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A competitive firm sells its product at a price of $0.10 per unit.  Its total and marginal cost functions are:
   TC = 5 - 0.5Q + 0.001Q2
   MC = -0.5 + 0.002Q,       
where TC is total cost ($) and Q is output rate (units per time period).

a.   Determine the output rate that maximizes profit or minimizes losses in the shortterm.
b.   If input prices increase and cause the cost functions to become
    TC = 5 - 0.10Q + 0.002Q2
    MC = -0.10 + 0.004Q,
what will the new equilibrium output rate be?  Explain what happened to the profit maximizing output rate when input prices were increased.

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Thanks, very pleased with your answer

a) For competitive firm at equilibrium, P (Price) = MC (Marginal Cost) occur. So,

=> 0.10 = -0.5 + 0.002Q

=> 0.10 + 0.5 = 0.002Q

=> 0.6 = 0.002Q

=> Q = 0.6 / 0.002

=> Q = 300

TR (Total Revenue) = P * Q = 0.10 * 300

=> TR = 30

TC = 5 - (0.5 * 300) + 0.001 * (300)^2 = 5 - 150 + 90 

=> TC = - 55

Profit = TR - TC = 30 - (- 55) = 30 + 55

=> Profit = $85

Therefore, here the profit maximizing output level is, Q = 300 and profit is $85.

b) For competitive firm at equilibrium, P (Price) = MC (Marginal Cost) occur. So,

=> 0.10 =  - 0.10 + 0.004Q

=> 0.10 + 0.10 = 0.004Q

=> 0.20 = 0.004Q

=> Q = 0.20 / 0.004

=> Q = 50

TR = P * Q = 0.10 * 50 = 5

TC =  5 - (0.10 * 50) + 0.002 * (50)^2 = 5 - 5 + 5 = 5

Profit = TR - TC = 5 - 5 = $0.

Therefore, here the new equilibrium output level is, Q = 50 and profit is $0.

Here by comparing (a) and (b)'s profit maximizing output level and profit level we can see that after increase in input price the profit maximizing output level has decreased from 300 to 50 and the profit level falls from $85 to $0.