MORE ECON

Externalities

100

A competitive cement industry faces demand

P=500−2Q

and constant marginal cost MC=80

Pollution damage from cement production is

D(Q)=20Q+0.1Q^2

Tasks:

a) Find the competitive equilibrium, CS, PS, total damage, and net social surplus.
b) Find the socially optimal quantity where MB=MC+MDMB=MC+MD.
c) Compute the Pigouvian tax that achieves the efficient quantity and recalculate welfare.

(a) Q = 210, P = 80, CS = 44,100; PS = 0

(b) Q = 181.81, P= 136.36; CS = 33056; PS = 10246, Damage= 6941, TS: 36,362

(c) T = $56.36 / unit; CS= 330567, PS = 0, Govt = 10246

200

Two electronics plants, A and B, each emit 150 units of waste. The city wants total emissions reduced to 180 units.

Marginal abatement costs:

• MACA=4+0.2RAMACA=4+0.2RA

• MACB=2+0.6RBMACB=2+0.6RB

Each plant receives 90 tradable permits.

Tasks:

a) Compute abatement and cost for each plant with no trading.
b) Find the cost-effective (equilibrium) trading outcome, trades, and permit price.
c) Compare total costs with and without trading.

(a) No trading

• CA=600; CB=1200

• Total cost = 1800

(b) With trading (cost-effective allocation)

• Abatement:

  • RA∗=87.5

  • RB∗=32.5

• Final emissions:

  • A: 62.5, B: 117.5

• Permit trades:

  • A sells 27.5 permits

  • B buys 27.5 permits

• Permit price = 21.5

(c) Total abatement cost with trading

• Costs:

  • CA∗=1115.625

  • CB∗=381.875

  • Total cost with trading = 1497.5

• Cost savings from trading = 1800 − 1497.5 = 302.5

300

A city is considering subsidizing flu vaccinations.

• Demand (private marginal benefit): P=60−Q

• Marginal cost (supply): MC=20

• Each vaccination creates a positive external benefit: MEB(Q)=5+0.1Q

1. Find the private market equilibrium quantity and price, assuming no regulation and perfect competition.

2. Find the socially efficient quantity.

3. Suppose the government offers a per-unit subsidy to reach the socially efficient quantity. What is the subsidy? Find the consumer price, government spending, and compare net social surplus with and without the subsidy.
   
   

1. Private market (no subsidy)

• Quantity: Qp=40

• Price: Pp=20

2. Socially efficient quantity

• Efficient quantity: Q∗=50, P=20

3. With subsidy to producers

• Subsidy per vaccination: s=10

• Consumer price with subsidy: Pc=10

• Government expenditure: G=500

Net social surplus:

• Without subsidy: SSp=1,080

• With subsidy: SSs=1,125

• Increase in SS (DWL removed): ΔSS=45

400

A mining company produces a metal in a competitive market.

• Demand: P=200−Q

• Private marginal cost: MC=20+0.5Q

• Marginal external damage from pollution: MD=0.5Q

(a) Ignoring the externality, find the competitive equilibrium quantity and price.

(b) Find the socially efficient quantity.

(c) If the government uses tradable permits to reach the efficient quantity, how many permits should it issue? What will the permit price be?

a) Q = 120, P = 80

b) Q* = 90, P* = 110

c) 90 permits, $45/permit

500

A city is considering building a solar-powered water purification plant.

• Construction cost: $500,000 million (paid once, now)

• Annual operating cost: $20 million

• Discount rate: 5%

• Demand for purified water: Q=200,000−1,000P

  (Price PP in dollars per year; quantity = households served)

Each household using purified water avoids groundwater extraction, creating external environmental benefits equal to:

EB=$150 per household per year

(a) If the city charges no price for water, how many households use the system? What are the: Annual consumer surplus; Annual operating cost Present value of benefits and costs? Should the city build the plant?

(b) Now include the external benefit: Compute the additional benefit per year and the NPV including externalities. Does the conclusion change?

(c) Suppose the city instead charges households $100 per year.
Compute all the same? Does the conclusion change?

