MICRO-Suppose Laundrette is an industrial laundry

Suppose Laundrette is an industrial laundry service that emits smoke during its operation. The smoke affects a nearby garden centre, named Gardenia. Both firms are local monopolies. The daily demand curve of Laundrette is given by 𝑝1 = 90 − 𝑞1, where 𝑞1 is the number of linens washed. Total daily smoke emitted by Laundrette is given by 𝑧 = 𝑞1/4 . The marginal cost of production forLaundrette is 10 (i.e., 𝑀𝐶1 = 10).Gardenia’s daily market demand curve is 𝑝2 = 82 − 𝑞2, where 𝑞2 is the number of flower plants sold. The marginal cost of production of Gardenia, which is affected by the smoke coming from Laundrette, is given by 𝑀𝐶2 = 10 + 𝑧. a. Derive two firms (privately optimal) outputs, profits and emission, assuming there is no regulation over emission. Explain your answer. [15 marks] b. Derive the Pareto optimal outputs, profits and emission. [15 marks] c. Explain how the Pareto optimal outcome can be achieved. Where applicable you may work out appropriate numbers.

Answer

Laundrette's privately optimal output is 40 linens, with a profit of

1,200 and emissions of 10 units. Gardenia's privately optimal output is 36 flower plants, with a profit of

1,296 and emissions of 10 units. The Pareto optimal outputs are 20 linens for Laundrette and 46 flower plants for Gardenia, with a profit of

800 for Laundrette and

1,692 for Gardenia, and emissions of 5 units. The Pareto optimal outcome can be achieved through negotiation or regulation that internalizes the externality.

Solution

a

To find Laundrette's privately optimal output, we set marginal revenue (MR1) equal to marginal cost (MC1). For Gardenia, we account for the externality cost by adding the smoke damage to its MC

b

The Pareto optimal output is found by maximizing the total surplus, considering the externality. We set the marginal social cost (MSC) of Laundrette equal to the marginal social benefit (MSB), which is the demand curve of Gardenia

c

The Pareto optimal outcome can be achieved through a Pigouvian tax, subsidies, or negotiation between the two firms, leading to an internalization of the externality

Key Concept

Private Optimum vs Pareto Optimum

Explanation

The private optimum does not account for externalities, while the Pareto optimum does, leading to a socially efficient outcome.

Key Concept

Marginal Social Cost (MSC)

Explanation

MSC includes both the private costs of production and the external costs imposed on others.

Key Concept

Internalizing Externalities

Explanation

Internalizing externalities involves adjusting incentives to reflect the true social cost or benefit of production or consumption.

Suppose Laundrette is an industrial laundry service that emits smoke during its operation. The smoke affects a nearby garden centre, named Gardenia. Both firms are local monopolies. The daily demand curve of Laundrette is given by 𝑝1 = 90 − 𝑞1, where 𝑞1 is the number of linens washed. Total daily smoke emitted by Laundrette is given by 𝑧 = 𝑞1/4 . The marginal cost of production forLaundrette is 10 (i.e., 𝑀𝐶1 = 10).Gardenia’s daily market demand curve is 𝑝2 = 82 − 𝑞2, where 𝑞2 is the number of flower plants sold. The marginal cost of production of Gardenia, which is affected by the smoke coming from Laundrette, is given by 𝑀𝐶2 = 10 + 𝑧. a. Derive two firms (privately optimal) outputs, profits and emission, assuming there is no regulation over emission. Explain your answer. [15 marks

Answer

Laundrette's privately optimal output is 40 linens, with a profit of

1,600 and emissions of 10 units of smoke. Gardenia's privately optimal output is 36 flower plants, with a profit of

1,296 and additional costs due to emissions.

Solution

a

Laundrette's Profit Maximization: To find the privately optimal output for Laundrette, we set marginal revenue (MR1) equal to marginal cost (MC1). The inverse demand function is

p_1 = 90 - q_1

. To find MR1, we take the derivative of total revenue (TR1), which is

p_1 \times q_1

. MR1 is then

90 - 2q_1

. Setting MR1 equal to MC1, we get

90 - 2q_1 = 10

. Solving for

q_1

, we find that Laundrette's optimal output is 40 linens

b

Laundrette's Profit Calculation: Profit for Laundrette is total revenue minus total cost (TR1 - TC1). With

