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Question 2 NAT · 1.0 marks
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Q1*………..
Published solution
**1. Write simultaneous profits:** Inverse demand is \(P=80-Q_1-Q_2\); each firm’s constant average/marginal cost is 8. Thus \(\pi_i=(72-Q_i-Q_j)Q_i\).
**2. Derive best responses:** Differentiating each profit in its own quantity gives \(72-2Q_i-Q_j=0\). The two best responses are \(Q_i=(72-Q_j)/2\), for positive outputs. Own second derivative is -2, confirming a maximum.
**3. Solve equilibrium:** Symmetry gives \(Q_1=Q_2=24\); total output is 48 and \(P=80-48=32\).
**4. Compute profits:** Each firm earns \((32-8)(24)=576\). The quantities are mutual best responses, so neither firm gains by changing its quantity alone.
**Requested value:** Firm 1 output = 24.
**Answer: 24.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Q2*………………
Published solution
**1. Write simultaneous profits:** Inverse demand is \(P=80-Q_1-Q_2\); each firm’s constant average/marginal cost is 8. Thus \(\pi_i=(72-Q_i-Q_j)Q_i\).
**2. Derive best responses:** Differentiating each profit in its own quantity gives \(72-2Q_i-Q_j=0\). The two best responses are \(Q_i=(72-Q_j)/2\), for positive outputs. Own second derivative is -2, confirming a maximum.
**3. Solve equilibrium:** Symmetry gives \(Q_1=Q_2=24\); total output is 48 and \(P=80-48=32\).
**4. Compute profits:** Each firm earns \((32-8)(24)=576\). The quantities are mutual best responses, so neither firm gains by changing its quantity alone.
**Requested value:** Firm 2 output = 24.
**Answer: 24.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
P*=............
Published solution
**1. Write simultaneous profits:** Inverse demand is \(P=80-Q_1-Q_2\); each firm’s constant average/marginal cost is 8. Thus \(\pi_i=(72-Q_i-Q_j)Q_i\).
**2. Derive best responses:** Differentiating each profit in its own quantity gives \(72-2Q_i-Q_j=0\). The two best responses are \(Q_i=(72-Q_j)/2\), for positive outputs. Own second derivative is -2, confirming a maximum.
**3. Solve equilibrium:** Symmetry gives \(Q_1=Q_2=24\); total output is 48 and \(P=80-48=32\).
**4. Compute profits:** Each firm earns \((32-8)(24)=576\). The quantities are mutual best responses, so neither firm gains by changing its quantity alone.
**Requested value:** Market price = 32.
**Answer: 32.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
π1…………
Published solution
**1. Write simultaneous profits:** Inverse demand is \(P=80-Q_1-Q_2\); each firm’s constant average/marginal cost is 8. Thus \(\pi_i=(72-Q_i-Q_j)Q_i\).
**2. Derive best responses:** Differentiating each profit in its own quantity gives \(72-2Q_i-Q_j=0\). The two best responses are \(Q_i=(72-Q_j)/2\), for positive outputs. Own second derivative is -2, confirming a maximum.
**3. Solve equilibrium:** Symmetry gives \(Q_1=Q_2=24\); total output is 48 and \(P=80-48=32\).
**4. Compute profits:** Each firm earns \((32-8)(24)=576\). The quantities are mutual best responses, so neither firm gains by changing its quantity alone.
**Requested value:** Firm 1 profit = 576.
**Answer: 576.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
π2………
Published solution
**1. Write simultaneous profits:** Inverse demand is \(P=80-Q_1-Q_2\); each firm’s constant average/marginal cost is 8. Thus \(\pi_i=(72-Q_i-Q_j)Q_i\).
**2. Derive best responses:** Differentiating each profit in its own quantity gives \(72-2Q_i-Q_j=0\). The two best responses are \(Q_i=(72-Q_j)/2\), for positive outputs. Own second derivative is -2, confirming a maximum.
**3. Solve equilibrium:** Symmetry gives \(Q_1=Q_2=24\); total output is 48 and \(P=80-48=32\).
**4. Compute profits:** Each firm earns \((32-8)(24)=576\). The quantities are mutual best responses, so neither firm gains by changing its quantity alone.
