Question

In Example 17.5 each individual has an initial endowment of 500 units of each good. a. Express the demand for Smith and Jones for goods Xand Fas functions of \(\mathrm{P}_{\text {xand }} \mathrm{Pj}\) and their initial endowments. b. Use the demand functions from part (a) together with the observation that total de mand for each good must be 1000 to calculate the equilibrium price ratio, \(P_{x} / P_{v}\) in this situation. What are the equilibrium consumption levels of each good by each person?

Short answer

Answer: In order to find the equilibrium consumption levels of goods X and Y for Smith and Jones, we need to first find the equilibrium price ratio (Px/Py) using the provided demand functions and the fact that the total demand for each good is 1000 units. Unfortunately, the specific demand functions, as well as the equilibrium price ratio, are not provided in the problem. Once these are known, we can plug the equilibrium price ratio back into the demand functions to find the equilibrium consumption levels for goods X and Y for each individual.

Step-by-step solution

Express the demand functions for Smith and Jones

Since we don't have any specific information regarding the preferences of these two individuals, we are going to assume that their demands follow the general form of a demand function. Let \(U_i\) represent the utility of individual \(i\) (Smith or Jones), where \(x_i\) and \(y_i\) are the amounts of goods X and Y demanded by that individual respectively. The demand function for Smith would look like: $$ D_{S_X}(P_X, P_Y, X_e, Y_e) = x_{S_X}(P_X, P_Y, X_e, Y_e) $$ $$ D_{S_Y}(P_X, P_Y, X_e, Y_e) = y_{S_Y}(P_X, P_Y, X_e, Y_e) $$ Similarly, the demand function for Jones would look like: $$ D_{J_X}(P_X, P_Y, X_e, Y_e) = x_{J_X}(P_X, P_Y, X_e, Y_e) $$ $$ D_{J_Y}(P_X, P_X, X_e, Y_e) = y_{J_Y}(P_X, P_Y, X_e, Y_e) $$

Distribute Endowments

Since each individual has an initial endowment of 500 units of each good, we can substitute these values into their demand functions: Smith: $$ D_{S_X}(P_X, P_Y, 500, 500) = x_{S_X}(P_X, P_Y, 500, 500) $$ $$ D_{S_Y}(P_X, P_Y, 500, 500) = y_{S_Y}(P_X, P_Y, 500, 500) $$ Jones: $$ D_{J_X}(P_X, P_Y, 500, 500) = x_{J_X}(P_X, P_Y, 500, 500) $$ $$ D_{J_Y}(P_X, P_Y, 500, 500) = y_{J_Y}(P_X, P_Y, 500, 500) $$

Calculate the Equilibrium Price Ratio

Since the total demand for each good must be 1000 units, we can write the following equations: 1. \(x_{S_X} + x_{J_X} = 1000\) 2. \(y_{S_Y} + y_{J_Y} = 1000\) Now, in order to solve for the equilibrium price ratio \(P_X/P_Y\), we need to express the demand functions in terms of the prices and endowments. Here, we will assume that the demands functions follow the general form: $$ x_i = f_i(P_X, P_Y, X_e, Y_e) $$ $$ y_i = g_i(P_X, P_Y, X_e, Y_e) $$ Using these general functions, we can rewrite the equations as: 1. \(f_S(P_X, P_Y, 500, 500) + f_J(P_X, P_Y, 500, 500) = 1000\) 2. \(g_S(P_X, P_Y, 500, 500) + g_J(P_X, P_Y, 500, 500) = 1000\) We can then use these equations to find the equilibrium price ratio, \(P_X/P_Y\). This step will require some algebra and problem-specific knowledge, which hasn't been provided in the exercise. However, once you find the equilibrium price ratio, you can use it to solve for the equilibrium consumption levels of goods X and Y for each individual.

Calculate the Equilibrium Consumption Levels

Once you have the equilibrium price ratio \(P_X/P_Y\), you can plug it back into the general demand functions: $$ x_{S_X} = f_S(P_X, P_Y, 500, 500) $$ $$ y_{S_Y} = g_S(P_X, P_Y, 500, 500) $$ $$ x_{J_X} = f_J(P_X, P_Y, 500, 500) $$ $$ y_{J_Y} = g_J(P_X, P_Y, 500, 500) $$ Solve these equations to find the equilibrium consumption levels of goods X and Y for Smith and Jones.

Key concepts

Demand Functions

Demand functions are fundamental to understanding consumer behavior in microeconomics. They express the relationship between the quantity of a good that consumers are willing to buy and several key variables, primarily the price of the good itself, the prices of related goods, consumer income, and preferences. In the context of the exercise, we consider the demand for goods X and F as functions of their prices \(P_X\) and \(P_Y\), as well as the initial endowments of each individual.

In simpler terms, a demand function can be represented as \(D(P_X, P_Y, W)\), where \(D\) is the quantity demanded, \(P_X\) is the price of good X, \(P_Y\) is the price of good Y, and \(W\) represents the wealth or endowment of the consumer. For Smith and Jones, this translates to how many units of goods X and Y they will demand based on the prices and their respective endowments of 500 units each. Understanding how to derive and manipulate these demand functions is crucial for students and academics alike to predict consumer behavior within the market. It’s important to note that demand curves typically slope downwards, indicating an inverse relationship between price and quantity demanded.

Equilibrium Price Ratio

The equilibrium price ratio is where the quantity of goods demanded equals the quantity supplied, at the intersection of the demand and supply curves. This ratio is pivotal in gauging the relative value of one good in terms of another in a barter economy. In our exercise, the equilibrium price ratio is \(P_X/P_Y\), signifying how many units of good Y are equivalent to one unit of good X in trade.

Finding the equilibrium price ratio involves setting up equations that equate total demand for goods X and Y to total supply, which is 1000 units. It is assumed that market forces will naturally move towards this point, stabilizing the price levels such that there is no excess demand or supply. For students grappling with this concept, it's essential to understand that finding this ratio necessitates using tools like algebra to solve the set equations, here, representing the combined demand of Smith and Jones for goods X and Y to their endowments. In practice, the equilibrium price ratio helps businesses in setting prices and making supply decisions in response to market conditions.

Utility Maximization

Utility maximization is the process by which consumers allocate their income to purchase a combination of goods and services that maximizes their overall satisfaction. It lies at the heart of consumer choice theory, assuming that consumers are rational and aim to get the most utility for their money. The challenge here involves balancing the marginal utility per dollar, which is the additional satisfaction received from spending an extra dollar on any good, with the consumer's budget constraint.

In the context of our exercise, Smith and Jones each strive to maximize their utility given their endowment of 500 units of each good and the prices of goods X and Y. The calculus behind this concept could be explored by employing techniques like Lagrangian multipliers, but the essential idea remains that utility maximization principles guide the demand functions. As consumers are faced with various choices, they compare the additional satisfaction from more of good X against that from good Y, choosing the combination that satisfies their preferences to the greatest extent within their budget constraint.

Endowment Distribution

Endowment distribution refers to the initial allocation of resources among individuals in an economy. In our exercise, Smith and Jones each start with an endowment of 500 units of goods X and Y. The way these goods are initially distributed will greatly affect their subsequent trading behavior and the market equilibrium. Endowments can take many forms including income, property, and other assets.

Students must recognize that changes in initial endowments can shift demand functions and potentially alter the equilibrium price ratio. This notion ties back to understanding the concept of income effect, where a change in a consumer’s wealth can lead to a change in the quantity of a good purchased. Considering how redistribution of endowments impacts consumer choices and market outcomes highlights the interconnectedness of economic principles and their practical implications.