Question

A medical device manufacturer sells its sterilization equipment in a market with an inverse demand curve of P = 6,000 – 400Q, where Q measures the number of sterilizers in thousands and P is the price per unit. The marginal cost of production is constant at $4,000. a. Solve for the profit-maximizing price and quantity. b. The Patient Protection and Affordable Care Act signed into law by President Barack Obama levies a tax on medical devices. Suppose the tax raises the marginal cost of production from $4,000 to $4,400. What are the new profit-maximizing price and quantity? c. The law calls for a 2.3% tax on a firm's total revenue, which leaves the marginal cost of production unchanged. What are the profit-maximizing price and quantity under this scenario?

Answer 1

a) given P (AR) = 6000-400Q

TR = AR*Q   = 6000Q-400 [Q^{^{2}}]

MR = dTR/dQ = 6000 - 800Q

given MC= $4,000

Equilibrium condition is MR=MC

6000-800Q = 4000

Q= 2000/800 = 2.5 (equilibrium quantity)

if you substitute this value in the given equation, P = 6000-400Q = 6000-400(2.5) = $5,000 (equilibrium price)

b)

given P (AR) = 6000-400Q

TR = AR*Q   = 6000Q-400 [Q^{^{2}}]

MR = dTR/dQ = 6000 - 800Q

now, the new MC= $4,400

Equilibrium condition is MR=MC

6000-800Q = 4400

Q= 1600/800 = 2 (equilibrium quantity)

if you substitute this value in the given equation, P = 6000-400Q = 6000-400(2) = $5,200 (equilibrium price)

3) After 2.3% tax, the total revenue is, TR = AR*Q   = .977 * (6000Q-400 [Q^{^{2}}] )

MR= .977 * (6000 - 800Q)

MC= 4000

then MR=MC means 5862- 781.6Q = 4000

Q= 1862/781.6 = 2.38 (rounded) ----- equlibrium quantity

Equilibrium Price: 6000-800Q = 6000-800*2.38 = 6000-1904= 4,096