Question

A monopoly firm faces a demand curve given by the following equation - P = $500 - 10Q, where Q equals quantity per day. Its marginal cost curve is MC = $100 per day. Assume that the firm faces no fixed cost. Please answer each question mathematically.

a). Now suppose a tax of $100 per unit is imposed. How will this affect the firms price?

b). How would a $100 per unit tax affect the firms profit maximizing output per day?

c). How would the $100 per unit tax affect the firms profit per day?

Answer 1

Answer:-

Given That:-

A monopoly firm faces a demand curve given by the following equation - P = $500 - 10Q, where Q equals quantity per day. Its marginal cost curve is MC = $100 per day. Assume that the firm faces no fixed cost. Please answer each question mathematically.

Demand Curve eqn P = $500 - 10Q,.

Q = quantity sold per day

X10 fixed cost

Sol:-

Total Revenue (TR) =

= 500 Q - 10 Q²

^{Marginal Revenue (MR) = First derivative of TR}

⁼

^{= 500 - 20 Q}

^{* MR = MC}

^{500 - 20 Q = 100 Q = 400/20 = 20}

^{Price = P = 500 - 200 = $ 300}

^(a)

^{$100 per day tax - Impact on price}

Advalorem taxes : are charged as a proportion of the price

eg: VAT

Two types of prices

firm price

consumer price

Relation between firm price & consumer price

= tax(Advaloremt)

P_c = consumed price

P_f = firm price

= 33.33 %

1.333 Pf = 500 - 10Q

P_f = 375 - 7.5Q

TR = 375 Q - 7.5 Q²

^{MR = 375 - 15 Q}

MR = MC

375 - 15 Q = 100

275 = 15 Q Q = 275/15 = 18.33

then price = 375 - 7.5(18.33)

= 375 - 137.5

= 237.5

The Affect of &100 per day tax is that price of firm decreases to $ 237.5 from $ 300

(b)

Specific taxes : Charged on output here consumer demand function remains constant ie

P_c = $ 500 - 10 Q but

P_f = $ 500 - 10Q - y y = specific tax

P_f = $ 500 - 10Q - 100/20

P_f = 495 - 10Q

TR = 495 Q - 10 Q²

^{MR = 495 - 20 Q}

^{MR = MC}

^{495 - 20 Q = 100}

^{395 = 20 Q}

^{Q = 395/20 = 19.75}

^{So the firm produces output of 19.75 units}

^(c)

^{Affect on profit}

^{We see that with $100 per day tax affect the price from $ 300 - 237.5}

output from $ 20 units - 19.75 units

Profit decreases to the firm.