Question

XL Corp has estimated its demand and cost functions to be as follows:
P = 60 - 0:2Q
C = 200 + 4Q +1.2Q 2
where Q is in units, P is in $ and C is in $.
a. Calculate the profit-maximizing price and output.
b. Calculate the size of the profit.
c. Calculate the price elasticity of demand at the above price.
d. If there is a $14 tax placed on the good, so that the producer has to pay the government $14 for every unit sold, calculate the new profit maximizing price and output.
e. What would happen to profit if the firm tried to pass on all the tax to the consumer in the form of a higher price?
f. If fixed costs rise by $200 how would this affect the firm’s situation?

Answer 1

a. Calculate the profit-maximizing price and output.

The profit maximizing price is computed using the following steps:

First find the total revenue function:

$\text{Revenue=Price}\times \text{Quantity}$

$\text{Revenue}=(60 - 0.2Q)Q$

$\text{Revenue}=60Q - 0.2Q^2$

The cost function is as follows:

$\text{Cost}=200 + 4Q +1.2Q 2$

The profit function will be :

$\text{Profit function}=60Q - 0.2Q^2-(200 + 4Q +1.2Q^2 )$

$\text{Profit function}=56Q - 1.4Q^2-200$

Find the first derivative of the profit function:

$\dfrac{\Delta \text{Profit function}}{\Delta Q}=56-2.8Q$

Equate the marginal revenue to be equal to 0 and solve for Q.

$56-2.8Q=0$

$Q=\dfrac{56}{2.8}=20$

The revenue maximizing quantity is 150 units.

The revenue maximizing price is:

$P = 60 - 0.2Q$

$P = 60 - 0.2*20=56$

b. Calculate the size of the profit.

$\text{Profit function}=56Q - 1.4Q^2-200$

$\text{Profit function}=56(20) - 1.4(20^2)-200$

$\text{Profit function}=360$

c. Calculate the price elasticity of demand at the above price.

$P = 60 - 0.2Q$

$Q=300-5P$

$\dfrac{\Delta Q}{\Delta p}=-5$

$\text{Elasticity of price}=-5 \times \dfrac{56}{20}=-14$

d. If there is a $14 tax placed on the good, so that the producer has to pay the government $14 for every unit sold, calculate the new profit maximizing price and output.

$\text{Profit function}=60Q - 0.2Q^2-(200 + 4Q +1.2Q^2 +14Q)$

$\text{Profit function}=42Q-1.4Q^2-200$

$\dfrac{\Delta \text{Profit function}}{\Delta Q}=42-2.8Q$

$42-2.8Q=0$

$Q=\dfrac{42}{2.8}=15$

Profit maximizing quantity is 15 units.

$P = 60 - 0.2*15=57$

The price maximizing profit is $57.

e. What would happen to profit if the firm tried to pass on all the tax to the consumer in the form of a higher price?

As a result of the increase in the price of the the quantity sold will decrease leading to decrease in the level of profits too.

f. If fixed costs rise by $200 how would this affect the firm’s situation?

The fixed cost is the cost that is not related to the level of the output. As a result once it is incurred, the firm will not incur it again in the future. Therefore, the profit will reduce by a similar amount as the increase in the fixed cost.