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Question 2 (15 points) Consider the same market characterized by demand $Q=144-...
Jan 23, 2025
Solution
a
To find the equilibrium price and total quantity for the monopoly case, we first need to determine the marginal cost (MC) and marginal revenue (MR). The cost function is given by C(q)=32q2+3q+272 C(q) = \frac{3}{2} q^{2} + 3q + \frac{27}{2} . The marginal cost is calculated as MC=dCdq=3q+3 MC = \frac{dC}{dq} = 3q + 3 . The demand function is Q=144−3P Q = 144 - 3P , which can be rearranged to find the inverse demand function: P=48−13Q P = 48 - \frac{1}{3}Q . The total revenue (TR) is TR=P⋅Q=(48−13Q)Q=48Q−13Q2 TR = P \cdot Q = (48 - \frac{1}{3}Q)Q = 48Q - \frac{1}{3}Q^{2} . The marginal revenue is MR=dTRdQ=48−23Q MR = \frac{dTR}{dQ} = 48 - \frac{2}{3}Q . Setting MR=MC MR = MC gives us 48−23Q=3Q+3 48 - \frac{2}{3}Q = 3Q + 3 . Solving for Q Q yields Q=30 Q = 30 . Substituting Q Q back into the demand function gives P=48−13(30)=42 P = 48 - \frac{1}{3}(30) = 42 . Thus, the equilibrium price is P=42 P = 42 and total quantity is Q=30 Q = 30
b
The monopolist's markup can be calculated as the difference between the price and marginal cost at equilibrium. First, we find the marginal cost at Q=30 Q = 30 : MC=3(30)+3=93 MC = 3(30) + 3 = 93 . The price at equilibrium is P=42 P = 42 . The markup is given by Markup=P−MC=42−93=−51 \text{Markup} = P - MC = 42 - 93 = -51 . To find the percentage of marginal costs, we use the formula Markup %=P−MCMC×100% \text{Markup \%} = \frac{P - MC}{MC} \times 100\% . Substituting the values gives Markup %=42−9393×100%≈−54.84% \text{Markup \%} = \frac{42 - 93}{93} \times 100\% \approx -54.84\%
c
Consumer surplus (CS) is calculated as the area of the triangle formed by the demand curve above the price level up to the quantity sold. The formula for consumer surplus is CS=12×Base×Height CS = \frac{1}{2} \times \text{Base} \times \text{Height} . The base is the quantity sold, Q=30 Q = 30 , and the height is the difference between the maximum price consumers are willing to pay (when Q=0 Q = 0 , P=48 P = 48 ) and the equilibrium price P=42 P = 42 . Thus, the height is 48−42=6 48 - 42 = 6 . Therefore, CS=12×30×6=90 CS = \frac{1}{2} \times 30 \times 6 = 90
Answer
Equilibrium Price: 4242, Total Quantity: 3030; Markup: approximately −54.84%-54.84\%; Consumer Surplus: 9090
Key Concept
Monopoly equilibrium involves determining the price and quantity where the monopolist maximizes profit, considering marginal costs and revenues.
Explanation
The solution outlines the steps to find the equilibrium price, quantity, markup, and consumer surplus in a monopoly market, illustrating the relationship between demand, marginal cost, and consumer welfare.
Solution
a
To determine the values of F F for which the firm will enter the market, we need to find the break-even point where total revenue equals total cost. The inverse demand function is given by p(Q)=30−Q2 p(Q) = 30 - \frac{Q}{2} . The total revenue TR TR is TR=p(Q)⋅Q=(30−Q2)Q=30Q−Q22 TR = p(Q) \cdot Q = \left(30 - \frac{Q}{2}\right)Q = 30Q - \frac{Q^2}{2} . The total cost TC TC is TC=C(Q)=Q2+F TC = C(Q) = Q^2 + F . Setting TR=TC TR = TC gives us 30Q−Q22=Q2+F 30Q - \frac{Q^2}{2} = Q^2 + F . Rearranging leads to 3Q22−30Q+F=0 \frac{3Q^2}{2} - 30Q + F = 0 . The firm will enter if the discriminant of this quadratic equation is non-negative, which leads to the condition F≤225 F \leq 225
b
For F=145 F = 145 , we first find the monopolist's output by maximizing profit. The marginal revenue MR MR is derived from the total revenue function: MR=30−Q MR = 30 - Q . The marginal cost MC MC is MC=dCdQ=2Q MC = \frac{dC}{dQ} = 2Q . Setting MR=MC MR = MC gives 30−Q=2Q 30 - Q = 2Q or Q=10 Q = 10 . The price at this quantity is p(10)=30−102=25 p(10) = 30 - \frac{10}{2} = 25 . Total welfare is the sum of consumer surplus and producer surplus. Consumer surplus is 12×(30−25)×10=25 \frac{1}{2} \times (30 - 25) \times 10 = 25 and producer surplus is (25−(102+145))×10=−45 (25 - (10^2 + 145)) \times 10 = -45 . Thus, total market welfare is 25−45=−20 25 - 45 = -20
c
The welfare gains of the monopoly can be compared to a perfectly competitive market where price equals marginal cost. In perfect competition, MC=2Q MC = 2Q leads to Q=15 Q = 15 and p(15)=22.5 p(15) = 22.5 . Consumer surplus in perfect competition is 12×(30−22.5)×15=56.25 \frac{1}{2} \times (30 - 22.5) \times 15 = 56.25 and producer surplus is (22.5−(152+145))×15=−45 (22.5 - (15^2 + 145)) \times 15 = -45 . Total welfare in perfect competition is 56.25−45=11.25 56.25 - 45 = 11.25 . Comparing the two, the monopoly results in a total welfare of -20, while perfect competition yields 11.25, indicating that the market does not benefit from the firm operating as a monopoly
Answer
a: F≤225 F \leq 225 ; b: Total market welfare = -20; c: Monopoly welfare is worse than perfect competition (11.25).