(a) Q = 200,000; CS = 20 million; PVcs = 400 million; Cost = 900 million; NET BENEFIT = -500 million --> Do not build

(b) Benfit = 30 million; PV benefit = 600 million; Total Benefit = 1,000 million; NET BENEFIT = +100 million --> "Build!"

(c) Q = 100,000; CS = 5 million; PVcs = 100million; Benefit = 15 million; PV benefit = 300 million; NET BENEFIT = -500 million --> Do not build

TRADE

100

A country has the following domestic market for steel:

QD=100−2P,   QS=−20+3P

The world price of steel is Pw = 20.

a) Under free trade, calculate domestic production, domestic consumption, and imports.
b) Compute consumer surplus and producer surplus under free trade.
c) Suppose a tariff raises the domestic price to P = 25. Compute the new production, consumption, and imports.
d) Calculate the change in CS, PS, tariff revenue, and deadweight loss.

• Free Trade:

  • QS=40, QD=80 imports =40

  • CS=1600, PS=800

• Tariff:

  • QS=45, QD=70, imports =25

  • CS=1225, PS=1012.5, revenue =125

  • DWL = 37.5

200

Two workers, each with 40 hours, can produce food or clothing.
Producing either good requires:

• 10 hours setup

• 1 hour per unit produced

a) If each worker splits time evenly (20 hours per good), how many units of each good does each worker produce?
b) If Worker A specializes in food and Worker B specializes in clothing, compute total food and clothing output.
c) If they trade to consume equal bundles, how many units of each good does each person consume?

• Per worker:

  • Food = 10

  • Clothing = 10

(b)

• Total (with specialization):

  • Worker A Food = 30

  • Worker B Clothing = 30

(c)

• Per worker after trade:

  • Each have Food = 15 Clothing = 15

300

A firm has production function:

Q=K^0.5 * L^0.5

The firm faces:

• Wage for labor: w=20

• Rental rate of capital: r=5

The firm wants to produce Q=100 units at minimum cost.

(a) Find MPL and MPK
(b) Find the MRTS.
(c) Derive the optimal capital–labor ratio K/L
(d) For Q=100, find the cost-minimizing KK and LL, and the total cost.

(a)

• MPL=0.5 K^0.5 * L^−0.5

• MPK=0.5 L^0.5* K^−0.5

(b)

• MRTS=MPL/MPK=LK

(c)

• K=4L

(d)
Using Q=100

• L=50

• K=200

• Total cost C=20L+5K=2000

400

A firm allocates 60 units of labor across two plants. The production functions are:

QA=50LA−LA^2,    QB=40LB−0.5LB^2

and total labor is constrained by:

LA+LB=60

(a) How much labor should the firm allocate to each plant?

(b) What is the optimal quantity for each plant to produce?

(a) La = 23.333; Lb = 36.666

(b) Qa = 622.22; Qb = 794.444

500

[DO 400 FIRST] 

A firm allocates 60 units of labor across two plants. The production functions are:

QA=50LA−LA^2,    QB=40LB−0.5LB^2

and total labor is constrained by:

LA+LB=60

BUT, labor costs have increased. La = $10/hr, Lb = $12, and the firm can only afford to spend $400 on labor. What are the updated labor quantities and output quantity. 

Ratio of MPLa/MPLb stays the same.

LA≈16.08, LB≈22.12

Q = 1185.5

Monopoly , Oligoply & Everything in Between

100

For many years, MetroWater, a publicly owned utility, has been the sole provider of water purification tablets. Its demand curve is:

P=100−2Q

Operating costs are constant at MC=20

Then the government announces privatization, allowing competition in the market.

