Question

Two of the simplest utility functions are: 1\. Fixed proportions: \(U(x, y)=\min [x, y]\) 2\. Perfect substitutes: \(U(x, y)=x+y\) a. For each of these utility functions, compute the following: \(\bullet\) Demand functions for \(x\) and \(y\) \(\bullet\) Indirect utility function \(\bullet\) Expenditure function b. Discuss the particular forms of these functions you calculated-why do they take the specific forms they do?

Short answer

Short answer: For a fixed proportions utility function, the demand functions for goods x and y are equal, indicating a constant proportion between the two goods: \(x = \frac{I}{p_x + p_y}\) and \(y = \frac{I}{p_x + p_y}\). The indirect utility function is \(V(p_x, p_y, I) = \frac{I}{p_x + p_y}\), and the expenditure function is \(E(p_x, p_y, u) = u(p_x + p_y)\). For a perfect substitutes utility function, the consumer only consumes the good with a lower price per unit, resulting in the following demand functions: \(x = \frac{I}{p_x}\) if \(p_x \leq p_y\) and \(0\) otherwise, and \(y = \frac{I}{p_y}\) if \(p_y \leq p_x\) and \(0\) otherwise. The indirect utility function for this case is \(V(p_x, p_y, I) = \max\left(\frac{I}{p_x}, \frac{I}{p_y}\right)\), and the expenditure function is \(E(p_x, p_y, u) = u \min(p_x, p_y)\). The specific forms of these functions result from the particular properties of their respective utility functions and the optimization of utility under budget constraints.

Step-by-step solution

1. Fixed proportions utility function calculations:

Let's start with the fixed proportions utility function (\(U(x, y) = \min[x, y]\)). To find the demand functions for goods \(x\) and \(y\), we need to consider the budget constraint, which is given by \(p_x x + p_y y = I\), where \(p_x\) and \(p_y\) are the prices of goods \(x\) and \(y\) respectively, and \(I\) is the consumer's income. The following steps will help us compute the demand functions for \(x\) and \(y\):

1.1 Optimal choice of goods x and y:

To achieve the highest utility, the consumer will expend all their income on both goods, such that \(x = y\). Thus, \(p_x x + p_y x = I \Rightarrow x (p_x + p_y) = I\) Now, we can find the demand function for good \(x\) as: \(x = \frac{I}{p_x + p_y}\). Similarly, the demand function for good \(y\) is given by: \(y = \frac{I}{p_x + p_y}\).

1.2 Indirect utility function:

Using the demand functions, we can compute the indirect utility function by replacing \(x\) and \(y\) in the utility function with their demand functions: \(V(p_x, p_y, I) = \min \left[\frac{I}{p_x + p_y}, \frac{I}{p_x + p_y}\right] = \frac{I}{p_x + p_y}\).

1.3 Expenditure function:

The expenditure function \(E(p_x, p_y, u)\) is the minimal cost needed to achieve a specific utility level \(u\). We can use the demand function to find this by solving for \(I\) using the indirect utility function: \(I = u(p_x + p_y)\) Substitute this back into the demand functions for goods \(x\) and \(y\), and we get: \(E(p_x, p_y, u) = up_x\frac{u}{p_x + p_y} + up_y\frac{u}{p_x + p_y} = u(p_x + p_y)\).

2. Perfect substitutes utility function calculations:

Next, let's move on to the perfect substitutes utility function (\(U(x, y) = x + y\)). We'll follow the same steps as earlier:

2.1 Optimal choice of goods x and y:

For this utility function, we observe that increasing the quantity of either \(x\) or \(y\) while maintaining the budget constraint will increase utility. Hence, the consumer will only consume the good with the lower price per unit. In case of equal prices, they can consumer any combination of the two goods. Therefore, the demand functions for goods \(x\) and \(y\) are: \(x = \frac{I}{p_x}\) if \(p_x \leq p_y\) and \(0\) otherwise \(y = \frac{I}{p_y}\) if \(p_y \leq p_x\) and \(0\) otherwise

2.2 Indirect utility function:

By plugging the demand functions back into the utility function, we can compute the indirect utility function: \(V(p_x, p_y, I) = \max\left(\frac{I}{p_x}, \frac{I}{p_y}\right)\).

2.3 Expenditure function:

Finally, we can find the expenditure function \(E(p_x, p_y, u)\) by solving for \(I\) in the indirect utility function: \(I = u \min(p_x, p_y)\) Hence, the expenditure function is given by: \(E(p_x, p_y, u) = u \min(p_x, p_y)\).

