Question

There are two firms (1 & 2) which use a quantity x of a polluting chemical. The total benefit that each firm derives from the use of this compound (excluding the purchasing costs for the chemical) is given by:

₁ = 50x₁ − 0.25x₁² and ₂ = 100x₂ − x₂²
The supply curve for the chemical is:
p = 10+0.75x
where x refers to the total quantity supplied (x = x₁ + x₂), and p is the price (marginal cost of production of the supplier).
Please solve the following tasks both graphically and mathematically. a) Find the aggregate demand curve for the good, and plot it on an Excel chart. Hint: recall the relationship between demand curves (price ~ quantity) and benefit/utility functions
b) Solve for the chemical’s price and aggregate quantity demanded. How is this quantity distributed among the firms?
Now suppose that the EPA levies a tax of $20 on the supply of each unit of the input.
c) Solve for the new price and total quantity. How is this quantity distributed among firms?
d) What is the change in producer and consumer surplus? How much revenue does the EPA receive in taxes? What is the deadweight loss?

Answer 1

a) To find the aggregate demand curve for the given chemical, we need to determine the quantity demanded at different price levels. By equating the benefit/utility functions of the two firms to the price and substituting the supply curve equation, we can solve for the aggregate quantity demanded and plot it on an Excel chart.

b) The chemical's price and aggregate quantity demanded can be found by determining the equilibrium point where the aggregate demand curve intersects the supply curve. The quantity is distributed among the firms based on their market shares and the price elasticity of demand.

c) When the EPA levies a tax on the supply of each unit of the input, the supply curve shifts upward, and the new price and total quantity can be determined by finding the new equilibrium point.

d) The change in producer and consumer surplus can be calculated by comparing the surplus before and after the tax. The tax revenue received by the EPA is the product of the tax rate and the total quantity traded. Deadweight loss represents the loss of economic efficiency due to market distortions like taxes.

Step-by-step explanation:

a) Aggregate Demand Curve:

The aggregate demand curve represents the total quantity of goods and services demanded by all buyers in an economy at different price levels. It is derived from the individual demand curves of consumers and firms. The aggregate demand curve slopes downward due to the following factors:

1. Wealth Effect: As prices decrease, the real value of wealth increases, leading to higher consumption and demand.
2. Interest Rate Effect: Lower prices reduce the demand for money, leading to lower interest rates and increased investment and consumption.
3. International Trade Effect: Lower prices make domestic goods relatively cheaper compared to foreign goods, increasing exports and decreasing imports.

To find the aggregate demand curve for the given chemical, we need to determine the quantity demanded at different price levels. Using the benefit/utility functions of the two firms, we can equate their benefits to the price:

For Firm 1: