Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (X) or chicken soup (I'). Smith's utility function is given by \\[ U_{r}=X_{-}^{3} Y \\] whereas Jones' is given by \\[ U_{j}=X^{s} Y^{s} \\] The individuals do not care whether they produce \(X\) or \(Y\), and the production function for cach good is given by \\[ \begin{array}{l} X=2 L \\ Y=3 L \end{array} \\] where \(L\) is the total labor devoted to production of each good. Using this information, a. What must the price ratio, \(P_{x} / P_{Y},\) be? b. Given this price ratio, how much Xand Fwill Smith and Jones demand? (Hint: Set the wage equal to 1 here. c. How should labor be allocated between \(X\) and \(Y\) to satisfy the demand calculated in part (b)?

Smith and Jones have different utility functions: Smith: \[U_r = X^{-3}Y\] Jones: \[U_j = X^sY^s\] The production functions for each good are: Ice Cream (X): \[X = 2L_x\] Chicken Soup (Y): \[Y = 3L_y\]

To find the price ratio, we first need to find the marginal rates of substitution (MRS) for both individuals. Smith: \[MRS_r = \frac{\partial U_r}{\partial X} / \frac{\partial U_r}{\partial Y} = \frac{-3X^{-4}Y}{X^{-3}} = -3X^{-1}\] Jones: \[MRS_j = \frac{\partial U_j}{\partial X} / \frac{\partial U_j}{\partial Y} = \frac{sX^{s-1}Y^s}{X^sY^{s-1}} = s\]

Since they do not care whether they produce X or Y, their marginal rates of substitution need to be equal. Therefore: \[-3X^{-1} = s\] \[s = -3X^{-1}\] Now, we can use their MRS to find the price ratio between the two goods: \[P_x / P_y = -3X^{-1}\]

Given the price ratio, we can determine how much Smith and Jones will demand for each product. We set the wage equal to 1, the constraint becomes: \[L_x + L_y = 10\] X: \(2L_x = X\) Y: \(3L_y = Y\) Now, we can find the optimal demand for X and Y by setting the MRS equal to the price ratio: \[-3X^{-1} = \frac{P_x}{P_y}\] \[P_x = -3X^{-1}P_y\] Replace the production functions for X and Y into the constraints \[2L_x + 3L_y = 10\] Substitute the given price ratio in \(P_x\) and \(P_y\): \[L_x = \frac{X}{2}\] \[L_y = \frac{Y}{3}\] Now, we can solve for X and Y: \[X = -3Y^{-1} \implies X = -\frac{3}{Y}\] \[L_x = \frac{X}{2} = \frac{-3}{2Y}\] \[L_y = \frac{Y}{3}\] Substitute \(L_x\) and \(L_y\) into the labor constraint \(L_x + L_y = 10\): \[\frac{-3}{2Y} + \frac{Y}{3} = 10\] Solve for Y: \[Y = 6\] Plug Y back into the relationship between X and Y: \[X = -\frac{3}{Y} = -\frac{3}{6} = -\frac{1}{2}\] Smith and Jones demand: Smith demand for X: \(X^{-3} = (-\frac{1}{2})^{-3} = -8\) Jones demand for X: \(X^s = (-\frac{1}{2})^s\) Smith demand for Y: 6 Jones demand for Y: 6

To satisfy the demand calculated in part (b), we allocate labor between X and Y. \[L_x = \frac{X}{2} = (-\frac{1}{2}) / 2 = -\frac{1}{4}\] \[L_y = \frac{Y}{3} = \frac{6}{3} = 2\] Smith and Jones should allocate: \- 1/4 hours of labor to produce ice cream 2 hours of labor to produce chicken soup