Discussion:

In both cases, the specific forms of the demand, indirect utility, and expenditure functions are derived from the properties of the utility functions. For fixed proportions, since the utility is based on the minimum of \(x\) and \(y\), the demand functions are equal, indicating a constant proportion between the two goods. In contrast, for perfect substitutes, the consumer is only concerned with the total number of goods, so they will only consume the good with a lower price per unit. This is reflected in the demand functions for perfect substitutes utility function. Overall, the forms of these functions are a direct consequence of the properties of each utility function and their subsequent optimization and budget constraints.

Key concepts

Fixed Proportions Utility

In the fixed proportions utility model, consumers derive satisfaction based on the minimum quantity of two goods, represented as \(U(x, y) = \min[x, y]\). This concept models preferences where goods are usually consumed together, like left and right shoes. Here, the consumption ratios are fixed; if one good is in short supply, it limits overall utility.
To determine the demand functions for these goods under a fixed budget, consumers adjust spending so both goods are consumed equally. This balance is reflected in demand functions \(x = \frac{I}{p_x + p_y}\) and \(y = \frac{I}{p_x + p_y}\). Both goods have the same calculated demand, showing a fixed ratio in purchases.

• The indirect utility function, \(V(p_x, p_y, I) = \frac{I}{p_x + p_y}\), indicates the highest utility possible given prices and income.
• The expenditure function, \(E(p_x, p_y, u) = u(p_x + p_y)\), computes the minimum cost to achieve a specific utility level, emphasizing the constant proportion.

Perfect Substitutes Utility

The perfect substitutes utility function, denoted as \(U(x, y) = x + y\), models goods that can be replaced by each other without any loss in satisfaction, such as coffee from different brands. Consumers choose the cheaper good, maximizing quantity through cost-efficiency rather than the mix of goods.
In this model, if \( p_x \leq p_y \), the consumer will choose entirely \(x\) and ignore \(y\), making the demand function \(x = \frac{I}{p_x}\) and \(y = 0\). Conversely, if \(p_y \leq p_x\), \(y\) is chosen entirely instead.

• The indirect utility function created is \(V(p_x, p_y, I) = \max\left(\frac{I}{p_x}, \frac{I}{p_y}\right)\), showing the optimal satisfaction achievable by choosing one good over the other based on lower prices.
• The expenditure function simplifies to \(E(p_x, p_y, u) = u \min(p_x, p_y)\), indicating minimum spending required for a desired utility.

Demand Functions

Demand functions articulate how consumers opt for quantities of two goods under given prices and income. For each utility type, calculating the demand function involves optimizing satisfaction given these constraints.
In fixed proportions, demand reflects a need for balanced consumption, leading to equal expressions, \(x = \frac{I}{p_x + p_y}\) and \(y = \frac{I}{p_x + p_y}\). On the other hand, perfect substitutes emphasize picking the cheaper option, which reveals different paths based on relative pricing:

• If \(p_x \leq p_y\), the demand is \(x = \frac{I}{p_x}\) and \(y = 0\).
• If \(p_y \leq p_x\), \(y = \frac{I}{p_y}\) and \(x = 0\).

The demand function is a fundamental tool in understanding how consumers decide between goods given financial limitations.

Indirect Utility Function

The indirect utility function relates optimal utility to prices and income, representing the highest satisfaction a consumer can obtain considering budget constraints. For fixed proportions, it is calculated as \(V(p_x, p_y, I) = \frac{I}{p_x + p_y}\). This conveys that utility is directly tied to how well-rounded the consumption can be, given equal importance to both goods.
With perfect substitutes, the function becomes \(V(p_x, p_y, I) = \max\left(\frac{I}{p_x}, \frac{I}{p_y}\right)\), which reflects that the highest utility is from focusing on the more cost-effective item, emphasizing saving money.

• In every scenario, the indirect utility function reveals the strategic approach a consumer will use to maximize their satisfaction within budget limits.

Expenditure Function

Expenditure functions determine the minimum spending necessary for a consumer to reach a specific utility level. Understanding these functions helps in discerning consumer behavior under different utility frameworks.
For fixed proportions, the expenditure function \(E(p_x, p_y, u) = u(p_x + p_y)\) signifies a balanced purchase demand fitting within the utility goal. Conversely, in perfect substitutes, \(E(p_x, p_y, u) = u \min(p_x, p_y)\) portrays spending optimization by focusing on the cheaper good.

• This demonstrates consumers stretch their budgets as much as possible while maintaining desired utility, managing their funds based on the nature of their individual preferences.