**Requested value:** Firm 2 profit = 576.
**Answer: 576.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Now suppose that the two firms are competing in the Stackelberg manner where one of the firms
makes its output decision before the other. Suppose Firm 1 is the Stackelberg leader (i.e., makes
its output decisions before Firm 2). Calculate the equilibrium quantity for the firms, equilibrium
market price and profits (π1, π2) of both the firms in equilibrium.
Q1*………..
Published solution
**1. Solve the follower’s problem:** After observing Q1, firm 2 maximizes \((72-Q_1-Q_2)Q_2\). Its best response is \(Q_2=(72-Q_1)/2\) when positive.
**2. Anticipate that response:** Firm 1’s profit becomes \[\pi_1=\left[72-Q_1-\frac{72-Q_1}{2}\right]Q_1=36Q_1-\frac12Q_1^2.\]
Its derivative is \(36-Q_1\), giving leader output 36. Negative second derivative confirms a maximum.
**3. Compute the follower and price:** \(Q_2=(72-36)/2=18\), total output is 54 and \(P=80-54=26\).
**4. Compute profits:** \(\pi_1=(26-8)(36)=648\), and \(\pi_2=(26-8)(18)=324\).
**Requested value:** Leader output = 36.
**Answer: 36.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Now suppose that the two firms are competing in the Stackelberg manner where one of the firms
makes its output decision before the other. Suppose Firm 1 is the Stackelberg leader (i.e., makes
its output decisions before Firm 2). Calculate the equilibrium quantity for the firms, equilibrium
market price and profits (π1, π2) of both the firms in equilibrium.
Q2*………………
Published solution
**1. Solve the follower’s problem:** After observing Q1, firm 2 maximizes \((72-Q_1-Q_2)Q_2\). Its best response is \(Q_2=(72-Q_1)/2\) when positive.
**2. Anticipate that response:** Firm 1’s profit becomes \[\pi_1=\left[72-Q_1-\frac{72-Q_1}{2}\right]Q_1=36Q_1-\frac12Q_1^2.\]
Its derivative is \(36-Q_1\), giving leader output 36. Negative second derivative confirms a maximum.
**3. Compute the follower and price:** \(Q_2=(72-36)/2=18\), total output is 54 and \(P=80-54=26\).
**4. Compute profits:** \(\pi_1=(26-8)(36)=648\), and \(\pi_2=(26-8)(18)=324\).
**Requested value:** Follower output = 18.
**Answer: 18.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Now suppose that the two firms are competing in the Stackelberg manner where one of the firms
makes its output decision before the other. Suppose Firm 1 is the Stackelberg leader (i.e., makes
its output decisions before Firm 2). Calculate the equilibrium quantity for the firms, equilibrium
market price and profits (π1, π2) of both the firms in equilibrium.
P*=............
Published solution
**1. Solve the follower’s problem:** After observing Q1, firm 2 maximizes \((72-Q_1-Q_2)Q_2\). Its best response is \(Q_2=(72-Q_1)/2\) when positive.
**2. Anticipate that response:** Firm 1’s profit becomes \[\pi_1=\left[72-Q_1-\frac{72-Q_1}{2}\right]Q_1=36Q_1-\frac12Q_1^2.\]
Its derivative is \(36-Q_1\), giving leader output 36. Negative second derivative confirms a maximum.
**3. Compute the follower and price:** \(Q_2=(72-36)/2=18\), total output is 54 and \(P=80-54=26\).
**4. Compute profits:** \(\pi_1=(26-8)(36)=648\), and \(\pi_2=(26-8)(18)=324\).
**Requested value:** Market price = 26.
**Answer: 26.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Now suppose that the two firms are competing in the Stackelberg manner where one of the firms
makes its output decision before the other. Suppose Firm 1 is the Stackelberg leader (i.e., makes
its output decisions before Firm 2). Calculate the equilibrium quantity for the firms, equilibrium
market price and profits (π1, π2) of both the firms in equilibrium.
π1…………
Published solution
**1. Solve the follower’s problem:** After observing Q1, firm 2 maximizes \((72-Q_1-Q_2)Q_2\). Its best response is \(Q_2=(72-Q_1)/2\) when positive.
**2. Anticipate that response:** Firm 1’s profit becomes \[\pi_1=\left[72-Q_1-\frac{72-Q_1}{2}\right]Q_1=36Q_1-\frac12Q_1^2.\]
Its derivative is \(36-Q_1\), giving leader output 36. Negative second derivative confirms a maximum.
