Question

Consider an economy with three individuals. Individual 1 has inverse demand function for a public good given as P1=50-2Q1, Individual 2 has inverse demand function for the public good given as P2=80-5Q2 and Individual 3 has inverse demand function for the public good given as P3=100-4Q3 The prices are measured in dollars per unit. Suppose the marginal cost of producing the public good is $30 per unit.

a) Determine the efficient level of the public good. b) Determine the Lindhal price of the public for each individual. c) Who pays more for the public good?

Answer 1

Explanation:To determine the optimal level of public good provision in this economy, we need to find the quantity of the public good demanded by each individual and then aggregate their demands. The optimal level of provision occurs when the aggregate quantity demanded equals the quantity supplied at the given marginal cost.

Let's find the quantity demanded by each individual first:

Individual 1's inverse demand function: P1 = 50 - 2Q1

Solving for Q1: 2Q1 = 50 - P1

Q1 = (50 - P1) / 2

Individual 2's inverse demand function: P2 = 80 - 5Q2

Solving for Q2: 5Q2 = 80 - P2

Q2 = (80 - P2) / 5

Individual 3's inverse demand function: P3 = 100 - 4Q3

Solving for Q3: 4Q3 = 100 - P3

Q3 = (100 - P3) / 4

Now, let's find the aggregate demand by summing up the quantities demanded by each individual:

Aggregate demand: Q = Q1 + Q2 + Q3

Substituting the expressions for Q1, Q2, and Q3:

Q = (50 - P1) / 2 + (80 - P2) / 5 + (100 - P3) / 4

To find the optimal level of provision, we need to set the aggregate quantity demanded equal to the quantity supplied at the given marginal cost:

Q = Qs

Given that the marginal cost of producing the public good is $30 per unit, we have:

Q = 30

Now, we can solve for the equilibrium quantities and prices.

Substitute Q = 30 into the aggregate demand equation:

30 = (50 - P1) / 2 + (80 - P2) / 5 + (100 - P3) / 4

Substitute the inverse demand functions into the aggregate demand equation:

30 = (50 - P1) / 2 + (80 - P2) / 5 + (100 - P3) / 4

= (50 - (50 - P1)) / 2 + (80 - (80 - 5Q2)) / 5 + (100 - (100 - 4Q3)) / 4

= P1 / 2 + (5Q2) / 5 + (4Q3) / 4

Simplify the equation:

30 = P1 / 2 + Q2 + Q3

Now, we have a system of equations:

P1 / 2 + Q2 + Q3 = 30 (Equation 1)

P1 = 50 - 2Q1 (Equation 2)

P2 = 80 - 5Q2 (Equation 3)

P3 = 100 - 4Q3 (Equation 4)

To solve this system, we need to substitute equations 2, 3, and 4 into equation 1 and solve for Q2 and Q3. Then we can find Q1 and the corresponding prices P1, P2, and P3.

However, it seems that you have not provided any information about the quantities Q2 and Q3 or their relationship to Q1. Without this information, it is not possible to determine the specific quantities demanded by each individual and the equilibrium prices.