Question

Suppose the utility function for goods \(X\) and \(Y\) is given by \\[ \text { utility }=U(X, Y)=X Y+Y \\] a. Calculate the uncompensated (Marshallian) demand functions for \(X\) and \(Y\) and describe how the demand curves for \(X\) and \(Y\) are shifted by changes in \(I\) or in the price of the other good. b. Calculate the expenditure function for \(X\) and \(Y\). c. Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods \(X\) and \(Y\). Describe how the compensated demand curves for \(X\) and \(Y\) are shifted by changes in income or by changes in the prices of the other good.

Short answer

Question: Derive the Marshallian and Hicksian demand functions for the utility function \(U(X, Y) = XY + Y\), and explain how the demand curves are shifted for each. Answer: The Marshallian demand functions are: 1. \(X^M = \frac{I - p_Y}{p_X^2}\) 2. \(Y^M = I - p_X \left(\frac{I - p_Y}{p_X^2}\right)\) For the Marshallian demand curves, the change in demand for \(X\) or \(Y\) will be determined by changes in income (\(I\)) or in the price of the other good (\(p_X\) or \(p_Y\)). The Hicksian demand functions are: 1. \(X^H = \frac{E-\frac{p_Y^2 U}{p_X^2 + p_Y}}{p_X^2}\) 2. \(Y^H = E - p_X \left(\frac{E-\frac{p_Y^2 U}{p_X^2 + p_Y}}{p_X^2}\right)\) For the Hicksian demand curves, changes in income have no effect, but changes in the prices of the other goods would. An increase in \(p_Y\) would lead to a rightward shift in the compensated demand curve for \(X\), while a decrease in \(p_Y\) would lead to a leftward shift. An increase in \(p_X\) would lead to a leftward shift in the compensated demand curve for \(Y\), while a decrease in \(p_X\) would lead to a rightward shift.

Step-by-step solution

a. Uncompensated (Marshallian) demand functions for \(X\) and \(Y\)

First, we'll derive the Marshallian demands for \(X\) and \(Y\). Given the utility function \(U(X, Y) = XY + Y\), we will set up the Lagrangian: \( L = XY + Y + \lambda(I - p_X X - p_Y Y) \) To find the Marshallian demands, we'll get the first order conditions (FOCs) by taking the partial derivatives with respect to \(X\), \(Y\), and \(\lambda\): 1. \(\frac{\partial L}{\partial X} = Y - \lambda p_X\) 2. \(\frac{\partial L}{\partial Y} = X + 1 - \lambda p_Y\) 3. \(\frac{\partial L}{\partial \lambda} = I - p_X X - p_Y Y\) Now we solve the 2 first equations to obtain \(\lambda\), obtaining \(\lambda = \frac{Y}{p_X} = \frac{X + 1}{p_Y}\), so we have: \(Y = p_X (X + 1)\) or \(X = \frac{Y - p_X}{p_X}\) Substituting \(X\) in the budget constraint, we have: \(I - p_X X - p_Y Y = I - p_X\left(\frac{Y - p_X}{p_X}\right) - p_Y Y = 0\) \(I - p_Y(p_X + 1) = 0\) Solving this equation, we obtain the Marshallian demands: \(X^M = \frac{I - p_Y}{p_X^2}\) \(Y^M = I - p_X \left(\frac{I - p_Y}{p_X^2}\right)\) The change in the demand curve for \(X\) or \(Y\) will be determined by changes in income (\(I\)) or in the price of the other good (\(p_X\) or \(p_Y\)).

b. Expenditure function for \(X\) and \(Y\)

The expenditure function is calculated as the minimum amount of money spent to reach a given utility level \(U\). We can derive this by substituting the demand functions for \(X\) and \(Y\) into the budget constraint, and then solving for \(I\): \(E(p_X,p_Y,U) = p_X X^M + p_Y Y^M\) \(E(p_X,p_Y,U) = p_X \left(\frac{I - p_Y}{p_X^2}\right) + p_Y \left(I - p_X \left(\frac{I - p_Y}{p_X^2}\right)\right)\) Now, we solve this equation for \(I\): \(I=E(p_X,p_Y,U)=\frac{p_Y^2 U}{p_X^2 + p_Y}\)

c. Compensated (Hicksian) demand functions for \(X\) and \(Y\)

To find the Hicksian demands, we need to find the optimal bundle (\(X^H\) and \(Y^H\)) that maximizes utility while keeping the expenditure fixed at the specific level \(E\). In order to do that, we replace \(I\) in the Marshallian demands by \(E(p_X, p_Y, U)\): \(X^H = \frac{E-\frac{p_Y^2 U}{p_X^2 + p_Y}}{p_X^2}\) \(Y^H = E - p_X \left(\frac{E-\frac{p_Y^2 U}{p_X^2 + p_Y}}{p_X^2}\right)\) Hicksian demand curves for \(X\) and \(Y\) will be shifted by changes in income or by changes in the prices of the other good. In this specific utility function, the Hicksian demands are not affected by changes in income, but are affected by changes in the prices of the other goods. The reasons are: - For \(X^H\), when \(p_Y\) increases, the denominator term \(p_X^2 + p_Y\) will increase as well, resulting in a higher overall value for \(X^H\), which means the compensated demand curve for \(X\) would shift rightward as the price for \(Y\) increases. Conversely, if \(p_Y\) decreases, the compensated demand curve for \(X\) would shift leftward. - For \(Y^H\), when \(p_X\) increases, the term inside the parenthesis in the second part will increase as well, resulting in a lower overall value for \(Y^H\), which means the compensated demand curve for \(Y\) would shift leftward as the price for \(X\) increases. Conversely, if \(p_X\) decreases, the compensated demand curve for \(Y\) would shift rightward.