1. As a monopolist, what were MetroWater’s optimal price and quantity?

2. Under this new (perfect) competition, what will the new price and quantity be after privatization?

3. Calculate the change in consumer surplus from privatization.

Problem 1 — MetroWater (Monopoly → Competition)

• Monopoly:

  • QM=200

  • PM=60

• Perfect competition:

  • QC=40

  • PC=20

• Consumer surplus:

  • CSM=400

  • CSC=1600

  • ΔCS=+1200

• Deadweight loss under monopoly:

  • DWL=400

200

Two streaming companies, VStream and CineNow, release bundled plans and compete by choosing the number of new shows to launch. Market demand for new content (in “content units”) is:

P=80−Q

Both have identical marginal costs of production: MC=20

1. Derive each platform’s best-response function.

2. Find the Cournot equilibrium number of shows each company launches.

3. What is the resulting subscription price?

4. Compare the total content produced to the amount a single monopoly studio would create.

• Best responses:

  • q1=(60−q2)/2

  • q2=(60−q1)/ 2

• Cournot equilibrium:

  • q1∗=20,  q2∗=20

  • Q=40

  • P=40

  • Profit per firm: π1=π2=400

• Monopoly benchmark:

  • QM=30,  PM=50

300

Two gas stations at a busy intersection—FuelCo and PumpPlus—sold gasoline at nearly the same costs for years. After a refinery upgrade, FuelCo’s marginal cost drops to 20, while PumpPlus remains at 30.

Consumers always buy from the station offering the lower price (Bertrand).

1. What is the Nash equilibrium set of prices after the cost change?

2. What profits does each station earn?

3. If both firms had identical costs before the shock, how would the equilibrium differ?

• Nash prices: FuelCo sets price just below 30 (≈30), PumpPlus sets 30.

• Profits: FuelCo positive, PumpPlus 0 (exact numbers need a demand curve).

• If both had MC=20:

  • Bertrand equilibrium: p1=p2=20, both earn 0 profit.

400

A coalition of rare-earth mineral producers attempts to form a cartel to keep global prices high.
Demand for rare-earth metals is:

P=200−4QP=200−4Q

Each country’s marginal cost is MC=40MC=40, and they agree to split output evenly.

1. If the cartel successfully behaves cooperates, what total output and price maximize joint profits? What is the Q and P?

2. The cartel collapses, and the two countries work independently. What is their best response function, each equilibrium output and market price?

3. Compare the two outcomes. Why should the cartel cooperate or cheat?

(a) Cartel (monopoly)

• Total cartel output: QC=20

• Cartel price: PC=120

• Per-country output (2 members): q=10

(b) Cournot duopoly (after breakup)

• Best responses:

  • q1=20−0.5q2

  • q2=20−0.5q1

• Cournot equilibrium:

  • q1∗=q2∗=40/3≈13.33

  • QD=80/3≈26.67

  • PD=280/3≈93.33

(c) Comparison (just signs/directions)

• Price: PC=120 > PD≈93.33

• Output: QC=20< QD≈26.67

• Total profit: higher under cartel than under Cournot.

500

A low-cost startup, EcoPower, enters a market currently dominated by GridGen. They compete in quantities (Cournot).
Market demand:

P=90−Q

Costs:

• GridGen: MC=30

• EcoPower (innovator): MC=10

1. Write the reaction functions for both firms.

2. Solve for the Cournot equilibrium outputs.

3. Explain economically why EcoPower captures a larger share of the market.

• qA∗≈13.33

• qB∗≈33.33

• Q≈46.67

• P≈43.33

Games & Decision under uncertainty

100

HomeHub, a dominant smart-device brand, is considering entering a new market for smart thermostats. A smaller competitor, HeatWave, can respond after seeing HomeHub’s choice.

Payoffs (HomeHub, HeatWave):

• If both enter: (−10,−10)(−10,−10)

• If HomeHub enters and HeatWave stays out: (30,0)(30,0)

• If HomeHub stays out and HeatWave enters: (0,20)(0,20)

1. Draw the game tree with HomeHub moving first and HeatWave observing its move.

2. Use backward induction to find the subgame-perfect Nash equilibrium (SPNE).

3. Compare this to the simultaneous-move version of the game (where they choose Enter/Stay Out at the same time). How does first-mover advantage change the outcome?

• Sequential game (HomeHub moves first):

  • SPNE: HomeHub Enter, HeatWave Stay Out

  • Payoffs: (30,0)

• Simultaneous-move version:

  • Nash equilibria (two):

    • (Enter, Stay Out) with payoffs (30,0)

    • (Stay Out, Enter) with payoffs (0,20)

200

Two smartphone brands, Apex and Nova, must decide whether to run a High or Low advertising campaign for their next flagship launch.

Both High: (5, 5) A Low & B High: (8, 3) A High & B Low: (3, 8) Both Low: (6, 6)

Payoffs are profits (millions of dollars).

1. Does either firm have a dominant strategy?

2. Identify all Nash equilibria.

3. In one sentence, say whether this looks more like a coordination game, a prisoner’s dilemma, or something else, and why.

• Dominant strategies:

  • Apex: High

  • Nova: High

• Nash equilibrium:

  • (High,High) with payoffs (5,5)

• It has a prisoner’s dilemma flavor because (Low,Low)=(6,6) is better for both than (5,5), but they move to (High,High).

300

The coastal city of Marina Vista is considering building a tsunami early-warning siren system. Once installed, everyone hears the signal—making it non-rival and non-excludable.

Two neighborhoods benefit from the system. Their marginal benefit (MB) schedules are:

• Neighborhood A: MBₐ = 20 – Q

• Neighborhood B: MBᵦ = 10 – 0.5Q
  where Q is the number of sirens installed across the city.

The marginal cost (MC) of each siren is constant at MC = 12.

Questions

a) Compute the social marginal benefit curve (vertical sum).
b) Find the efficient number of sirens.
c) Suppose each neighborhood tries to “free ride” and reports only half its true marginal benefit. What quantity would be provided under voluntary contributions?

a) Social MB (vertical sum):

MBtotal=(20−Q)+(10−0.5Q)=30−1.5Q

b) Efficient quantity (MC = 12):

12=30−1.5Q⇒Q∗=12 sirens

c) With each reporting only half their MB:
Reported MBₐ = 10 – 0.5Q
Reported MBᵦ = 5 – 0.25Q
Total reported MB = 15 – 0.75Q

Set = MC:

12=15−0.75Q⇒Q=4 sirens

400

SkyJet Airlines is considering purchasing a new fuel-efficient aircraft.

• If fuel prices stay high, the aircraft generates $12 million in savings.

• If fuel prices fall low, the aircraft leads to only $3 million in savings, not enough to justify the purchase cost.

• The probability fuel prices remain high is 40%.

SkyJet can buy an energy market forecast report for $200,000.
Historically:

• When the report predicts high prices, fuel prices end up high 70% of the time.

• When the report predicts low prices, fuel prices end up high 10% of the time.

• The report predicts high prices 50% of the time.

SkyJet must decide whether to:

1. Buy the aircraft immediately, or

2. Purchase the forecast report, then decide whether to buy the aircraft based on the forecast.

Tasks

(a) Compute the expected value of buying the aircraft without purchasing the forecast.
(b) Compute the expected value of buying the forecast, assuming SkyJet purchases the aircraft only when the forecast predicts high prices.
(c) Which strategy has the higher expected value?
(d) What assumption about risk preferences is implicit in your recommendation?

(a) Expected value of buying aircraft without forecast:

EVno forecast=0.4(12)+0.6(3)=6.6 million

(b) Expected value with forecast (buy aircraft only if forecast = “high”):

• Given “high” forecast: EV=0.7(12)+0.3(3)=9.3 million

• Occurs with probability 0.5 → overall: 0.5×9.3=4.65 million

• Subtract forecast cost: 4.65−0.2=4.45 million

So:

EVwith forecast=4.45 million

(c) Better strategy: Buy the aircraft immediately and skip the forecast, since
6.6>4.45 (in millions).

(d) Assumed risk preference: The decision rule uses expected value, so SkyJet is treated as risk-neutral.

500

A rural county is debating whether to build a satellite-linked wildfire detection grid. Once active, all residents receive the benefit of faster emergency alerts.

Demand for detection quality (think of Q as “coverage quality units”) is
P = 40 – 2Q, where P is the monthly willingness to pay per household.

There are 100,000 households.

The public system would be free to use (P = 0), with annual maintenance costs of $12 million and a one-time installation cost of $80 million.
Discount rate: 5%.

A private firm could offer a subscription version with constant marginal cost MC = 4, and would price competitively so that P = MC.

Questions

a) Compute annual consumer surplus under private provision.
b) Compute annual consumer surplus under public (free) provision.
c) Compute the present value of net social benefits of the public system. Should the county build it?

Private provision (P = MC = 4)

• Quantity:

  4=40−2Q⇒Q=184

• CS per household (monthly):

  CS=12(40−4)⋅18=324

• CS per household (annual): 324×12=3,888

• Total annual CS (100,000 households):

  3,888×100,000=388.8 million

Public provision (free, P = 0)

• Quantity:

  0=40−2Q⇒Q=20

• CS per household (monthly):

  CS=12(40−0)⋅20=400\

• CS per household (annual): 400×12=4,800

• Total annual CS:

  4,800×100,000=480 million

Public system: PV of net social benefits

• Annual net benefit flow: 480−12=4684

• PV of flow (perpetuity, r = 0.05):

  4680.05=9.36 billion

• Subtract installation cost:

  9.36 billion−0.08 billion=9.28 billion

Conclusion: The county should build the public system (PV ≈ $9.28 billion).

RANDOM

100

A competitive avocado farm has cost function

C(q)=100+4q+q^2

The market price is P = 24.

1. Find the firm’s profit-maximizing output.

2. Compute profit.

3. Should the firm shut down in the short run?

• MR = MC → 24 = 4 + 2q → q = 10*

• Profit = 24(10)−(100+4(10)+10^2) = $60

• VC(10)=4(10)+100=140? Wait—VC excludes fixed cost → VC=4q+q²=140; price × q = 240 ≥ VC → Do not shut down

200

A consumer has utility

u(x,y)=x^0.5*y^0.5,

and income M = 120, with prices p_x = 4, p_y = 6.

1. Write the budget constraint.

2. Solve for the optimal bundle (x, y)

1. Budget constraint: 4x+6y=120

2. MU/p rule:

(0.5x−0.5y^0.5)/4=(0.5y^−0.5x^0.5)/6

Simplifies to

y/x=4/6=2/3

Substitute into budget → x = 15*, y = 10*

300

The city wants to plant public shade trees along major roads. Each tree provides cooling and pollution reduction city-wide.

Three groups value the trees differently:

• Residents: MBᴿ = 15 – 0.5Q

• Businesses: MBᴮ = 10 – 0.25Q

• Tourists (collectively): MBᵀ = 5 – 0.1Q

Marginal cost of planting and maintaining each tree is MC = 8.

Questions

a) Compute the total marginal benefit curve (vertical sum).
b) Find the efficient number of trees.
c) If only residents were counted in the political process, what quantity would be chosen? What is the deadweight lossfrom ignoring business and tourist benefits?

MB curves:

• Residents: MBR=15−0.5Q

• Businesses: MBB=10−0.25Q

• Tourists: MBT=5−0.1Q

• Total MB:

  MBtotal=15−0.5Q+10−0.25Q+5−0.1Q=30−0.85Q
• MC = 8.

a) Total MB: 30−0.85Q

b) Efficient number of trees (MC = total MB):

8=30−0.85Q⇒0.85Q=22⇒Q∗≈25.9≈26 trees

c) If only residents count:

• Residents’ MB: 15−0.5Q

  8=15−0.5Q⇒0.5Q=7⇒Q=14 trees

• DWL from ignoring businesses & tourists (area between MB_total and MC from Q = 14 to 26): DWL = 60

400

A metal producer operates two plants with marginal product functions:

MPA=40−LA; MPB=30−0.5LB.

The firm has 60 total units of labor to allocate.

1. Find the labor allocation (L_A, L_B) that equalizes marginal products.

2. How much labor goes to each plant?

L_A = 20, L_B = 40

500

MALLORY TO MAKE UP

TA. da