**3. Compute the follower and price:** \(Q_2=(72-36)/2=18\), total output is 54 and \(P=80-54=26\).
**4. Compute profits:** \(\pi_1=(26-8)(36)=648\), and \(\pi_2=(26-8)(18)=324\).
**Requested value:** Leader profit = 648.
**Answer: 648.**
Suppose two firms operate in a market and can produce at a constant average (and marginal) cost
of AC = MC = $8. Let Q1 be the output of the first firm and Q2 be the output of the second. Market
demand is given by Q1+ Q2 = 80 - P. Assuming that the firms are competing as per the Cournot
competition, find out the equilibrium quantity for each firm, equilibrium price and profit (π1, π2) of
each firm.
Based on the above data, answer the given subquestions.
Now suppose that the two firms are competing in the Stackelberg manner where one of the firms
makes its output decision before the other. Suppose Firm 1 is the Stackelberg leader (i.e., makes
its output decisions before Firm 2). Calculate the equilibrium quantity for the firms, equilibrium
market price and profits (π1, π2)) of both the firms in equilibrium.
π2………
Published solution
**1. Solve the follower’s problem:** After observing Q1, firm 2 maximizes \((72-Q_1-Q_2)Q_2\). Its best response is \(Q_2=(72-Q_1)/2\) when positive.
**2. Anticipate that response:** Firm 1’s profit becomes \[\pi_1=\left[72-Q_1-\frac{72-Q_1}{2}\right]Q_1=36Q_1-\frac12Q_1^2.\]
Its derivative is \(36-Q_1\), giving leader output 36. Negative second derivative confirms a maximum.
**3. Compute the follower and price:** \(Q_2=(72-36)/2=18\), total output is 54 and \(P=80-54=26\).
**4. Compute profits:** \(\pi_1=(26-8)(36)=648\), and \(\pi_2=(26-8)(18)=324\).
**Requested value:** Follower profit = 324.
**Answer: 324.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The equilibrium quantity and price before incidence of tax are, P*=…..
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**Requested value:** The original price is INR 300.
**Answer: 300.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The equilibrium quantity and price before incidence of tax are, Q*=…..
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**Requested value:** Original quantity is 120 pizzas.
**Answer: 120.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
Consumer surplus before incidence of tax is, C.S = …..
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**4. Compute original consumer surplus:** Demand intercept is 420. The triangle above price 300 and below demand has area \(CS_0=\tfrac12(420-300)(120)=7200\) INR.
**Answer: 7200.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
Producer surplus before incidence of tax is, P.S = …….
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**4. Compute original producer surplus:** Supply intercept is 60. The triangle below price 300 and above supply has area \(PS_0=\tfrac12(300-60)(120)=14400\) INR.
**Answer: 14400.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The equilibrium quantity after the incidence of tax is, Q\(^{t}\) = ……
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**Requested value:** Post-tax quantity = 116.
**Answer: 116.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The new price that the buyer pays after incidence of tax is, Pb= ……
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**Requested value:** Price paid by buyers = 304.
**Answer: 304.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The net price received by the seller after the incidence of tax is, Ps= …….
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**Requested value:** Net price received by sellers = 292.
**Answer: 292.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The loss in consumer surplus due to incidence of tax is, ΔCS= ….
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**4. Compute consumer-surplus loss:** The original surplus is \(\tfrac12(420-300)(120)=7200\). New surplus is \(\tfrac12(420-304)(116)=6728\). The loss is \(7200-6728=472\) INR. Equivalently, it is the rectangle \(4(116)\) plus triangle \(\tfrac12(4)(4)\).
**Answer: 472.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The loss in producer surplus due to incidence of tax is, ΔPS= …..
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**4. Compute producer-surplus loss:** Original surplus is \(\tfrac12(300-60)(120)=14400\). New surplus is \(\tfrac12(292-60)(116)=13456\). The loss is \(14400-13456=944\) INR, equal to \(8(116)+\tfrac12(8)(4)\).
**Answer: 944.**
The market demand curve for pizzas is P=420-Q and the market supply curve is P=60+2Q. A tax of
t=12 INR is imposed on one unit of pizza sold. Answer the given subquestions.
The total tax revenue is, R= ……
Published solution
**1. Find the untaxed equilibrium:** Equating \(420-Q=60+2Q\) gives \(3Q=360\), so \(Q_0=120\) and \(P_0=300\) INR per pizza.
