Question

A medical device manufacturer sells its sterilization equipment to consumers 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) Suppose instead the law calls for a 20% tax on a firm's total revenue and leaves their marginal cost unchanged, what are the profit- maximizing price and quantity under this scenario?

Answer 1

Answer

A) Price = $5,000 per unit ; Quantity = 2.5 thousand units.

The inverse demand curve, the medical device manufacturer faces is as follows,

P = 6,000 – 400Q.......(1) , where Q is the number of sterilizers in thousands and, P is the price per unit.

The marginal cost(MC) of production is constant at $4,000

MC = $4,000.......(2)

The firm maximizes its profit at the quantity of output at which its marginal revenue and marginal cost are same.

Multiplying equation(1) by 'Q', we get,

P * Q = 6,000 * Q - 400Q * Q

PQ = 6,000Q - 400

Or, TR = 6,000Q - 400 , where 'TR' = Total Revenue

Now, differentiating TR with respect to Q, we get,

d(TR) / dQ = d(6,000Q - 400 ) / dQ

Or, MR = 6,000 - 800Q, where MR = Marginal revenue = Change in TR / Change of an additional output sold

Now, for profit maximization, MR = MC.

  6,000 - 800Q = 4,000

Or, - 800Q = 4,000 - 6,000

Or, - 800Q = - 2,000

Or, 800Q = 2,000

Or, Q = 2000 / 800

Or, Q = 2.5

The firm produces the profit-maximizing level of output of 2.5 thousand units.

Now, putting the value of Q in equation(1), we get,

P = 6,000 – 400 * 2.5

Or, P = 6,000 - 1,000

Or, P = 5,000

The price, the firm charges for its profit-maximizing level of output is, $5,000 per unit.

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B) Price = $5,200 per unit ; Quantity = 2 thousand units.

The tax raises the marginal cost of production from $4,000 to $4,400.

MC = $4,400

The demand curve remains same. So, the marginal revenue is same as before.

For profit maximization, MR = MC.

  6,000 - 800Q = 4,400

Or, - 800Q = 4,400 - 6,000

Or, - 800Q = - 1,600

Or, 800Q = 1,600

Or, Q = 1600 / 800

Or, Q = 2

The firm will now produce the profit-maximizing level of output of 2 thousand units.

Now, putting the value of Q in equation(1), we get,

P = 6,000 – 400 * 2

Or, P = 6,000 - 800

Or, P = 5,200

The price, the firm will now charge for its present profit-maximizing level of output is, $5,200 per unit.

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C) Price = $5,500 per unit ; Quantity = 1.25 thousand units.

The profit(π) of a firm is the difference between total revenue(TR) and total cost(TC).

  π = TR - TC

When 20% tax is imposed on the firm's total revenue, the firn's revenue decreases by 20% of its revenue. So, the profit will also decrease by 20% of TR.

The profit after the tax(π1) on revenue is as follows;

π1 = TR - TC - 20% * TR

Or,  π1 = (1- 20%)TR - TC

Differentiating the above equation with respect to Q, we get,

d(π1) / dQ = (1- 20%) * d(TR) / dQ - d(TC) / dQ

Or, d(π1) / dQ = (1- 20%)MR - MC

Now, profit is maximized when, d(π1) / dQ = 0

At the profit-maximizing quantity,

(1- 20%)MR - MC = 0

Or,   (1- 20%)MR = MC

Now, putting the value of MR and MC in the above equation, we get,

(1- 20%) * (6,000 - 800Q) = 4,000

Or, 0.8 *(6,000 - 800Q) = 4,000

Or, 4,800 - 640Q = 4,000

Or,  - 640Q = 4,000 - 4,800

Or,   - 640Q = - 800

Or, Q = 1.25

After the tax is imposed on total revenue, the firm will produce the profit-maximizing level of output of 1.25 thousand units.

Now, putting the value of Q in equation(1), we get,

P = 6,000 – 400 * (1.25)

Or, P = 6,000 - 500

Or, P = 5,500

After the tax is imposed on total revenue, the price, the firm will charge for its profit-maximizing level of output is, $5,500 per unit.

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