Question 1: Find the expenditure function for a consumer with each functions. ? a. U(X_(1),X_(2))=min X_(1),X_(2)) U(X_(1),X_(2))=sqrt (X_(1)X_(2)) C U(X_(1),X_(2))=X_(1)+X_(2)

To find the expenditure function for a consumer with each utility function, we need to first find the demand functions for each good and then derive the expenditure function.<br /><br />a. $U(X_{1},X_{2})=min\{ X_{1},X_{2}\}$<br /><br />The utility function is a Cobb-Douglas function with equal weights for both goods. The demand functions for each good can be found by maximizing the utility function subject to the budget constraint. Let the prices of good 1 and good 2 be $P_1$ and $P_2$, respectively, and let the consumer's income be $I$. The demand functions are:<br /><br />$X_{1} = \frac{I}{2P_{1}}$<br /><br />$X_{2} = \frac{I}{2P_{2}}$<br /><br />The expenditure function is the sum of the products of the quantities demanded and their respective prices:<br /><br />$E(P_{1}, P_{2}, I) = P_{1}X_{1} + P_{2}X_{2} = \frac{I}{2} + \frac{I}{2} = I$<br /><br />b. $U(X_{1},X_{2})=\sqrt {X_{1}X_{2}}$<br /><br />The utility function is a Cobb-Douglas function with equal weights for both goods. The demand functions for each good can be found by maximizing the utility function subject to the budget constraint. Let the prices of good 1 and good 2 be $P_1$ and $P_2$, respectively, and let the consumer's income be $I$. The demand functions are:<br /><br />$X_{1} = \frac{I}{P_{1}\sqrt{\frac{I}{P_{2}}}}$<br /><br />$X_{2} = \frac{I}{P_{2}\sqrt{\frac{I}{P_{1}}}}$<br /><br />The expenditure function is the sum of the products of the quantities demanded and their respective prices:<br /><br />$E(P_{1}, P_{2}, I) = P_{1}X_{1} + P_{2}X_{2} = \frac{I}{\sqrt{P_{1}P_{2}}} + \frac{I}{\sqrt{P_{1}P_{2}}} = \frac{2I}{\sqrt{P_{1}P_{2}}}$<br /><br />c. $U(X_{1},X_{2})=X_{1}+X_{2}$<br /><br />The utility function is a linear function with equal weights for both goods. The demand functions for each good can be found by maximizing the utility function subject to the budget constraint. Let the prices of good 1 and good 2 be $P_1$ and $P_2$, respectively, and let the consumer's income be $I$. The demand functions are:<br /><br />$X_{1} = \frac{I}{P_{1}}$<br /><br />$X_{2} = \frac{I}{P_{2}}$<br /><br />The expenditure function is the sum of the products of the quantities demanded and their respective prices:<br /><br />$E(P_{1}, P_{2}, I) = P_{1}X_{1} + P_{2}X_{2} = I + I = 2I$<br /><br />Therefore, the expenditure functions for each utility function are:<br /><br />a. $E(P_{1}, P_{2}, I) = I$<br /><br />b. $E(P_{1}, P_{2}, I) = \frac{2I}{\sqrt{P_{1}P_{2}}}$<br /><br />c. $E(P_{1}, P_{2}, I) = 2I$