**2. Insert the tax wedge:** With tax 12, buyer price Pb and seller net price Ps satisfy \(P_b=P_s+12\). Thus \(420-Q=60+2Q+12\), giving \(Q_t=116\).
**3. Find the two new prices:** \(P_b=420-116=304\), while \(P_s=60+2(116)=292\). Their difference is exactly 12. Consumers bear a 4-rupee price increase and sellers an 8-rupee net-price decrease.
**4. Calculate revenue:** Tax is collected on the 116 pizzas still sold, so \(R=12(116)=1392\) INR. The total private-surplus loss is \(472+944=1416\); subtracting revenue leaves deadweight loss 24, matching \(\tfrac12(12)(120-116)\).
**Answer: 1392.**
Consider the two lotteries A and B. With lottery A, there is a 0.90 chance that you receive a payoff
of $0 and a 0.10 chance that you receive a payoff of $400. With lottery
B, there is a 0.5 chance that you receive a payoff of $30 and 0.50 chance that you receive a payoff
of $50.
Based on the above data, answer the given subquestions.
What is the expected value of Lottery A…………..
Published solution
**1. Read Lottery A:** A pays zero with probability 0.90 and 400 dollars with probability 0.10.
**2. Weight each outcome:** \[E[I_A]=0.90(0)+0.10(400)=40.\]
**3. Interpret:** The expected payoff is $40. It is a probability-weighted average, not a guaranteed payment.
**Answer: 40.**
Consider the two lotteries A and B. With lottery A, there is a 0.90 chance that you receive a payoff
of $0 and a 0.10 chance that you receive a payoff of $400. With lottery
B, there is a 0.5 chance that you receive a payoff of $30 and 0.50 chance that you receive a payoff
of $50.
Based on the above data, answer the given subquestions.
What is the expected value of Lottery B………….
Published solution
**1. Read Lottery B:** The two equally likely payoffs are $30 and $50.
**2. Average them using probabilities:** \[E[I_B]=0.5(30)+0.5(50)=15+25=40.\]
**3. Compare risk and mean:** Its expected payoff is $40, matching Lottery A, even though its payoff dispersion is much smaller.
**Answer: 40.**
Consider the two lotteries A and B. With lottery A, there is a 0.90 chance that you receive a payoff
of $0 and a 0.10 chance that you receive a payoff of $400. With lottery
B, there is a 0.5 chance that you receive a payoff of $30 and 0.50 chance that you receive a payoff
of $50.
Based on the above data, answer the given subquestions.
Suppose that John’s utility function is U= 100I
What is John’s expected utility from Lottery A…………
Published solution
**1. Apply utility to outcomes:** With U(I)=100I, Lottery A’s utilities are 0 at payoff zero and 40,000 at payoff 400.
**2. Average utilities:** \[E[U_A]=0.9(0)+0.1(40{,}000)=4000.\]
**3. Check linearity:** This also equals \(100E[I_A]=100(40)=4000\). It is a utility value, not a dollar payoff.
**Answer: 4000.**
Consider the two lotteries A and B. With lottery A, there is a 0.90 chance that you receive a payoff
of $0 and a 0.10 chance that you receive a payoff of $400. With lottery
B, there is a 0.5 chance that you receive a payoff of $30 and 0.50 chance that you receive a payoff
of $50.
Based on the above data, answer the given subquestions.
Suppose that John’s utility function is U= 100I
What is John’s expected utility from Lottery B…………
Published solution
**1. Compute state utilities:** Lottery B gives utility \(100(30)=3000\) or \(100(50)=5000\), each with probability 0.5.
**2. Average:** \[E[U_B]=0.5(3000)+0.5(5000)=1500+2500=4000.\]
**3. Check:** Its expected utility matches Lottery A because utility is linear and both expected monetary payoffs equal $40.
**Answer: 4000.**
Consider the two lotteries A and B. With lottery A, there is a 0.90 chance that you receive a payoff
of $0 and a 0.10 chance that you receive a payoff of $400. With lottery
B, there is a 0.5 chance that you receive a payoff of $30 and 0.50 chance that you receive a payoff
of $50.
Based on the above data, answer the given subquestions.
Suppose that John’s utility function is U= 100I
What investment will John choose?
