Question

Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup \((y) .\) Smith's utility function is given by \\[ U_{s}=x^{0.3} y^{07} \\] whereas Jones" is given by \\[ U_{I}=x^{0.5} y^{0.5} \\] The individuals do not care whether they produce \(x\) or \(y\), and the production function for each good is given by \\[ x=2 l \text { and } y=3 l \\] where \(l\) is the total labor devoted to production of each good. a. What must the price ratio, \(p_{x} / p_{y}\) be? b. Given this price ratio, how much \(x\) and \(y\) will Smith and Jones demand? Hint: Set the wage equal to 1 here. c. How should labor be allocated between \(x\) and \(y\) to satisfy the demand calculated in part (b)?

Short answer

b. Calculate the quantity demanded for both ice cream (x) and chicken soup (y) for Smith and Jones. c. Find how labor should be allocated between the two goods to satisfy the demand from part (b). a. The price ratio of ice cream (x) to chicken soup (y) is 3/7. b. The quantity demanded for ice cream (x) is (3/7)(y_s + y_j) and for chicken soup (y) is y_s + y_j. c. Labor should be allocated as follows: 25/6 hours for producing ice cream (x) and 35/6 hours for producing chicken soup (y).

Step-by-step solution

Find the Marginal Rate of Substitution (MRS) for Both Individuals

Calculate the MRS for both individuals, as it reflects the rate at which they are willing to trade one good for another. Define the MRS as the ratio of the marginal utility (MU) for each good: - For Smith: \(MU_{x_s} = \frac{\partial U_s}{\partial x} = 0.3x^{-0.7}y^{0.7}\) \(MU_{y_s} = \frac{\partial U_s}{\partial y} = 0.7x^{0.3}y^{-0.3}\) \(MRS_s = \frac{MU_{x_s}}{MU_{y_s}} = \frac{0.3x^{-0.7}y^{0.7}}{0.7x^{0.3}y^{-0.3}} = \frac{0.3}{0.7}\frac{y}{x}\) - For Jones: \(MU_{x_j} = \frac{\partial U_j}{\partial x} = 0.5x^{-0.5}y^{0.5}\) \(MU_{y_j} = \frac{\partial U_j}{\partial y} = 0.5x^{0.5}y^{-0.5}\) \(MRS_j = \frac{MU_{x_j}}{MU_{y_j}} = \frac{0.5x^{-0.5}y^{0.5}}{0.5x^{0.5}y^{-0.5}} = \frac{y}{x}\)

Determine the Price Ratio of x and y

Since both individuals only care about their utility (not whether they produce x or y), the MRS for both must be equal to the price ratio, i.e., \(\frac{MRS_s}{MRS_j} = \frac{p_x}{p_y}\). \(\frac{\frac{0.3}{0.7}\frac{y}{x}}{\frac{y}{x}} = \frac{p_x}{p_y}\) Thus, the price ratio is: \(p_x / p_y = \frac{0.3}{0.7} = \frac{3}{7}\)

Find the Demand for x and y for Smith and Jones

To find the demands, we will use their utility maximization conditions: - For Smith: \(\frac{MU_{x_s}}{p_x} = \frac{MU_{y_s}}{p_y}\) \(\frac{0.3x^{-0.7}y^{0.7}}{3l} = \frac{0.7x^{0.3}y^{-0.3}}{7l}\) \(x_s = \frac{3}{7} y_s\) - For Jones: \(\frac{MU_{x_j}}{p_x} = \frac{MU_{y_j}}{p_y}\) \(\frac{0.5x^{-0.5}y^{0.5}}{3l} = \frac{0.5x^{0.5}y^{-0.5}}{7l}\) \(x_j = \frac{3}{7} y_j\) Summing up the demands by both individuals, we get: \(x = x_s + x_j = \frac{3}{7}(y_s + y_j) = \frac{3}{7}y\)

