Question

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 \(s\).

Short answer

In summary: a. We proved that the quantity of strudel Heidi buys is not affected by the price of goat's milk (\(\partial s / \partial p_{m} = 0\)). b. We also showed that the quantity of goat's milk Heidi buys is not affected by the price of strudel (\(\partial m / \partial p_{s} = 0\)). c. We used the Slutsky equation to prove that the income effects for goat's milk and strudel are identical. The key result being that the ratio (\(\frac{\partial s / \partial I}{\partial m / \partial I}\)) is equal to 1. d. Finally, we explicitly proved the same result in part (c) using the Marshallian demand functions for \(m\) and \(s\), confirming the identical income effects for both goods.

Step-by-step solution

a. Show that \(\partial s / \partial p_{m} = 0\).

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.

b. Show that \(\partial m / \partial p_{s} = 0\).

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.

c. Use the Slutsky equation to prove that the income effects are identical.

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.

d. Prove part (c) explicitly using the Marshallian demand functions for \(m\) and \(s\).

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.

Key concepts

Marshallian Demand Functions

The concept of Marshallian Demand Functions is crucial in understanding how consumers choose to allocate their income across different goods to maximize utility. These demand functions are derived from the consumer's utility maximization problem, subject to a budget constraint. In simpler terms, they represent the quantity of each good that maximizes consumer satisfaction given prices and the consumer's income.

• Mathematically, the Marshallian demand functions are determined by solving the consumer's utility maximization problem using their utility function (for instance, the utility function for goods like goat’s milk and strudel).
• By using these functions, we can study how the demand for goods changes when prices or income change.

In the given exercise, Heidi's Marshallian demand functions were derived through the Lagrangian method, providing insights into her purchasing habits at different price levels.

Slutsky Equation

The Slutsky Equation is a fundamental concept which helps in decomposing the total change in demand for a good into its substitution and income effects when the price of a good changes. Understanding this equation is key for analyzing how changes in price influence buying behavior.

• The Slutsky equation reads: \[\frac{\partial x_i}{\partial p_j} = \frac{\partial h_i}{\partial p_j} - x_i \frac{\partial x_i}{\partial I},\] where \(x_i\) is the quantity of a good, \(p_j\) is the price, and \(I\) is the income.
• Essentially, the equation separates the total effect into a substitution effect (the change in quantity due to a relative price change, holding utility constant) and an income effect (the change in quantity due to the change in real income, that is, purchasing power).

In this exercise, the Slutsky equation reveals that price changes for goat's milk and strudel have identical implications on income effects for Heidi, emphasizing the symmetry of net substitution effects.

Budget Constraint

A budget constraint represents all combinations of goods and services that a consumer can afford given their income and the prices of those goods. It is a basic yet powerful concept that limits the choices available to consumers.

• In the utility maximization problem, the budget constraint is given as \(p_m \cdot m + p_s \cdot s = I\), where \(p_m\) and \(p_s\) are the prices and \(m\) and \(s\) are the quantities of goat's milk and strudel, respectively.
• The budget line shows the trade-off between the two goods, where any increase in the consumption of one good implies a decrease in the consumption of the other, provided the income and price remain constant.

In the exercise related to Heidi, this constraint ensures that her consumption stays within her budget, guiding the allocation of her expenditures on goat's milk and strudel.

Lagrangian Method

The Lagrangian Method is a technique used in microeconomics to solve optimization problems with constraints, such as maximizing utility given a budget constraint, as seen in Heidi's exercise. This method is incredibly useful for finding the optimal combination of goods that maximizes utility.

• The Lagrangian function is constructed by combining the utility function and the budget constraint, with a Lagrange multiplier (\(\lambda\)) that indicates the shadow price of the constraint.
• The process involves taking partial derivatives of the Lagrangian with respect to all variables, setting them to zero, and solving the resulting set of equations.

For Heidi’s utility function of goat's milk and strudel, the Lagrangian method helped derive demand functions and demonstrated that price changes in one good did not affect the quantity consumed of the other. This approach provides insight into consumer behavior and the trade-offs made under budgetary limitations.