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Investment A
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Investment B
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John is indifferent
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Cannot be determined
Published solution
**1. Compute expected payoffs:** Lottery A has \(0.9(0)+0.1(400)=40\). Lottery B has \(0.5(30)+0.5(50)=40\).
**2. Apply linear utility:** U(I)=100I gives expected utility 4000 for both.
**3. Compare choices:** John is indifferent between the lotteries under this utility. Lottery B is less variable, but linear utility assigns no separate penalty for that risk.
**Answer: C — Selected: John is indifferent.**
A: **Incorrect:** A does not have higher expected utility than B.
B: **Incorrect:** B’s lower variance does not create a utility advantage for this linear-utility investor.
C: **Correct:** Both lotteries have equal expected utility.
D: **Incorrect:** The supplied utility and payoffs determine the comparison.
Consider the two lotteries A and B. With lottery A, there is a 0.90 chance that you receive a payoff
of $0 and a 0.10 chance that you receive a payoff of $400. With lottery
B, there is a 0.5 chance that you receive a payoff of $30 and 0.50 chance that you receive a payoff
of $50.
Based on the above data, answer the given subquestions.
Suppose that John’s utility function is U= 100I
Choose the correct alternative
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With this utility function John is risk averse
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With this utility function John is risk loving
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With this utility function John is risk neutral
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Cannot be determined
Published solution
**1. Inspect utility curvature:** U(I)=100I has constant marginal utility 100 and second derivative zero.
**2. Apply the risk classification:** Concave utility indicates risk aversion; convex utility indicates risk loving. Linear utility makes the decision maker indifferent among lotteries with the same mean.
**3. Conclude:** John is risk neutral under the supplied utility.
**Answer: C — Selected: With this utility function John is risk neutral.**
A: **Incorrect:** Risk aversion requires concavity rather than a linear function.
B: **Incorrect:** Risk loving requires convex curvature, which is absent.
C: **Correct:** Constant marginal utility gives risk neutrality.
D: **Incorrect:** The utility function is enough to classify the attitude.
The market for paper in southern region of India is characterized by the following demand and
supply curves : QD=3400-20P and QS=800+20P Where QD is the quantity demanded in 100 pound
lots and QS is the quantity supplied in 100 pound lots and P is the price per 100 pound lot.
Currently there is no attempt to regulate the dumping of effluent into streams and rivers by paper
mills. As a result, dumping is widespread. The Marginal External Cost (MEC) associated with the
production of paper is given by the curve MEC= 0.05QS
Based on the above data, answer the given subquestions.
Calculate the output of paper if it is produced under competitive conditions and no attempt is
made to monitor or regulate the dumping of effluent Q=..............
Published solution
**1. Use private competitive supply:** Without regulation, equate \(Q_D=3400-20P\) and \(Q_S=800+20P\), excluding external pollution costs from producer decisions.
**2. Solve price:** \(3400-20P=800+20P\) gives \(2600=40P\), so P=65.
**3. Find quantity:** \(Q=3400-20(65)=2100\) lots of 100 pounds. Ignoring pollution creates the private equilibrium requested.
**Answer: 2100.**
The market for paper in southern region of India is characterized by the following demand and
supply curves : QD=3400-20P and QS=800+20P Where QD is the quantity demanded in 100 pound
lots and QS is the quantity supplied in 100 pound lots and P is the price per 100 pound lot.
Currently there is no attempt to regulate the dumping of effluent into streams and rivers by paper
mills. As a result, dumping is widespread. The Marginal External Cost (MEC) associated with the
production of paper is given by the curve MEC= 0.05QS
Based on the above data, answer the given subquestions.
Calculate the price of paper if it is produced under competitive conditions and no attempt is made
to monitor or regulate the dumping of effluent P=..............
Published solution
**1. Equate demand and private supply:** \(3400-20P=800+20P\).
**2. Collect terms:** \(2600=40P\), giving \(P=65\).
**3. Verify:** At 65, demand and supply both equal 2100 lots. The price is 65 per 100-pound lot; pollution’s external cost is ignored in this unregulated equilibrium.
**Answer: 65.**
The market for paper in southern region of India is characterized by the following demand and
supply curves : QD=3400-20P and QS=800+20P Where QD is the quantity demanded in 100 pound
lots and QS is the quantity supplied in 100 pound lots and P is the price per 100 pound lot.
