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Corrections Maximum Price for Gasoline

Problem 1.

Part A: A firm with "market power" is operating two plants and selling its product in two markets. The demand and cost configurations it faces are:

Demand: Costs:
Market 1: P = 484 - 20Q1 Plant 1: TC1 = 961 - 10 Q1 + Q12

Market 2: P = 484 - 5Q2 Plant 2: TC2 = 324 - 10 Q2 +3Q22

On the demand side, Q1 represents the amounts sold in market 1 and Q2 represents the amounts sold in market 2. On the cost side. TC1 represents the total cost of producing Q1 level of output in plant 1 and TC2 represents the total cost of producing Q2 level of output in plant 2.

Given the demand and cost figures above,

1. How much will this firm sell in each of the two markets?
2. What will the product price be in each market?
3. How much will be produced in each plant?
4. How much of the firm's profit can be attributed to each plant?

Hint: The output from both plants can be sold in either market.

Part B: After the firm learns it is to be regulated to make a profit equal only to a normal rate of return, it decides to consolidate production into a single plant (but it will still sell in both markets). After consolidation, total cost is represented by the equation:

TC = 8940 - 10Q + 0.75Q2,

where Q represents the total output produced in this single (consildated) plant. Market demand is as described in Part A of this problem. Given this information,

1. How much will this firm sell in each of the two markets?
2. What will the product price be in each market?
3. Are consumers better or worse off before or after the regulation? If so, by how much?

Problem 2. Suppose the utility function of the average consumer of gasoline is:

U = X0.15Y0.75

where X represents the number of gallons of gasoline purchased by this person per week and Y represents the number of "units" of all other goods consumed by this person per week. If this person's weekly income is $300, then:

1. What is the per gallon price of gasoline if this person purchases $50 worth of gas per week?

2. If a 50 per cent per gallon tax is imposed on gasoline, what would be this person's weekly, after tax consumption of gasoline?

3. How much would this person have to be reimbursed in order to make him/her feel as well off as s/he did prior to imposition of the 50 per cent per gallon tax?

4. If a 50 per cent per gallon tax is imposed on gasoline but the full amount of the tax collected (on all gallons purchased) is rebated as a cash payment to this person, what would be this person's weekly consumption of gasoline after (s)he receives the rebate?

5. What is the maximum amount this person would be willing to pay to avoid the 50 per cent per gallon tax on gasoline?

6. If a 50 per cent per gallon tax is imposed on gasoline but the full amount of the tax is rebated in the form of coupons that can be used to purchase anything but gasoline, what would be this person's daily consumption of gasoline after (s)he receives the rebate?

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Solution Preview

1.How much will this firm sell in each of the two markets?

Since the output from both plants can be sold in either market, the Qi on the demand side is different from Qi level of output in plant i, we define that:
In market 1, P = 484 - 20Qa; and in Market 2: P = 484 - 5Qb,
while the labels on the supply side doesn't change.

In market 1, P = 484 - 20Qa
Total Revenue 1 = P1*Qa = Qa (484 - 20Qa) = 484 Qa - 20Qa2
MR1 = dTR1 / dQa = 484- 40Qa

In market 2, P = 484 - 5Qb
Total Revenue 2 = P2*Qb = Qa (484 - 5Qb) = 484 Qb - 5Qb2
MR2 = dTR2 / dQb = 484- 10Qb

Since the firm is to maximizing its profit, it will sell the product to the two markets at a level where MR1 = MR2. Because otherwise, the firm will sell more goods to the market with higher marginal revenue.
So we set: MR1 = MR2.
484- 40Qa= 484- 10Qb
Qb = 4Qa
So the total quantity demanded is Qa + Qb = Qa + 4Qa = 5Qa

In plant 1, TC1 = 961 - 10 Q1 + Q12
Marginal cost is MC1 = 2Q1-10

Plant 2: TC2 = 324 - 10 Q2 +3Q22
Marginal cost is MC2 = 6Q2-10

Since the firm is to minimizing its cost, it will produce at a level where MC1 = MC2. Because otherwise, the firm will produce more goods in the plant with lower marginal cost.
So we set: MC1 = MC2.
2Q1-10=6Q2-10
Q1 = 3Q2
So the total quantity demanded is Q1 +Q2=3Q2 + Q2 = 4Q2

Since in the firm the total output = total sales volume
Then 5Qa = 4Q2
So Q2 = 5/4 Qa
And Q1 = 3Q2 = 15/4 Qa

Now the firm's total profit form is:
Profit = TR1 +TR2 - TC1 - TC2
= 484 Qa - 20Qa2 + 484 Qb- 5Qb2 - (961 - 10 Q1 + Q12) - (324 - 10 Q2 +3Q22 )
= 484 Qa - 20Qa2 + 484 (4Qa) - 5(4Qa) 2 - (961 - 10 (15/4 Qa) + (15/4 Qa) 2) - (324 - 10 (5/4 Qa) +3(5/4 Qa) 2 )
= 484 Qa - 20Qa2 + 1936 Qa - 20Qa 2 - 961 + 75/2 Qa - 125/16 Qa 2 -324 + 25/2 Qa +75/16 Qa 2
= -105/2 Qa 2 + 2470Qa -1285

Then the marginal profit w.r.t. Qa is dProfit / dQa = -105Qa + 2470 = 0
Solve for Qa = 23.52, which should be sold in market 1;
Then Qb = ...

Solution Summary

The expert calculates the corrections for the maximum price of gasoline.

$2.19