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.

Key concepts

Marshallian Demand

The Marshallian Demand, often called uncompensated demand, is derived when the consumer aims to maximize utility given a set budget. Using the utility function \(U(X, Y) = XY + Y\), we establish the Lagrangian to incorporate the budget constraint, set by income \(I\) and the prices of goods \(p_X\) for \(X\) and \(p_Y\) for \(Y\). This allows us to form a problem that balances the trade-off between spending resources and maximizing satisfaction.
The Lagrangian function is created as follows:

• \( L = XY + Y + \lambda(I - p_X X - p_Y Y) \).

From this, we derive the Marshallian demands. The first-order conditions (FOCs) obtained from taking partial derivatives with respect to \(X\), \(Y\), and \(\lambda\) help us find optimal choices. Solving these equations gives us:

• \(X^M = \frac{I - p_Y}{p_X^2}\)
• \(Y^M = I - p_X \left(\frac{I - p_Y}{p_X^2}\right)\)

Marshallian demand reflects changes in both income N\(I\) and the prices \(p_X\), \(p_Y\), illustrating how these factors shift the demand curve for each good. If income rises, demand usually increases, shifting the demand curve outward. If the price of a good rises, all else equal, demand for that good typically falls.

Hicksian Demand

Hicksian Demand, or compensated demand, concerns itself with the specific utility level maintenance at minimal cost. Here, rather than responding to income, demand responses focus solely on price changes, thus removing the income effect.To derive Hicksian demand, we substitute the expenditure function in place of income in the Marshallian demand formula:

• \(X^H = \frac{E-\frac{p_Y^2 U}{p_X^2 + p_Y}}{p_X^2}\)
• \(Y^H = E - p_X \left(\frac{E-\frac{p_Y^2 U}{p_X^2 + p_Y}}{p_X^2}\right)\)

In the specific context of our utility function, the Hicksian demand curves are shifted only by changes in the prices of other goods rather than income fluctuations. This compensatory approach looks at the amount needed to keep utility constant despite price changes. For instance, if the price of \(Y\) increases, the Hicksian demand for \(X\) could rise if \(X\) substitutes for \(Y\), showing the demand curve shifting. Similarly, an increase in \(p_X\) can cause \(Y\)'s demand curve to shift left, reflecting a reduced quantity demanded at the same utility level.

Expenditure Function

The expenditure function is a powerful tool in microeconomics, used to determine the minimum level of spending needed to achieve a certain utility. It captures how much income is necessary to maintain a utility level \(U\) given the prices \(p_X\) and \(p_Y\).For our example, the expenditure function is derived by plugging the Marshallian demand back into the budget constraint and solving for \(I\):

• \(E(p_X, p_Y, U) = \frac{p_Y^2 U}{p_X^2 + p_Y}\)

This expression tells us how expenditure varies with price changes while achieving the same utility level. If prices rise, naturally the expenditure required to maintain the same utility level will also increase.Understanding the expenditure function is crucial because it underpins how consumers adjust their spending in response to price changes. It also forms a basis for calculating Hicksian demand. Tracking how expenditure shifts due to prices provides insights into consumer behavior and resource allocation in markets.