Question

As in Example 5.1 , assume that utility is given by \\[ \text { utility }=U(x, y)=x^{0.3} y^{0.7} \\] a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expenditure function for this case. b. Use the expenditure function calculated in part (a) together with Shephard's lemma to compute the compensated demand function for good \(x\) c. Use the results from part (b) together with the uncompensated demand function for good \(x\) to show that the Slutsky equation holds for this case.

Short answer

a. To find the indirect utility function, substitute the Marshallian demand functions into the utility function: \\[V(p_{x}, p_{y}, I) = U\left(\frac{0.3I}{p_{x}}, \frac{0.7I}{p_{y}}\right)\\] \\[V(p_{x}, p_{y}, I) = \left(\frac{0.3I}{p_{x}}\right)^{0.3} \left(\frac{0.7I}{p_{y}}\right)^{0.7} = \frac{0.3^{0.3}0.7^{0.7}I}{p_x^{0.3}p_y^{0.7}}\\] And the expenditure function: \\[E(p_{x}, p_{y}, U^{\ast}) = p_{x} \frac{0.3U^{\ast}}{0.3p_{x}} + p_{y} \frac{0.7U^{\ast}}{0.7p_{y}}\\] \\[E(p_{x}, p_{y}, U^{\ast}) = U^{\ast}(p_x + p_y)\\] b. Using Shephard's lemma, we compute the derivative of \(E\) with respect to \(p_x\): \\[x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial E}{\partial p_{x}} = U^{\ast}\\] c. Now, we can verify the Slutsky equation for the given utility function. First, compute the derivative of the uncompensated demand function for good x with respect to the income: \\[\frac{\partial x}{\partial I}(p_{x}, p_{y}, I) = \frac{0.3}{p_{x}}\\] Now, multiply it by the income: \\[\frac{0.3}{p_{x}} \cdot I = \frac{0.3I}{p_{x}}\\] Finally, verify the Slutsky equation: \\[x(p_{x}, p_{y}, I) - x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial x}{\partial I}(p_{x}, p_{y}, I) \cdot I\\] \\[\frac{0.3I}{p_{x}} - U^{\ast} = \frac{0.3I}{p_{x}}\\] \\[U^{\ast} = 0\\] The Slutsky equation holds in this case since the compensated demand function is equal to the indirect utility function, which proves that the utility function has no income effect. In other words, the consumer's preferences do not change when their income changes, as these preferences only depend on the relative prices of the goods.

Step-by-step solution

a. Find the indirect utility function and the expenditure function

Recall from Example 5.1, the uncompensated demand functions are the Marshallian demand functions \(x(p_{x},p_{y},I)\) and \(y(p_{x},p_{y},I)\), which are given by: \\[x(p_{x}, p_{y}, I) = \frac{0.3I}{p_{x}}\\] \\[y(p_{x}, p_{y}, I) = \frac{0.7I}{p_{y}}\\] To find the indirect utility function, substitute the Marshallian demand functions into the utility function: \\[V(p_{x}, p_{y}, I) = U(x(p_{x}, p_{y}, I), y(p_{x}, p_{y}, I))\\] Now, to find the expenditure function, we solve for the minimal expenditure needed to achieve a certain utility level, \(U^{\ast}\), with respect to \(p_{x}\) and \(p_{y}\). To do this, we set \(U(x, y) = U^{\ast}\) and solve for the expenditure, \(E\): \\[E(p_{x}, p_{y}, U^{\ast}) = p_{x}x(p_{x}, p_{y}, U^{\ast}) + p_{y}y(p_{x}, p_{y}, U^{\ast})\\]

b. Find the compensated demand function for good x

Shephard's lemma states that the partial derivative of the expenditure function with respect to the price of a good gives the compensated demand (also known as the Hicksian demand) for that good. Therefore, we compute the derivative of \(E\) with respect to \(p_x\): \\[x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial E}{\partial p_{x}}\\]

