A bakery produces bread. The production function of bread is q(e)=24e, where is

Question

A bakery produces bread. The production function of bread is q(e)=24e, where is e is the baker’s effort.

The baker can sell each loaf of bread (i.e. each q) for 2.
Baking bread creates effort costs for the baker which are given by c(e)=2e2.

The bread also produces a very nice smell, which creates a positive externality worth s(q) = 0.5 q to the local residents.

a. What number of loaves (i.e. q) maximizes the baker’s profits and what are the baker’s profits in this case?

b. What number of loafs (i.e. q) maximizes social welfare and what are the baker’s profits in this case?

c. What is – in general – the Pigouvian solution to undersupply of positive externalities?

d. The local government decides to tax or subsidize the production of bread in order to get the baker to produce the social welfare maximizing number of loaves (i.e. q). How high is the tax or subsidy per loaf of bread?

e. What are the baker’s profits with the tax / subsidy from d)?

Please show method and steps for the answer, thank you

Explanation / Answer

a. In order to maximize the personal profit, Baker will produce till

Marginal Private cost = Marginal Private benefit

  revenue = p*q = 2*24e = 48e

Marginal Private benefit = drevenue/de = 48

Margiinal Private cost = dcost/de = 2*2e = 4e

Puting MPC =MPB

4e = 48

e = 12

So, Number of Loafs = 24*12 = 288

Baker's Profit = Total Revenue - Total Cost = 48*12 - 2(12)^2 = 576 - 288 = 288

b.

In case of social maximizing , Social welfare Maximizes when

Marginal Private cost = Marginal Social Benefit

Marginal Social Benefit = Marginal Private benefit + Marginal External Benefit

    External benefit = 0.5q = 0.5*2e = 12e

Marginal External Benefit = 12

Marginal Social Benefit = 48 + 12 = 60

Putting MPC = MSB

4e = 60

e = 15

So, Number of Loafs = 24*15 = 360

Baker's Profit = Total Revenue - Total Cost = 2*360 - 2(15)^2 = 720 - 450 = 270

c.

The Pigovian solution is to provide the subsidy to the baker which is sufficient to compensate the loss incured by the baker to produce socially optimal quantity instead of private optimal.

d.

Subsidy given = Difference in profit in two cases = 288 - 270 = 18

Subsidy per loaf = Subsidy/q = 18/360 = 0.05

e.

Banker's profit with tax = baker's profit in case of private maximizing = 270 + 0.05*360 = 288

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