Currently there is no attempt to regulate the dumping of effluent into streams and rivers by paper
mills. As a result, dumping is widespread. The Marginal External Cost (MEC) associated with the
production of paper is given by the curve MEC= 0.05QS
Based on the above data, answer the given subquestions.
Determine the socially efficient price of the paper PS=............
Published solution
**1. Write inverse curves:** Marginal benefit is \(MB=170-0.05Q\). Private marginal cost is \(MPC=0.05Q-40\). External marginal cost is \(MEC=0.05Q\).
**2. Include external harm:** \(MSC=MPC+MEC=0.10Q-40\). Set MB=MSC: \(170-0.05Q=0.10Q-40\), giving \(210=0.15Q\) and \(Q^*=1400\).
**3. Read the demand price:** \(P^*=170-0.05(1400)=100\). At that quantity MPC=30 and MEC=70, so the socially relevant consumer price is their sum 100.
**Answer: 100.**
The market for paper in southern region of India is characterized by the following demand and
supply curves : QD=3400-20P and QS=800+20P Where QD is the quantity demanded in 100 pound
lots and QS is the quantity supplied in 100 pound lots and P is the price per 100 pound lot.
Currently there is no attempt to regulate the dumping of effluent into streams and rivers by paper
mills. As a result, dumping is widespread. The Marginal External Cost (MEC) associated with the
production of paper is given by the curve MEC= 0.05QS
Based on the above data, answer the given subquestions.
Determine the socially efficient quantity of paper QS=...............
Published solution
**1. Account for pollution:** Inverse supply gives private cost \(0.05Q-40\); add marginal external cost 0.05Q to get social marginal cost \(0.10Q-40\).
**2. Match benefit and social cost:** \[170-0.05Q=0.10Q-40.\]
**3. Solve:** \(210=0.15Q\) gives \(Q^*=1400\) hundred-pound lots. It is lower than unregulated output 2100 because efficient production includes the external damage.
**Answer: 1400.**
Based on the above data, answer the given subquestions.
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4
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1/4
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1/3
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3
Published solution
**1. Differentiate production:** The source gives \(Q=120\sqrt{LK}\). Marginal products satisfy \(MP_L/MP_K=K/L\).
**2. Use least-cost tangency:** Wage w=10 and rental price r=40 require \(MP_L/MP_K=w/r=10/40\).
**3. Conclude:** \(K/L=1/4\). A unit of capital is more expensive, so the least-cost combination uses four times as much labor as capital.
**Answer: B — Selected: 1/4.**
A: **Incorrect:** This reverses the required ratio.
B: **Correct:** The tangency gives K/L=10/40=1/4.
C: **Incorrect:** One third does not match the input-price ratio.
D: **Incorrect:** Three is not the marginal-product/input-price condition.
Based on the above data, answer the given subquestions.
What is the cost minimizing input combination if the firm wants to produce 720 units per year?
L*=.............
Published solution
**1. Impose the output target:** \(120\sqrt{LK}=720\) implies \(\sqrt{LK}=6\), so LK=36.
**2. Use the cost-minimizing ratio:** From \(MP_L/MP_K=K/L=w/r=1/4\), capital is K=L/4.
**3. Solve for labor:** \(L(L/4)=36\) gives L-squared=144. The positive solution is L=12, with K=3. Both inputs are nonnegative as required.
**Answer: 12.**
Based on the above data, answer the given subquestions.
What is the cost minimizing input combination if the firm wants to produce 720 units per year?
K*=..............
Published solution
**1. Meet production:** \(Q=120\sqrt{LK}=720\) gives LK=36.
**2. Match marginal products per dollar:** Wage 10 and capital price 40 give K/L=1/4, or L=4K.
**3. Solve for capital:** \(4K^2=36\), so K=3 and L=12. Check output: \(120\sqrt{12(3)}=720\). The cost at this least-cost bundle is \(10(12)+40(3)=240\) per year under the stated units.
**Answer: 3.**
A monopolist faces the demand curve P=25-Q, where P is measured in dollars per unit and Q in
thousands of units. The monopolist has a constant average cost of $5 per unit. Answer the given
subquestions.
What are the monopolist’s profit-maximizing price and quantity, P*= …….
Published solution
**1. Derive marginal revenue:** Demand is P=25-Q, with Q in thousands. Revenue in thousands is \(TR=25Q-Q^2\), so MR=25-2Q.