c. Show that the Slutsky equation holds

The Slutsky equation states that the difference between the uncompensated and compensated demand functions is equal to the product of the income effect and the derivative of the uncompensated demand function for good x with respect to the income: \\[x(p_{x}, p_{y}, I) - x^c(p_{x}, p_{y}, U^{\ast}) = \frac{\partial x}{\partial I}(p_{x}, p_{y}, I) \cdot I\\] We have found the uncompensated demand function, \(x(p_{x}, p_{y}, I)\), and the compensated demand function, \(x^c(p_{x}, p_{y}, U^{\ast})\). We will now calculate the derivative of the uncompensated demand function for good x with respect to the income, multiply it by the income, and show that the Slutsky equation holds true for our given utility function.

Key concepts

Indirect Utility Function

The indirect utility function provides a way to look at how much utility a consumer can achieve given their income and the prices of goods. It is derived by substituting the Marshallian demand functions, also known as uncompensated demand functions, into the utility function.
To find the indirect utility function for our problem, we take the demand functions: \(x(p_{x}, p_{y}, I) = \frac{0.3I}{p_{x}}\) and \(y(p_{x}, p_{y}, I) = \frac{0.7I}{p_{y}}\), and plug them into the utility function \(U(x, y) = x^{0.3} y^{0.7}\).
This gives us the indirect utility function: \(V(p_{x}, p_{y}, I) = \left(\frac{0.3I}{p_{x}}\right)^{0.3} \left(\frac{0.7I}{p_{y}}\right)^{0.7}\).
This tells us how much utility a consumer can expect to derive based on their income \(I\) and the prices \(p_x\) and \(p_y\). This function is a crucial element when analyzing consumer behavior in different economic environments.

Expenditure Function

The expenditure function is crucial for understanding the minimum cost required to achieve a certain utility level given the prices of goods. This function takes the form \(E(p_{x}, p_{y}, U^{\ast})\), where \(U^{\ast}\) is a specific utility level.
To derive this function, we solve the equation \(U(x, y) = U^{\ast}\) for cost, leading to \(E = p_{x}x + p_{y}y\). By substituting compensated demand functions where they equate to \(U^{\ast}\), and setting \(x\) and \(y\) in terms of \(p_{x}\), \(p_{y}\), and \(U^{\ast}\), we can calculate the expenditure.
For the utility function \(U(x, y) = x^{0.3} y^{0.7}\), this function tells us how much income a consumer needs to achieve a specific level of happiness when prices shift.
In application, economists use it in various analyses, especially when considering scenarios like changes in policy or rising costs.

Compensated Demand

Compensated demand, or Hicksian demand, represents how much of a good a consumer will buy when compensated for price changes, such that utility levels remain constant. This is determined using Shephard's lemma.
In practice, the compensated demand function \(x^c(p_{x}, p_{y}, U^{\ast})\) is found by taking the derivative of the expenditure function \(E\) with respect to the price of the good in question.
This derivative gives us the rate at which expenditure must change to maintain a specific utility level when the price of a good changes, resulting in the compensated quantity demanded.
This concept plays a significant role in understanding consumer preference shifts devoid of income changes, making it pivotal in economic analyses related to price elasticity and welfare economics.

Uncompensated Demand

Uncompensated demand functions, often termed Marshallian demand functions, are a way of examining how much of a good a consumer will demand based purely on their income and prevailing prices, without any compensation for price changes.
In our case study, these functions are represented as \(x(p_{x}, p_{y}, I) = \frac{0.3I}{p_{x}}\) and \(y(p_{x}, p_{y}, I) = \frac{0.7I}{p_{y}}\).
These functions illustrate the direct effects of price and income levels on consumer behavior. Although they don't account for price changes directly, they offer insight into how consumer demand responds as if part of income is used to neutralize price variations.
Moreover, when juxtaposed with compensated demand, these functions help illustrate the income and substitution effects, which are integral when using tools like the Slutsky equation to break down and understand total effects on demand when prices fluctuate.