Question

Think about a monopolist, the market demand function is: Q_D = 100/P², the monopolist’s cost function is: C(Q) = 2Q.

1. Based on the definition of price elasticity of demand: e_D,P= (dQ_D/Q_D​)/(dP/P), what is the price elasticity of the market demand curve in this case? 
2. What is the monopolist’s optimal price based on the inverse elasticity rule? 
3. If the government decide to impose a $1 per unit tax on the output, please use the inverse elasticity rule to decide if the monopolist’s optimal price will increase more than $1 after the tax?

Answer 1

Task #279342

Solution:

(1).

$Qd=\frac{100}{P^2} \\ P=10Qd^{-\frac{1}{2}}$

Inverse demand function will be:

$P=0.1Qd^{-\frac{1}{2}} \\MR=0.2Qd^{-\frac{1}{2}} \\TR=0.2Qd^{\frac{1}{2}} \\MC=2$

At profit maximizing point.

$MR=MC\\ 0.2Qd^{-\frac{1}{2}}=2\\ Qd^{-\frac{1}{2}}=10\\ Qd=0.01$

Price will be:

$P=0.1\times 0.01^{-\frac{1}{2}}=1\\ P=\$1$

Price elasticity of supply will be:

$eD,P= \frac{\frac{dQD}{QD​}}{\frac{dP}{P}}$

$Ep=-0.1\times \frac{1}{0.01}=-10\\Ep=10\\ Ep>1$

The good is elastic

2). Inverse elasticity formula:

$MR=P(1+\frac{1}{e})$

At profit maximization point.

From the inverse function

$P=10Qd^{-\frac{1}{2}}\\ e=-\frac{1}{2}$

$MR=MC\\ MC=2\\ 2=P(1+\frac{1}{-\frac{1}{2}})\\ 2=-P\\ P=-2\\ P=2$

3). If a tax of one dollar is imposed.

The monopolist’s marginal costs are

$MC(Q)=c$ , but after the tax they become $c+τ$ , where τ 

denotes the per‐unit tax.

But we know that ε<‐1 for the monopolist , the

monopolist produces in the elastic segment of the

demand curve.

Which becomes:

$\frac{\delta P}{\delta τ}=\frac{1}{{1+\frac{1}{\epsilon }}}$

$=\frac{1}{{1+{-\frac{1}{2} }}}=2\\ Ep=2 >0.5$

so the imposition of the tax increases optimal price by more than the tax.