**2. Set MR=MC:** Constant average cost 5 gives MC=5. Thus \(25-2Q=5\), so Q=10 thousand units.
**3. Read price:** \(P^*=25-10=15\) dollars per unit. Price comes from demand, not directly from the marginal-revenue value.
**Answer: 15.**
A monopolist faces the demand curve P=25-Q, where P is measured in dollars per unit and Q in
thousands of units. The monopolist has a constant average cost of $5 per unit. Answer the given
subquestions.
What are the monopolist’s profit-maximizing price and quantity, Q*= …
Published solution
**1. Optimize monopoly profit:** Demand P=25-Q implies MR=25-2Q. With MC=5, \(25-2Q=5\) gives Q=10.
**2. Check the scale:** The demand variable Q is measured in thousands, not individual units.
**3. Convert:** The profit-maximizing physical output is \(10(1000)=10{,}000\) units. The answer field uses physical units, matching the exported key.
**Answer: 10000.**
A monopolist faces the demand curve P=25-Q, where P is measured in dollars per unit and Q in
thousands of units. The monopolist has a constant average cost of $5 per unit. Answer the given
subquestions.
The resulting profit will be, π*= ……..
Published solution
**1. Find the monopoly choice:** MR=25-2Q equals MC=5 at Q=10 thousand units, and demand gives P=15.
**2. Find the unit margin:** \(P-AC=15-5=10\) dollars.
**3. Multiply by physical output:** \[\pi=10(10{,}000)=100{,}000\text{ dollars}.\] Equivalently, revenue is $150,000 and cost $50,000.
**Answer: 100000.**
A monopolist faces the demand curve P=25-Q, where P is measured in dollars per unit and Q in
thousands of units. The monopolist has a constant average cost of $5 per unit. Answer the given
subquestions.
The firm’s Lerner index is, L= ……
Published solution
**1. Define the Lerner index:** \(L=(P-MC)/P\), the price-cost margin divided by price.
**2. Use the optimum:** MR=25-2Q=5 gives Q=10 thousand and P=15, while MC=5.
**3. Calculate:** \[L=(15-5)/15=10/15=2/3\approx0.67.\] This differs from markup as a fraction of cost, which would use a different denominator.
**Answer: 0.67.**
**1. Relate marginal revenue to elasticity:** For a single-price monopolist facing downward-sloping demand, \(MR=P(1-1/|\epsilon|)\).
**2. Examine inelastic demand:** If \(|\epsilon|<1\), then MR is negative. Reducing output raises revenue, and with nonnegative marginal cost it also does not raise production cost.
**3. State the usual result:** An interior profit-maximizing single-price monopolist with nonnegative marginal cost does not operate in the strictly inelastic portion. With zero MC, the unit-elastic boundary can arise; exceptions require different assumptions such as negative marginal cost.
**Answer: B — Selected: The inelastic region of a market demand curve.**
A: **Incorrect:** The optimum ordinarily lies in the elastic region, or at the unit-elastic boundary when MC is zero.
B: **Correct:** Strictly inelastic demand gives negative marginal revenue and cannot be an interior optimum with nonnegative MC.
C: **Incorrect:** The monopolist can produce on part of the demand curve.
D: **Incorrect:** The standard forbidden region is available in B.
Quasi linear preferences are linear in one good and non-linear in the other
good
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U=min{ax, by} represents complements
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U= ax+by represents substitutes
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If a person always chooses three units of one good for one unit of another
good, then they have leontief preferences
Published solution
**1. Recognize the standard forms:** Quasilinear utility is linear in one good. \(\min(ax,by)\), with positive coefficients, represents perfect complements. \(ax+by\), with positive coefficients, represents perfect substitutes.
**2. Examine the inference in D:** Observing a person select a 3-to-1 bundle does not by itself establish L-shaped indifference curves. For example, smooth Cobb-Douglas preferences at fixed relative prices can also generate that demand ratio across incomes.
**3. Identify the unsupported conclusion:** Leontief preferences require goods to be useful in fixed proportions across the relevant preference tradeoffs, not merely an observed bundle ratio without sufficient price-variation information. D is therefore the intended incorrect inference. A genuinely fixed desired proportion under all relevant relative-price changes would instead be consistent with perfect complements.