q_1 = 40

, TR1 is

90 \times 40 - 40^2

, and TC1 is

10 \times 40

. The profit is then $1,600

c

Laundrette's Emission Calculation: Emissions for Laundrette,

z

, are given by

q_1/4

. With

q_1 = 40

, emissions are 10 units of smoke

d

Gardenia's Profit Maximization: Gardenia's profit maximization requires setting its marginal revenue (MR2) equal to its marginal cost (MC2), which includes the cost of emissions from Laundrette. The inverse demand function for Gardenia is

p_2 = 82 - q_2

. MR2 is then

82 - 2q_2

. Since MC2 is

10 + z

, and

z = 10

from Laundrette's emissions, MC2 is

20

. Setting MR2 equal to MC2, we get

82 - 2q_2 = 20

. Solving for

q_2

, we find that Gardenia's optimal output is 36 flower plants

e

Gardenia's Profit Calculation: Profit for Gardenia is total revenue minus total cost (TR2 - TC2). With

q_2 = 36

, TR2 is

82 \times 36 - 36^2

, and TC2 is

(10 + 10) \times 36

. The profit is then $1,296

Key Concept

Profit Maximization in the Presence of Externalities

Explanation

In the absence of regulation, each monopoly maximizes profit without considering the external costs imposed on the other firm. Laundrette does not account for the smoke's impact on Gardenia's costs, leading to a suboptimal allocation of resources from a societal perspective.

Suppose Laundrette is an industrial laundry service that emits smoke during its operation. The smoke affects a nearby garden centre, named Gardenia. Both firms are local monopolies. The daily demand curve of Laundrette is given by 𝑝1 = 90 − 𝑞1, where 𝑞1 is the number of linens washed. Total daily smoke emitted by Laundrette is given by 𝑧 = 𝑞1/4 . The marginal cost of production forLaundrette is 10 (i.e., 𝑀𝐶1 = 10).Gardenia’s daily market demand curve is 𝑝2 = 82 − 𝑞2, where 𝑞2 is the number of flower plants sold. The marginal cost of production of Gardenia, which is affected by the smoke coming from Laundrette, is given by 𝑀𝐶2 = 10 + 𝑧. question: Derive the Pareto optimal outputs, profits and emission.

Answer

Pareto optimal outputs for Laundrette and Gardenia are 64 and 58 units respectively, Pareto optimal profits for Laundrette and Gardenia are

1,536 and

1,452 respectively, and Pareto optimal emission is 16 units.

Solution

a

Pareto Efficiency: Pareto efficiency occurs when no individual or preference criterion can be better off without making at least one individual or preference criterion worse off

b

Social Optimum: To find the Pareto optimal outputs, we need to maximize the total surplus (sum of consumer surplus and producer surplus for both firms) minus the damage from the smoke

c

Total Surplus Maximization: The total surplus is maximized when the marginal social benefit equals the marginal social cost

d

Marginal Social Cost (MSC) for Laundrette: The MSC for Laundrette is the sum of its marginal cost and the marginal damage to Gardenia, which is

MC_1 + \frac{d(MC_2)}{d(q_1)}

e

Marginal Social Cost (MSC) for Gardenia: The MSC for Gardenia is simply its marginal cost,

MC_2

f

Setting up the Optimization Problem: To find the Pareto optimal outputs, we set the marginal revenue (MR) of each firm equal to its MSC

g

Calculating Marginal Revenue (MR) for Laundrette:

MR_1 = \frac{d(TR_1)}{d(q_1)} = \frac{d((90 - q_1)q_1)}{d(q_1)} = 90 - 2q_1

h

Calculating Marginal Revenue (MR) for Gardenia:

MR_2 = \frac{d(TR_2)}{d(q_2)} = \frac{d((82 - q_2)q_2)}{d(q_2)} = 82 - 2q_2

i

Equating MR and MSC for Laundrette:

90 - 2q_1 = 10 + \frac{d(MC_2)}{d(q_1)} = 10 + \frac{1}{4}

j

Equating MR and MSC for Gardenia:

82 - 2q_2 = 10 + \frac{q_1}{4}

k

Solving the Equations: Solving the system of equations from steps i and j gives us the Pareto optimal outputs for Laundrette and Gardenia

l

Pareto Optimal Outputs: Solving the equations, we find

q_1 = 64

and

q_2 = 58

m

Pareto Optimal Emissions: The Pareto optimal emission level is

z = \frac{q_1}{4} = 16

n

Pareto Optimal Profits: The Pareto optimal profits for Laundrette and Gardenia are calculated by subtracting total costs from total revenues using the Pareto optimal outputs

o

Profit Calculation for Laundrette:

\pi_1 = (90 - 64) \times 64 - 10 \times 64 = 1,536

p

Profit Calculation for Gardenia:

\pi_2 = (82 - 58) \times 58 - (10 + 16) \times 58 = 1,452

Key Concept

Pareto Efficiency and Social Optimum

Explanation

Pareto efficiency is achieved when the total surplus is maximized, considering both firms' profits and the externalities involved. The social optimum is found by equating marginal revenues to marginal social costs for both firms.

Alice and Bill live in an exchange economy with two goods, 𝑥1 and 𝑥2. Aliceowns 50 of both goods, and Bill owns 250 of both goods. Alice’s utility functionis 𝑢𝐴(𝑥1, 𝑥2) = 𝑥2 + 50𝑙𝑛𝑥1 and Bill’s utility function is 𝑢𝐵(𝑥1, 𝑥2) = 𝑥2 +100𝑙𝑛𝑥1 a. Derive the contract curve and illustrate it in an Edgeworth box. Explain yourwork. [20 marks] b. Determine the competitive equilibrium price and consumptions of the twogoods. Explain the differences, if any, in the consumption levels of the twoagents. Graphically illustrate your answer. [25 marks] c. Suppose the government transfers 100 units of Bill’s good 1 endowment toAlice. How is your answer to (b) is affected? Explain in words, and graphicallyillustrate your answer. [25 marks] d. How would a sufficiently large redistribution of the good 1 and good 2endowments alter your answer to (b)? Graphically illustrate your answer.Explain the intuition behind your answer.[30 marks

Answer

The contract curve is derived from the tangency of indifference curves, the competitive equilibrium price and consumptions are determined by the equality of the marginal rate of substitution and the price ratio, and government transfers and redistribution of endowments affect the equilibrium by altering the budget constraints and potentially leading to a new equilibrium with different consumption levels.

Solution

a

To derive the contract curve, we need to find the allocations where the indifference curves of Alice and Bill are tangent to each other, which implies that their marginal rates of substitution (MRS) are equal. For Alice,

MRS_{A} = \frac{dU_{A}/dx_{1}}{dU_{A}/dx_{2}} = \frac{50/x_{1}}{1}

and for Bill,

MRS_{B} = \frac{dU_{B}/dx_{1}}{dU_{B}/dx_{2}} = \frac{100/x_{1}}{1}

. Setting

MRS_{A} = MRS_{B}

gives us the contract curve equation

b

The competitive equilibrium price and consumptions are determined by setting the MRS of each individual equal to the price ratio

p_{1}/p_{2}

. We solve for the price ratio and the consumption bundles

(x_{1A}, x_{2A})

and

(x_{1B}, x_{2B})

that satisfy both individuals' budget constraints and the market clearing conditions

c

A government transfer of 100 units of good 1 from Bill to Alice changes their endowments and thus their budget constraints. This will shift the budget line for Alice to the right and for Bill to the left in the Edgeworth box, leading to a new competitive equilibrium with different consumption levels

d

A sufficiently large redistribution of goods will alter the initial endowments significantly, which can lead to a new competitive equilibrium. The new equilibrium will depend on the new endowments and how they affect the budget constraints and the MRS of each individual. The intuition is that redistributing goods can improve the utility of one individual without harming the other if the redistribution moves the economy closer to the contract curve

Key Concept

Contract Curve

Explanation

The contract curve represents all the allocations where the indifference curves of the two individuals are tangent to each other, indicating that there is no further possible mutually beneficial trade.

Key Concept

Competitive Equilibrium

Explanation

Competitive equilibrium occurs when the market clears, meaning that the quantity demanded equals the quantity supplied for both goods, and the MRS of each individual equals the price ratio, leading to an optimal allocation of resources.

Key Concept

Government Transfers

Explanation

Government transfers alter the endowments and budget constraints of individuals, which can lead to a new competitive equilibrium with different consumption levels, reflecting the change in individuals' purchasing power.

Key Concept

Redistribution of Endowments

Explanation

Redistribution of endowments can lead to a new competitive equilibrium by changing the budget constraints and potentially improving the allocation of resources if the redistribution moves the economy closer to the contract curve.

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