}{\delta} + \lambda(m - p_x x - p_y y).\\] To find the first-order conditions, we need to take the partial derivatives with respect to x, y, and λ, and set them equal to zero: \\[\frac{\partial L}{\partial x} = \frac{8 x^7}{\delta} - \lambda p_x = 0,\\] \\[\frac{\partial L}{\partial y} = \frac{8 y^7}{\delta} - \lambda p_y = 0,\\] \\[\frac{\partial L}{\partial \lambda} = m - p_x x - p_y y = 0.\\] Now, divide the first derivative with respect to x by the first derivative with respect to y: \\[\frac{\partial L / \partial x}{\partial L / \partial y} = \frac{\frac{8 x^7}{\delta} - \lambda p_x}{\frac{8 y^7}{\delta} - \lambda p_y}.\\] Setting the derivatives equal to zero, we get: \\[\frac{8 x^7 / \delta}{8 y^7 / \delta} = \frac{\lambda p_x}{\lambda p_y}\\] Now, cancel out the common terms: \\[\frac{x^7}{y^7} = \frac{p_x}{p_y}\\] Take the 7th root of both sides to get the proportion: \\[\frac{x}{y} = \left( \frac{p_x}{p_y} \right)^{\frac{1}{7}}.\\]

Part b: Cobb-Douglas case

When \(\delta = 0\), the utility function becomes a Cobb-Douglas utility function, which can be written as: \\[U(x,y) = x^{\frac{8}{7}}y^{\frac{8}{7}}.\\] Now, recall the proportion we derived in part (a): \\[\frac{x}{y} = \left(\frac{p_x}{p_y}\right)^{\frac{1}{7}}.\\] Multiplying both sides by \(y\) and then substituting \(x\) with \(\frac{m}{p_x + p_y}\), we get: \\[x = y \left(\frac{p_x}{p_y}\right)^{\frac{1}{7}} \\] \\[x = \frac{m}{p_x + p_y}\] So the consumer allocates half of their funds to good x and half to good y in the Cobb-Douglas case.

Part c: Ratio of px*x / py*y depending on δ

Recall the proportion we derived in part (a): \\[\frac{x}{y} = \left(\frac{p_x}{p_y}\right)^{\frac{1}{7}}.\\] Multiply both sides by \(p_y\) and divide by \(p_x\): \\[\frac{p_y x}{p_x y} = \frac{p_y}{p_x} \left(\frac{p_x}{p_y}\right)^{\frac{1}{7}}.\\] Now substitute \(\frac{p_x}{p_y}\) with \(\frac{p_y}{p_x}\) and simplify the expression: \\[\frac{p_y x}{p_x y} = \left(\frac{p_y}{p_x}\right)^{\frac{6}{7}}.\\] The result shows that the ratio \(\frac{p_x x}{p_y y}\) depends on \(\delta\) through the exponent \(\frac{6}{7}\). Intuitively, the larger the value of \(\delta\), the closer the preferences are to perfect complements. The smaller the value of \(\delta\), the closer the preferences are to perfect substitutes. Therefore, the ratio \(\frac{p_x x}{p_y y}\) captures the degree to which the consumer views goods x and y as substitutable or complementary.

Part d: Indirect Utility and Expenditure Functions

To derive the indirect utility function, we need to eliminate x and y from the utility function using the budget constraint and the proportion we derived in part (a). First, let's replace \(x\) in terms of \(y\): \\[x = y \left(\frac{p_x}{p_y}\right)^{\frac{1}{7}}\\] Now, substitute this expression into the budget constraint: \\[p_x y \left(\frac{p_x}{p_y}\right)^{\frac{1}{7}} + p_y y = m\\] Solve for \(y\): \\[y = \frac{m}{p_x\left(\frac{p_x}{p_y}\right)^{\frac{1}{7}}+p_y}\\] Next, substitute the value of \(y\) found above into the expression for \(x\): \\[x = \frac{m}{p_y\left(\frac{p_y}{p_x}\right)^{\frac{1}{7}}+p_x}\\] Now, substitute \(x\) and \(y\) into the utility function: \\[v(p_x, p_y, m) = \frac{\left(\frac{m}{p_y\left(\frac{p_y}{p_x}\right)^{\frac{1}{7}}+p_x}\right)^8}{\delta}+\frac{\left(\frac{m}{p_x\left(\frac{p_x}{p_y}\right)^{\frac{1}{7}}+p_y}\right)^8}{\delta}\\] To derive the expenditure function, we need to solve the system of first-order conditions for \(x\) and \(y\). Once we have expressions for \(x\) and \(y\) in terms of the prices and utility level, we can substitute them back into the budget constraint and solve for \(m\): \\[m(p_x, p_y, u)= p_x x(p_x, p_y, u) + p_y y(p_x, p_y, u).\\] The indirect utility function and the expenditure function are homogenous of degree zero and one in prices, respectively.