Heidi receives utility from two goods, goat's milk ( \(m\) ) and strudel (s), according to the utility function \\[U(m, s)=m \cdot s.\\] a. Show that increases in the price of goat's milk will not affect the quantity of strudel Heidi buys; that is, show that \(\partial s / \partial p_{m}=0\). b. Show also that \(\partial m / \partial p_{s}=0\). c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the derivatives in parts (a) and (b) are identical. d. Prove part (c) explicitly using the Marshallian demand functions for \(m\) and \(s\).

We are given Heidi's utility function \\[U(m, s) = m \cdot s.\\] The expenditure for \(m\) and \(s\) are \(p_m \cdot m\) and \(p_s \cdot s\), respectively. Let's assume that Heidi's income is \(I\). Then her budget constraint can be written as \\[p_m \cdot m + p_s \cdot s = I.\\] To find the optimal consumption of \(m\) and \(s\), we need to maximize the utility function subject to the budget constraint. To do this, we can use the Lagrange method with the Lagrangian as follows: \\[\mathcal{L} = U(m, s) + \lambda (I - p_m \cdot m - p_s \cdot s).\\] Now we need to find the partial derivatives of the Lagrangian with respect to \(m\), \(s\), and \(\lambda\) and set them equal to zero: 1. \\[\frac{\partial \mathcal{L}}{\partial m} = s - \lambda p_m = 0.\\] 2. \\[\frac{\partial \mathcal{L}}{\partial s} = m - \lambda p_s = 0.\\] 3. \\[\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_m \cdot m - p_s \cdot s = 0.\\] Now we need to solve these equations to find partial derivative \(\partial s / \partial p_{m}\): From equation (1), \\[\lambda = \frac{s}{p_m}.\\] From equation (2), \\[\lambda = \frac{m}{p_s}.\\] Combining both equations, we get \\[\frac{s}{p_m} = \frac{m}{p_s} \Rightarrow s = \frac{m p_m}{p_s}.\\] Now, let's find the partial derivative of \(s\) with respect to \(p_m\): \\[\frac{\partial s}{\partial p_{m}} = \frac{\partial}{\partial p_{m}} \left(\frac{m p_m}{p_s}\right) = \frac{m}{p_s}.\\] Since \(m\) and \(p_s\) are independent of \(p_m\), the partial derivative \(\partial s / \partial p_{m}\) is equal to zero. Hence, the quantity of strudel Heidi buys is not affected by the price of goat's milk.

Using the result from the previous part, we found that \(\lambda = \frac{s}{p_m} = \frac{m}{p_s}\). Now let's find the partial derivative of \(m\) with respect to \(p_s\): From the expression of \(\lambda\), we can rewrite it as: \\[m = \frac{s p_s}{p_m}.\\] Now, let's compute the partial derivative of \(m\) with respect to \(p_s\): \\[\frac{\partial m}{\partial p_{s}} = \frac{\partial}{\partial p_{s}} \left(\frac{s p_s}{p_m}\right) = \frac{s}{p_m}.\\] Since \(s\) and \(p_m\) are independent of \(p_s\), the partial derivative \(\partial m / \partial p_{s}\) is equal to zero. Hence, the quantity of goat's milk Heidi buys is not affected by the price of strudel.

The Slutsky equation states that the total effect on the demand for a good due to a price change can be decomposed into a substitution effect and an income effect: \\[\frac{\partial x_i}{\partial p_j} = \frac{\partial h_i}{\partial p_j} - \frac{\delta_{ij} x_i}{\partial I}.\\] In our case, \(i\) and \(j\) can be either \(m\) or \(s\). We know from parts (a) and (b) that \(\frac{\partial s}{\partial p_m} = \frac{\partial m}{\partial p_{s}} = 0\) and the substitution effects are also zero because the utility function exhibits a constant marginal rate of substitution. Therefore, we are left with: \\[\frac{\partial s / \partial I}{\partial m / \partial I} = 1.\\] This equality implies that the income effects for both goods are identical.

To prove part (c) explicitly, we need to find the Marshallian demand functions for \(m\) and \(s\) based on their respective expressions we derived earlier: \\[m = \frac{s p_s}{p_m},\\] \\[s = \frac{m p_m}{p_s}.\\] Using the budget constraint \(p_m \cdot m + p_s \cdot s = I\), we can substitute the expressions for \(m\) and \(s\) into the constraint equation: \\[p_m \cdot \frac{s p_s}{p_m} + p_s \cdot \frac{m p_m}{p_s} = I \Rightarrow s^2 + m^2 = \frac{I^2}{p_m^2 + p_s^2}.\\] Now, let's find the demand functions \(m(I, p_m, p_s)\) and \(s(I, p_m, p_s)\): 1. \\[m(I, p_m, p_s) = \frac{s(I, p_m, p_s) p_s}{p_m}.\\] 2. \\[s(I, p_m, p_s) = \frac{m(I, p_m, p_s) p_m}{p_s}.\\] Taking derivative of both the demand functions with respect to Income \(I\): 1. From part (a), we know that \(\frac{\partial s}{\partial p_m} = 0\). So, using the budget constraint, we find the derivative with respect to \(I\): \\[\frac{\partial m}{\partial I} = \frac{p_m}{p_m^2 + p_s^2}.\\] 2. Similarly, using the result from part (b) along with the budget constraint, we find the derivative with respect to \(I\): \\[\frac{\partial s}{\partial I} = \frac{p_s}{p_m^2 + p_s^2}.\\] The ratios of these income effects are: \\[\frac{\partial s / \partial I}{\partial m / \partial I} = \frac{\frac{p_s}{p_m^2 + p_s^2}}{\frac{p_m}{p_m^2 + p_s^2}} = \frac{p_s}{p_m} = 1.\\] Thus, we have proven explicitly that the income effects for goat's milk and strudel are identical.