Question

My company sells California t-shirts to Bay Area tourists, and this is a perfectly competitive market. My cost function is given by ?(?) = ? ? ? ? .

a) Find my profit-maximizing quantity produced if the market price for my California t-shirts is $25. What would my profits be?

b) The COVID-19 pandemic reduces willingness and ability to travel and visit the Bay Area. At the same time, t-shirt companies have to pay for personal protective equipment for their workers and hire more staff to make sure that customers aren’t entering shops without masks on. Use the model of supply and demand to make a prediction about expected changes in price and quantity in this market due to these changes.

c) As the economy slips into recession a significant amount of uncertainty is introduced. Suppose that now there is a 20% chance that the recession is deep and the market price falls to $10 per shirt, otherwise the recession is relatively mild and the price falls to just $20 per shirt. Given this uncertainty, what should be the firm’s profit-maximizing level of output? (assume their cost function is still ?(?) = 1 4 ? 2 )

Answer 1

a) Let the profit maximizing quantity be q

So Total revenue = 25q

Total cost = 14q^2

Total Profit Pr= 25q - 14q^2

At Maximum Profit, d/dq Pr = 0

or d/dq ( 25q - 14q^2) = 0

or 25 - 28q = 0

or q = 0.89

So the profit maximizing quantity is 0.89.

The maximum profit is = 25 * 0.89 - 14 * 0.89^2 = 11.16

b) The demand falls for the t-shirt since fewer people are visiting. Simultaneously the cost rises because of the additional manpower involved and PPE to be made. SSo the demand curve as well as the supply curve shifts to the left. As a result of this the equilibrium quantity will definitely fall. The price may rise on fall depending on the relative movement of the 2 curves. If the supply curve shifts more than the demand curve, then the price will rise but if the supply curve shifts less than the demand curve then the price will fall.

c) If the price falls to $10, profit = 10q - 14q^2

If the price falls to $20, profit = 20q - 14q^2

So considering 20% probability that recession is strong, we have expected profit = 0.2 * ( 10q - 14q^2 ) + 0.8 * (20q - 14q^2)

Pr = 2q - 2.8q^2 + 16q - 11.2q^2 = 18q - 14q^2

At profit maximizing situation,

d/dq (18q - 14q^2) = 0

or 18 - 28q = 0

or q = 0.643

So the firm's profit maximizing quantity should be 0.643.

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