Problem 25 - Applied Calculus

Question

If \(x_{1}\) and \(x_{2}\) are the number of items of two goods bought, a customer's utility is $$U\left(x_{1}, x_{2}\right)=2 x_{1} x_{2}+3 x_{1}$$ The unit cost is \(\$ 1\) for the first good and \(\$ 3\) for the second. Use Lagrange multipliers to find the maximum value of \(U\) if the consumer's disposable income is \(\$ 100 .\) Estimate the new optimal utility if the consumer's disposable income increases by \(\$ 6\)

Step-by-Step Solution

Answer

Initial utility is 1820; new utility is 2035 with extra income.

Step 1: Identify the Constraint and Utility Function

The utility function is given by \( U(x_1, x_2) = 2x_1x_2 + 3x_1 \). The constraint, based on the budget, is the equation representing the consumer's spending, which is \( x_1 + 3x_2 = 100 \) for the initial income.

Step 2: Set Up the Lagrangian

Formulate the Lagrangian function: \[ \mathcal{L}(x_1, x_2, \lambda) = 2x_1x_2 + 3x_1 + \lambda (100 - x_1 - 3x_2) \] where \( \lambda \) is the Lagrange multiplier.

Step 3: Find Partial Derivatives

Find the partial derivatives of the Lagrangian:- With respect to \( x_1 \): \( \frac{\partial \mathcal{L}}{\partial x_1} = 2x_2 + 3 - \lambda = 0 \)- With respect to \( x_2 \): \( \frac{\partial \mathcal{L}}{\partial x_2} = 2x_1 - 3\lambda = 0 \)- With respect to \( \lambda \): \( \frac{\partial \mathcal{L}}{\partial \lambda} = 100 - x_1 - 3x_2 = 0 \) These equations will be solved simultaneously.

Step 4: Solve the System of Equations

Using the equations from Step 3:1. From \( 2x_2 + 3 - \lambda = 0 \), we get \( \lambda = 2x_2 + 3 \).2. From \( 2x_1 - 3\lambda = 0 \), plug-in \( \lambda \): \[ 2x_1 - 3(2x_2 + 3) = 0 \] which simplifies to \( 2x_1 = 6x_2 + 9 \).3. From \( 100 - x_1 - 3x_2 = 0 \), rearranging gives \( x_1 = 100 - 3x_2 \).Substitute \( x_1 = 100 - 3x_2 \) into \( 2x_1 = 6x_2 + 9 \):\[ 2(100 - 3x_2) = 6x_2 + 9 \]This leads to \( 200 - 6x_2 = 6x_2 + 9 \), which solves to \( x_2 = 15.91666... \), approximately 16.Substitute \( x_2 = 16 \) back into \( x_1 = 100 - 3x_2 \): \( x_1 = 100 - 48 = 52 \).

Step 5: Calculate Maximum Utility at Initial Income

Substitute \( x_1 = 52 \) and \( x_2 = 16 \) into the utility function:\[ U(52, 16) = 2(52)(16) + 3(52) = 1664 + 156 = 1820 \]Therefore, the maximum utility is 1820 at an income of 100.

Step 6: Adjust for Increased Income

If income increases by \$6, the budget constraint becomes \( x_1 + 3x_2 = 106 \).Repeat Steps 2 to 4 with the new constraint:1. From \( 100 - x_1 - 3x_2 = 0 \), \( x_1 = 106 - 3x_2 \).2. Substitute into \( 2x_1 = 6x_2 + 9 \):\[ 2(106 - 3x_2) = 6x_2 + 9 \]This equation simplifies to: \( 212 - 6x_2 = 6x_2 + 9 \), solving gives \( x_2 = 16.91666... \), approximately 17.3. Then \( x_1 = 106 - 51 = 55 \) when \( x_2 = 17 \).

Step 7: Calculate New Maximum Utility

Using \( x_1 = 55 \) and \( x_2 = 17 \), calculate the utility:\[ U(55, 17) = 2(55)(17) + 3(55) = 1870 + 165 = 2035 \] The estimated new utility is 2035.

Key Concepts

Utility Function

Understanding the concept of a utility function is essential in consumer theory within economics. A utility function represents a consumer's preference by mapping the subjective wellbeing or satisfaction derived from consuming various quantities of goods and services. Typically, these functions are used to predict consumer behavior regarding purchasing decisions.

In the given problem, the utility function is expressed as \( U(x_1, x_2) = 2x_1x_2 + 3x_1 \), where \( x_1 \) and \( x_2 \) denote quantities of two goods. This function suggests that the utility increases with the consumption of both goods, favoring a balance between \( x_1 \) and \( x_2 \) while giving more weight to the first good due to the additive term \( 3x_1 \).

Some important aspects of utility functions include:

• They help determine the level of satisfaction or happiness a consumer will experience from different combinations of goods.
• Utility functions are used to assess how likely consumers are to switch preferences when faced with changes in price or income.
• The equation form and coefficients determine how each good impacts the total utility.

In most economic models, the goal is to maximize this utility, considering constraints such as budget limits.

Budget Constraint

A budget constraint represents a financial limit on a consumer's expenditure, dictated by their income and the prices of goods they plan to buy. It's an equation that shows possible combinations of goods that a consumer can purchase without exceeding their income.

In this exercise, the initial budget constraint is \( x_1 + 3x_2 = 100 \). This constraint illustrates that the consumer has \\(100 to spend, where the cost of purchasing one unit of the first good is \\)1, and for the second good, it is \\(3 per unit. The constraint is adjusted to \( x_1 + 3x_2 = 106 \) when the income increases by \\)6.

Key points about budget constraints include:

• The slope of the budget line reflects the relative prices of the two goods. It illustrates the trade-off or opportunity cost of consuming more of one good over the other.
• Changes in income or prices will shift or rotate the budget constraint, affecting the consumer's choices and optimal utility level.
• Solving the budget constraint gives insight into the combination of goods that maximizes utility within the given income.

Thus, understanding and working with budget constraints are critical steps in optimizing consumer utility while adhering to financial limits.

Partial Derivatives

Partial derivatives are a vital mathematical concept used in multivariable calculus. They allow us to understand how functions change with respect to one variable while keeping other variables constant, which is especially useful in optimizing problems like utility maximization.

In the problem, partial derivatives are used to derive conditions for optimality from the Lagrangian function. The Lagrangian is expressed as \( \mathcal{L}(x_1, x_2, \lambda) = 2x_1x_2 + 3x_1 + \lambda (100 - x_1 - 3x_2) \). By taking partial derivatives with respect to \( x_1 \), \( x_2 \), and the Lagrange multiplier \( \lambda \), we set up conditions that identify the maximum utility given the budget constraint.

The partial derivatives in this problem are:

• \( \frac{\partial \mathcal{L}}{\partial x_1} \) involves seeing how the utility changes as we slightly adjust the amount of the first good while keeping \( x_2 \) constant.
• \( \frac{\partial \mathcal{L}}{\partial x_2} \) examines changes in utility from altering the quantity of the second good while \( x_1 \) is constant.
• \( \frac{\partial \mathcal{L}}{\partial \lambda} \) captures the adherence to the budget constraint, ensuring optimal solutions do not exceed the consumer's income.

This approach helps economists and students understand how utility is optimized under constraints and provides a structured path to solving similar problems efficiently.