Answer to Question #186699 in Microeconomics for Ebenezer Addo

Question

You are the manager of a monopolistically competitive firm and your demand and cost functions are given by Q=20-2P and C(Q)=104-14Q+Q*2

i)find the inverse demand function

ii) determine the fixed cost and the variable cost of the total cost function above

iii)determine the profit maximizing price and level of production

iv) calculate the firms maximum profits

v)estimate the elasticity of the monopolist`s product

Answer 1

i) find the inverse demand function

Market demand is given as:

$Q=20-2P$

The inverse demand curve is:

$\boxed{\color{red}{P=10-0.5Q}}$

ii) determine the fixed cost and the variable cost of the total cost function above

Total cost function is given as:

$C=104-14Q+Q^2$

The fixed cost is:

$\boxed{\color{red}{FC=104}}$

The variable cost is:

$\boxed{\color{red}{VC=-14Q+Q^2}}$

iii)determine the profit-maximizing price and level of production

The marginal revenue is equal to:

$MR=10-Q$

The marginal cost is:

$MC=-14+2Q$

Setting MR=MC, we get:

$10-Q=-14+2Q\\[0.3cm] 24=3Q\\[0.3cm] \boxed{\color{red}{Q^*=8}}$

The optimal price is:

$P^*=10-0.5(8)\\[0.3cm] \boxed{\color{red}{P^*=\$4}}$

iv) calculate the firms maximum profits

Total cost at optimal production is:

$C=104-14(8)+8^2\\[0.3cm] C=\$56$

The total revenue is equal to:

$TR=\$4\times 8=\$32$

Profit is equal to:

$\rm Profit=TR-C\\[0.3cm] \rm Profit=\$32-\$56=\boxed{\color{red}{-\$24}}$

v)estimate the elasticity of the monopolist`s product

$MR=P\left(1+\dfrac{1}{e}\right)$

Marginal revenue at the optimal quantity is:

$MR=10-8=\$2$

Therefore:

$\\[0.3cm]$ $2=4\left(1+\dfrac{1}{e}\right)\\[0.4cm] 0.5=\left(1+\dfrac{1}{e}\right)\\[0.4cm] -0.5=\dfrac{1}{e}\\[0.4cm] e=\boxed{\color{red}{-2}}$