Question

As in Example 5.1 , assume that utility is given by \\[ \text { utility }=U(x, y)=x^{0.3} y^{0.7} \\] a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expenditure function for this case. b. Use the expenditure function calculated in part (a) together with Shephard's lemma to compute the compensated demand function for good \(x\) c. Use the results from part (b) together with the uncompensated demand function for good \(x\) to show that the Slutsky equation holds for this case.

Short answer

a. To find the indirect utility function, substitute the Marshallian demand functions into the utility function: \\[V(p_{x}, p_{y}, I) = U\left(\frac{0.3I}{p_{x}}, \frac{0.7I}{p_{y}}\right)\\] \\[V(p_{x}, p_{y}, I) = \left(\frac{0.3I}{p_{x}}\right)^{0.3} \left(\frac{0.7I}{p_{y}}\right)^{0.7} = \frac{0.3^{0.3}0.7^{0.7}I}{p_x^{0.3}p_y^{0.7}}\\] And the expenditure function: \\[E(p_{x}, p_{y}, U^{\ast}) = p_{x} \frac{0.3U^{\ast}}{0.3p_{x}} + p_{y} \frac{0.7U^{\ast}}{0.7p_{y}}\\] \\[E(p_{x}, p_{y}, U^{\ast}) = U^{\ast}(p_x + p_y)\\] b. Using Shephard's lemma, we compute the derivative of \(E\) with respect to \(p_x\): \\[x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial E}{\partial p_{x}} = U^{\ast}\\] c. Now, we can verify the Slutsky equation for the given utility function. First, compute the derivative of the uncompensated demand function for good x with respect to the income: \\[\frac{\partial x}{\partial I}(p_{x}, p_{y}, I) = \frac{0.3}{p_{x}}\\] Now, multiply it by the income: \\[\frac{0.3}{p_{x}} \cdot I = \frac{0.3I}{p_{x}}\\] Finally, verify the Slutsky equation: \\[x(p_{x}, p_{y}, I) - x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial x}{\partial I}(p_{x}, p_{y}, I) \cdot I\\] \\[\frac{0.3I}{p_{x}} - U^{\ast} = \frac{0.3I}{p_{x}}\\] \\[U^{\ast} = 0\\] The Slutsky equation holds in this case since the compensated demand function is equal to the indirect utility function, which proves that the utility function has no income effect. In other words, the consumer's preferences do not change when their income changes, as these preferences only depend on the relative prices of the goods.

Step-by-step solution

a. Find the indirect utility function and the expenditure function

Recall from Example 5.1, the uncompensated demand functions are the Marshallian demand functions \(x(p_{x},p_{y},I)\) and \(y(p_{x},p_{y},I)\), which are given by: \\[x(p_{x}, p_{y}, I) = \frac{0.3I}{p_{x}}\\] \\[y(p_{x}, p_{y}, I) = \frac{0.7I}{p_{y}}\\] To find the indirect utility function, substitute the Marshallian demand functions into the utility function: \\[V(p_{x}, p_{y}, I) = U(x(p_{x}, p_{y}, I), y(p_{x}, p_{y}, I))\\] Now, to find the expenditure function, we solve for the minimal expenditure needed to achieve a certain utility level, \(U^{\ast}\), with respect to \(p_{x}\) and \(p_{y}\). To do this, we set \(U(x, y) = U^{\ast}\) and solve for the expenditure, \(E\): \\[E(p_{x}, p_{y}, U^{\ast}) = p_{x}x(p_{x}, p_{y}, U^{\ast}) + p_{y}y(p_{x}, p_{y}, U^{\ast})\\]

b. Find the compensated demand function for good x

Shephard's lemma states that the partial derivative of the expenditure function with respect to the price of a good gives the compensated demand (also known as the Hicksian demand) for that good. Therefore, we compute the derivative of \(E\) with respect to \(p_x\): \\[x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial E}{\partial p_{x}}\\]

c. Show that the Slutsky equation holds

The Slutsky equation states that the difference between the uncompensated and compensated demand functions is equal to the product of the income effect and the derivative of the uncompensated demand function for good x with respect to the income: \\[x(p_{x}, p_{y}, I) - x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial x}{\partial I}(p_{x}, p_{y}, I) \cdot I\\] We have found the uncompensated demand function, \(x(p_{x}, p_{y}, I)\), and the compensated demand function, \(x^c(p_{x}, p_{y}, U^{\ast})\). We will now calculate the derivative of the uncompensated demand function for good x with respect to the income, multiply it by the income, and show that the Slutsky equation holds true for our given utility function.