Question

An economy has 2000 people. 1000 of them have utility functions U(x,y) = x+y and 1000 ofthem have utility functions U(x,y) = min{2x,y}. Everybody has an initial allocation of 1 unit of xand 1 unit of y. Find the competitive equilibrium prices and consumptions for each type of person

Answer 1

Step-by-step explanation:

To find the competitive equilibrium prices and consumptions for each type of person, we need to consider the utility functions and initial allocations of the individuals in the economy.

Let's break down the information given:

1. There are 2000 people in the economy, with 1000 people having the utility function U(x,y) = x+y and the other 1000 people having the utility function U(x,y) = min{2x,y}.

2. Everybody has an initial allocation of 1 unit of x and 1 unit of y.

To find the competitive equilibrium, we need to determine the prices at which supply equals demand for each type of person.

For the individuals with the utility function U(x,y) = x+y:

- The marginal rate of substitution (MRS) is equal to 1, which represents the rate at which they are willing to substitute one good for another while maintaining the same level of utility.

For the individuals with the utility function U(x,y) = min{2x,y}:

- The MRS is equal to 2, which represents the rate at which they are willing to substitute x for y.

In a competitive equilibrium, the prices of x and y will be such that the MRS for each individual is equal to the price ratio of the goods.

Let's assume the price of x is Px and the price of y is Py. Then, we can set up the following equations:

For individuals with U(x,y) = x+y:

MRS = Px/Py = 1

For individuals with U(x,y) = min{2x,y}:

MRS = Px/Py = 2

Since there are 1000 individuals of each type, the aggregate demand for x and y will be 1000 units each.

By solving the equations, we can find the competitive equilibrium prices:

Px/Py = 1 (from the first equation)

Px/Py = 2 (from the second equation)

From these equations, we can see that Px = 2Py.

To find the consumption for each type of person, we substitute the equilibrium prices into the utility functions.

For individuals with U(x,y) = x+y:

Since Px = 2Py, the optimal consumption is x = 2 and y = 1.

For individuals with U(x,y) = min{2x,y}:

Since Px = 2Py, the optimal consumption is x = 0.5 and y = 1.

Therefore, the competitive equilibrium prices are Px = 2 and Py = 1, and the consumptions for each type of person are as follows:

- Individuals with U(x,y) = x+y: x = 2 units and y = 1 unit

- Individuals with U(x,y) = min{2x,y}: x = 0.5 units and y = 1 unit

Answer 2

The competitive equilibrium prices for the two types of individuals in the economy are 2:1, with the price of x being twice the price of y. The consumptions for each type of person remain at 1 unit of x and 1 unit of y.

Step-by-step explanation:

To find the competitive equilibrium prices and consumptions for each type of person in this economy, we need to determine the equilibrium prices at which the demand and supply for the goods x and y equalize. Let's start by looking at the utility functions of the two types of people:

1. Type 1 individuals have the utility function U(x, y) = x + y.
2. Type 2 individuals have the utility function U(x, y) = min{2x, y}.

To find the competitive equilibrium prices, we need to determine the prices at which the marginal utility per unit of expenditure is equal for both types of individuals. This can be done by comparing the marginal utility per unit of expenditure for each type of person:

• For Type 1 individuals with the utility function U(x, y) = x + y, the marginal utility of x is 1 and the marginal utility of y is 1.
• For Type 2 individuals with the utility function U(x, y) = min{2x, y}, the marginal utility of x is 2 and the marginal utility of y is 1.

Since the marginal utility per unit of expenditure for Type 1 individuals is 1 and for Type 2 individuals is 2, the competitive equilibrium price ratio should be 2:1. This means that the price of x should be twice the price of y in order to equalize the marginal utility per unit of expenditure for both types of individuals. As for the consumptions, we can determine them by plugging in the equilibrium prices into the budget constraints of each type of person:

• For Type 1 individuals with the utility function U(x, y) = x + y, their initial allocation of 1 unit of x and 1 unit of y remains unchanged.
• For Type 2 individuals with the utility function U(x, y) = min{2x, y}, their initial allocation of 1 unit of x and 1 unit of y also remains unchanged.

Therefore, the competitive equilibrium prices are 2:1 (the price of x is twice the price of y) and the consumptions for each type of person remain at 1 unit of x and 1 unit of y.