Q18) 3) (450 points; 25 minutes) In the competitive market for medical scanning devices, there are 400 teaching hospitals in the US who would buy at most one device

Market demand in units is given by: Qd=1250 - (1/4)P up to the constraint that at most 400 units will be sold (i.e. Qd < 400). Market supply is given by: Qs= -3000 + P.

 (a) What is the equilibrium price and quantity of devices sold?

(b) The government seeks to restrain healthcare spending by taxing teaching hospitals $1000 per machine bought. What is the equilibrium quantity of devices sold, and what price do firms receive? Is there any deadweight loss, and, if so, how much?

(c) If, instead of imposing a tax, the government provided a subsidy, would there be any deadweight loss? (No calculation is necessary to answer this question.)

4. (450 points; 25 minutes) In a small country, Gelbium, yearly demand for mail delivery is Q=110,000- 10,000 p. where p is expressed in Gelbium Marks (GM) and Q is expressed as the number of pieces of mail. The marginal cost of delivering a piece of mail is constant and equals 1 Gelbium Mark. There is no fixed cost.

(a) Suppose the government runs the mail service. If the government wants to maximize the social surplus, how much should it charge for mail delivery?

(b) Suppose the government privatizes mail delivery. It sells the monopoly right to deliver mail to a commercial firm. The firm is free to set its price. How much will the firm charge for delivering a piece of mail? What is the maximum (yearly) amount the government can charge for the monopoly rights? What is the effect of this privatization on total welfare?

(c) Assume that the government, after having sold the monopoly rights, gives a subsidy of 2 Gelbium Marks per piece of mail that the firm delivers. What will be the impact of this subsidy on consumer surplus and on firm profits (relative to unsubsidized monopoly)? What will be the welfare effect?

 

Detailed solution is given below.

Step-by-step explanation

Ans (3) Given Supply Q_s = -3000 + P

From that we get P = 3000 + Q_s

Given Demand Q_d = 1250 - 0.25P

From that we get P = 5000 - Q_d

The max demand Q_{d_max} can be 400.

Using this information above, we can graph the market as:-

 

(a) What is the equilibrium price and quantity of devices sold?

At equilibrium, the market demand is equal to the market supply Q_d = Q_s

→ -3000 + P = 1250 - 0.25P

→ 1.25P = 4250

→ P* = $3400

Substituting this value in Q_d, we get:-

Q* = -0.25P + 1250 = 400 units.

Ans (a) The equilibrium price is of $3400 and the equilibrium quantity is of 400 medical scanners.

 

(b) The government seeks to restrain healthcare spending by taxing teaching hospitals $1000 per machine bought. What is the equilibrium quantity of devices sold, and what price do firms receive? Is there any deadweight loss, and, if so, how much?

→ Due to a tax imposition of $1000 by the government, the buyer now has to pay a price P_t = (P+ 1000)

Quantity of supply remains unchanged. We can find the new equilibrium from the relation Q_d = Q_s

→ -3000 + P = 1250 - 0.25P_t

→ -3000 + P = 1250 - 0.25(P+ 1000)

→ 1.25P = 4000

→ P* = $3200

Substituting this value in Q_s, we get:-

Q* = -3000 + 3200 = 200 units.

In order to calculate the DWL, we need to graph the shift in the supply curve.

 

Area of this DWL triangle gives us the required DWL due to the imposition of tax.

Area of DWL triangle = 0.5*(400-200)*(4200-3200) = $100,000

Ans (b) The new equilibrium price is of $3200 and the new equilibrium quantity is of 200 medical scanners. The total deadweight loss due to the imposition of tax is of $100,000.

 

(c) If, instead of imposing a tax, the government provided a subsidy, would there be any deadweight loss?

→ Incase of a subsidy of $1000, quantity of demand remains the same. The entire subsidy would go to the benefit of the buyers. So, there would be no deadweight loss.

 

Ans (4) Given Demand Q_d = 110000 - 1000P. Marginal Cost MC = 1 GM.

From this, we get the inverse demand function which gives us the price as P = 11 - 0.0001Q

(a) Suppose the government runs the mail service. If the government wants to maximize the social surplus, how much should it charge for mail delivery?

→ In order to maximize the social surplus, the price equals the MC. i.e. Price P = 1 GM.

This occurs at a quantity of Q = 110000 - 1000*1 = 100,000 units

Total Surplus at this point = 0.5*(10*100,000) = 500,000GM

Ans (a) The government should charge a price of 1 GM for mail delivery if it wants to maximize the social surplus.

 

(b) Suppose the government privatizes mail delivery. It sells the monopoly right to deliver mail to a commercial firm. The firm is free to set its price. How much will the firm charge for delivering a piece of mail? What is the maximum (yearly) amount the government can charge for the monopoly rights? What is the effect of this privatization on total welfare?

→ We know, Total Revenue TR = P*Q = (11 - 0.0001Q)Q = 11Q - 0.0001Q²

Marginal Revenue MR = ?dQd??(TR) = ?dQd??(11Q - 0.0001Q²) = 11 - 0.0002Q

For optimal quantity, MR = MC

→ 11 - 0.0002Q = 1

→ Q = 50,000 units

Hence, P = 11 - 0.0001Q = 11 - 0.0001*50000 = 11 - 5 = 6 GM

Hence, Profit = 50,000(6-1) = 250,000GM

This profit is the max amount the government should charge for giving monopoly rights to any firm.

We get consumer welfare = 0.5*(5*50,000) = 125,000GM

Total Surplus = 250,000+125,000 = 375,000GM

Hence, Welfare loss = Old surplus(Calculated in part a) - New surplus = 500,000 - 375,000 = 125,000GM

Ans (b) The government should charge a price of 1GM for delivering a piece of mail. The maximum yearly amount the the government should charge for selling monopoly rights is 250,000GM. There is a welfare loss of 125,000GM incase the government wants to privatize.

 

(c) Assume that the government, after having sold the monopoly rights, gives a subsidy of 2 Gelbium Marks per piece of mail that the firm delivers. What will be the impact of this subsidy on consumer surplus and on firm profits (relative to unsubsidized monopoly)? What will be the welfare effect?

→ If the government gives a subsidy of 2GM after selling the monopoly rights, this firms MC would drop by the same amount,

Hence, new MC = 1 - 2 = -1GM

For optimal quantity, MR = MC

→ 11 - 0.0002Q = -1

→ Q = 60,000 units

Hence, P = 11 - 0.0001Q = 11 - 0.0001*60000 = 11 - 6 = 5GM

Hence, Profit = 50,000(5-(-1)) = 60,000*6 = 360,000GM

New consumer surplus = 0.5*(6*60000) = 180,000GM

Government Subsidy = 2*60,000 = 120,000GM

New Total Surplus = (360,000 + 180,000 - 120,000)GM = 420,000GM

 

Increase in profits = 360,000 - 250,000 = 110,000GM

Increase in Consumer Surplus = 180,000 - 125,000 = 55,000GM

Increase in Total surplus = 420,000-375,000 = 45,000GM

Ans (c) The impact of a government subsidy of 2GM would increase consumer surplus by 55,000GM and profits by 110,000GM. The total welfare effect is an increase of 45,000GM.

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