**Answer: D — Selected: If a person always chooses three units of one good for one unit of another good, then they have leontief preferences.**
A: **Incorrect:** This is the defining quasilinear form, so it is not the incorrect statement sought.
B: **Incorrect:** The minimum form represents complementarity, so do not select it as incorrect.
C: **Incorrect:** The linear form represents perfect substitutes, so do not select it as incorrect.
D: **Correct:** Selected as incorrect: an observed fixed purchase ratio alone is insufficient to establish Leontief preferences.
A person takes more risks that are not known to the life insurance company
because he has life insurance
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A person buys life insurance because he has risky lifestyle that is not known to
the life insurance company
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A person is a risk lover
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Pregnant women with health insurance make more doctor visits than
uninsured pregnant women
Published solution
**1. Define adverse selection:** Before contracting, one side has hidden information about its type or risk, affecting who chooses to participate.
**2. Apply to insurance:** A person who already has a risky lifestyle and buys insurance without the insurer knowing that risk fits hidden-type selection.
**3. Distinguish moral hazard:** Increasing risk because insurance was obtained is a hidden-action/incentive effect after contracting. Risk-loving preferences alone do not establish either information problem.
**Answer: B — Selected: A person buys life insurance because he has risky lifestyle that is not known to the life insurance company.**
A: **Incorrect:** Changing behavior because of insurance describes moral hazard.
B: **Correct:** Pre-existing undisclosed risk that affects purchase is adverse selection.
C: **Incorrect:** Risk attitude alone does not establish hidden-type selection.
D: **Incorrect:** More use of covered care after insurance is not the defining hidden-type example here.
Moral hazard occurs when contracts are written in such a way that
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The interests of agent and principal conve
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The interests of agent and principal diverge
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Agents will wish to maximize the principal’s utility
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Production and risk bearing efficiency are achieved
Published solution
**1. Define the principal-agent issue:** A principal delegates an action to an agent. Moral hazard arises when the agent’s relevant behavior is not fully observed/contractible and incentives are misaligned.
**2. Match the offered condition:** Divergent interests can motivate the agent to take hidden actions favoring their own payoff over the principal’s objectives.
**3. State the qualification:** Divergence alone is not a complete definition; limited observability or enforceability is also needed. Among the options, B identifies the relevant incentive conflict rather than aligned interests or achieved efficiency.
**Answer: B — Selected: The interests of agent and principal diverge.**
A: **Incorrect:** Aligned interests reduce the stated incentive-conflict problem.
B: **Correct:** Misaligned interests, together with hidden or noncontractible actions, underlie moral hazard.
C: **Incorrect:** An agent maximizing the principal’s utility does not describe the harmful incentive problem.
D: **Incorrect:** Achieved efficiency is not the cause of moral hazard.
Which of the following best describes the free rider problem?
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When individuals contribute more than their fair share to a public good
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When individuals benefit from public good without contributing to its cost
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When firms charge excessively high prices for public goods
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When governments provide too many private goods.
Published solution
**1. Identify the public-good feature:** With non-excludability, people can benefit even if they do not contribute to provision.
**2. Explain the incentive:** Each person may prefer that others pay while they still enjoy the benefit. If many behave this way, voluntary funding can be insufficient.
**3. Name the problem:** Benefiting without contributing to cost is free riding. It is about contribution incentives, rather than simply high prices or the amount of private goods supplied.
**Answer: B — Selected: When individuals benefit from public good without contributing to its cost.**
A: **Incorrect:** Overcontributing is the opposite of the free-riding incentive.
B: **Correct:** Enjoying the benefit while avoiding its cost is the defining behavior.
C: **Incorrect:** High prices alone do not define a free rider.
D: **Incorrect:** The issue is funding a non-excludable benefit, not excessive private-good provision.
**1. Compare one more dollar of advertising:** An extra advertising dollar costs $1 and changes sales quantity by the supplied marginal amount delta Q.
**2. Value the additional sales:** Each extra unit contributes price minus marginal cost, P-MC. The marginal profit contribution before advertising cost is \((P-MC)\Delta Q\).
**3. Apply the interior optimum condition:** Set marginal benefit equal to the one-dollar marginal advertising cost: \[(P-MC)\Delta Q=1.\] This assumes the usual smooth interior choice; at a corner an inequality can replace equality.
**Answer: 1.**