Allocate Labor to Satisfy Demand

Use the production functions for each good to find the labor allocations required to satisfy demand: \(x = 2l_x\) \(y = 3l_y\) Since we found that \(x = \frac{3}{7}y\), we can rewrite the relationship as: \(2l_x = \frac{3}{7} * 3l_y\) \(l_x = \frac{3}{7} * l_y\) Total labor available is 10 hours, so: \(l_x + l_y = 10\) Substituting the equation above, we get: \(\frac{3}{7} * l_y + l_y = 10\) \(l_y = \frac{35}{6}\) \(l_x = 10 - l_y = \frac{25}{6}\)

Summary

To summarize the exercise, we have found the following results: a. The price ratio, \(p_x / p_y\), must be \(\frac{3}{7}\). b. Smith and Jones will demand \(x = \frac{3}{7}(y_s + y_j)\) and \(y = y_s + y_j\). c. Labor should be allocated as follows: \(l_x = \frac{25}{6}\) hours for producing ice cream (x) and \(l_y = \frac{35}{6}\) hours for producing chicken soup (y).

Key concepts

Utility Maximization

Utility maximization is a fundamental concept in economics that explains how individuals choose among various options to achieve the highest level of satisfaction or 'utility'. When faced with limited resources, such as time or money, individuals allocate these resources to maximize their utility subject to those constraints.

In the context of the provided exercise, Smith and Jones are maximizing their utility by choosing the combination of ice cream (x) and chicken soup (y) that gives them the greatest satisfaction. This is calculated using their utility functions: for Smith, the utility function is given by \(U_{s}=x^{0.3} y^{0.7}\) and for Jones, it is \(U_{I}=x^{0.5} y^{0.5}\). The demand for products x and y, given the price ratio, results from their attempt to balance the utility received from each good with respect to its cost, ultimately determining the most utility-efficient way to spend their 10 hours of labor.

Price Ratio

Price ratio plays a critical role in an individual's decision-making process when faced with multiple goods or services to consume. It represents the relative price of one good in terms of another, which can influence the allocation of resources like labor or capital. The notion is central to consumer choice theory and allows economists to understand trade-offs that consumers face.

Within our exercise, the price ratio \(p_{x} / p_{y}\) is determined by the marginal rate of substitution, which equals the slope of the indifference curve at any point. This explains how much of one good a person would give up to obtain an additional unit of another, keeping utility constant. Since Smith and Jones are indifferent to producing x or y, their willingness to trade one for the other is captured by this price ratio, which is calculated to be \(3/7\). This ratio, in turn, influences how much x and y Smith and Jones will demand.

Labor Allocation

Labor allocation refers to the distribution of labor resources across different tasks or productions. It is an important concept in both personal decision-making and in the wider economy as it affects the overall output and efficiency of goods or services. Efficient labor allocation means that labor is used in such a way that it produces the maximum possible output.

In the scenario described in the exercise, Smith and Jones must decide how to allocate their 10 hours of labor between producing ice cream (x) and chicken soup (y) to satisfy demand while maximizing utility. The optimal allocation must take into account the productivity of labor in each task (given by the production functions), and the demand for x and y (dependent on the utility functions and price ratio). The exercise walks us through calculating that labor allocation mathematically to meet the determined demands effectively.

Production Functions

Production functions are mathematical equations that describe the relationship between inputs used in production and the resulting output. In the context of our exercise, the production function for each good, ice cream (x) and chicken soup (y), is given by \(x=2l\) and \(y=3l\) respectively, where \(l\) is the labor devoted to producing that good.

The production functions enable Smith and Jones to understand how much of each good can be produced given a certain amount of labor. This knowledge, combined with the utility functions and price ratio, informs how labor should be allocated to maximize utility. By matching the allocation of labor to the demands for x and y, as determined by the utility maximization problem, Smith and Jones can ensure that their limited labor resources are used in the most efficient way to produce goods.