Key Concept
Market entry conditions and welfare analysis in monopoly vs. perfect competition
Explanation
The analysis shows that the firm will enter if fixed costs are low enough, and that monopoly leads to lower total welfare compared to perfect competition.
Question 4 Solution
a
To find the equilibrium price and quantity for the monopoly, we first need to determine the marginal cost (MC) and marginal revenue (MR). The cost function is given by C(q)=200+20q+0.02q2 C(q) = 200 + 20q + 0.02q^2 . The marginal cost is calculated as MC=dCdq=20+0.04q MC = \frac{dC}{dq} = 20 + 0.04q . The demand function is given by P(Q)=100−0.02Q P(Q) = 100 - 0.02Q . To find MR, we first express Q Q in terms of P P : Q=500−50P Q = 500 - 50P . Then, TR=P⋅Q=P(500−50P) TR = P \cdot Q = P(500 - 50P) and MR=d(TR)dQ=100−0.04Q MR = \frac{d(TR)}{dQ} = 100 - 0.04Q . Setting MR=MC MR = MC gives us the equilibrium quantity. Solving 100−0.04Q=20+0.04Q 100 - 0.04Q = 20 + 0.04Q leads to Q=1,000 Q = 1,000 and substituting back gives P=0 P = 0 . Thus, equilibrium price is P=0 P = 0 and quantity is Q=1,000 Q = 1,000
b
The price elasticity of demand at the market equilibrium can be calculated using the formula Ed=dQdP⋅PQ E_d = \frac{dQ}{dP} \cdot \frac{P}{Q} . From the demand function P(Q)=100−0.02Q P(Q) = 100 - 0.02Q , we find dQdP=−50 \frac{dQ}{dP} = -50 . At equilibrium P=0 P = 0 and Q=1,000 Q = 1,000 , substituting these values gives Ed=−50⋅01,000=0 E_d = -50 \cdot \frac{0}{1,000} = 0 . Thus, the price elasticity of demand is 0 0
c
The monopolist's markup can be calculated using the formula Markup=P−MC \text{Markup} = P - MC . From part (a), we found P=0 P = 0 and MC=20+0.04(1,000)=60 MC = 20 + 0.04(1,000) = 60 . Therefore, the markup is 0−60=−60 0 - 60 = -60
d
The Lerner Index is calculated as L=P−MCP L = \frac{P - MC}{P} . Substituting the values from part (a) gives L=0−600 L = \frac{0 - 60}{0} , which is undefined since we cannot divide by zero
Answer
Equilibrium Price: 00, Quantity: 1,0001,000; Price Elasticity: 00; Markup: −60-60; Lerner Index: Undefined
Key Concept
Monopoly equilibrium involves setting marginal revenue equal to marginal cost to determine price and quantity.
Explanation
The calculations show that in this scenario, the monopolist sets a price of 0,leadingtoaquantityof0, leading to a quantity of 1,000, with a price elasticity of demand of $0, indicating perfectly inelastic demand at this price. The negative markup indicates that the monopolist is not covering its costs.
Question 1 Solution
a
To find the equilibrium price and quantity in a perfectly competitive market, we set the demand equal to the total supply. The demand function is given by D(p)=144−3P D(p) = 144 - 3P . The total cost function for each firm is C(q)=32q2+3q+272 C(q) = \frac{3}{2}q^2 + 3q + \frac{27}{2} . First, we calculate the marginal cost (MC): MC=dCdq=3q+3 MC = \frac{dC}{dq} = 3q + 3 . In perfect competition, P=MC P = MC . Setting P P from the demand function equal to MC MC gives us the equilibrium price and quantity. Solving these equations yields P=30 P = 30 and total quantity Q=144−3(30)=54 Q = 144 - 3(30) = 54 . Since there are 6 firms, each firm produces q=Q6=9 q = \frac{Q}{6} = 9
b
In the long-run equilibrium, firms can enter or exit the market until economic profits are zero. The long-run equilibrium price will equal the minimum average cost (AC). The average cost is calculated as AC=C(q)q=32q2+3q+272q=32q+3+272q AC = \frac{C(q)}{q} = \frac{\frac{3}{2}q^2 + 3q + \frac{27}{2}}{q} = \frac{3}{2}q + 3 + \frac{27}{2q} . Setting P=AC P = AC and solving for q q gives us the long-run equilibrium output per firm, and thus the long-run equilibrium price will be P=30 P = 30 as well, with total quantity adjusting accordingly
Answer
Equilibrium Price: 3030, Total Quantity: 5454, Long-run Price: 3030
Key Concept
In a perfectly competitive market, equilibrium occurs where supply equals demand, and in the long run, firms enter or exit until economic profits are zero.
Explanation
The equilibrium price and quantity are determined by the intersection of the demand and supply curves, and in the long run, firms adjust their output to ensure no economic profits exist.
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