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.

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.