Answer to Question #157997 in Microeconomics for Syed Qasim

Question

Road Runner Co is a Pakistani manufacturer making Bicycles. It exports to two markets,Bangladesh and Sri Lanka. Demand for Bicycles in thesetwo markets is given by the following Functions:

Bangladesh                  Q₁ = 12 – P₁

                                                      

  Sri Lanka                     Q₂ =   8 – P₂

Where Q₁ and Q₂ are respective quantities sold (in thousands) andP₁ and P₂ are the respective prices (in Pak. Rupees per unit) in the two markets. Total cost function is

C = 5 + 2 (Q₁+ Q₂)

     a. Determine the company’s total profit function. Also,

(i) What are the profit maximizing levels of price and output for the two markets?

(ii) Calculate the marginal revenues in each market.

       b. Now consider two cases:

(i)  Company is effectively able to price discriminate in thetwo markets. What will 

             be the total profits?

(ii)  Suppose the company does not engage in price discrimination. By charging thesameprice in the two markets what are the profit maximizing levels of price,output, and the total profits?                                            

c. Analyze, with graphs, the two alternative pricing strategies available to the company. 

Answer 1

(a) Let's determine the company’s total profit function. By the definition of the profit, we have:

$$Profit=\pi=TR-TC.$$

Let's write the inverse demand functions:

$$P_1=12-Q_1,$$ $$P_2=8-Q_2.$$

By the definition of the TR, we have:

$$TR(Bangladesh)=P_1Q_1=(12-Q_1)Q_1,$$ $$TR(Sri\ Lanka)=P_2Q_2=(8-Q_2)Q_2,$$ $$TR(Total)=(12-Q_1)Q_1+(8-Q_2)Q_2.$$

Finally, we can write the company’s total profit function:

$$\pi(Q_1,Q_2)=(12-Q_1)Q_1+(8-Q_2)Q_2-5-2(Q_1+Q_2).$$

(i) Let's find the profit maximizing levels of price and output for the two markets. Let's solve the system of two differential equations:

$$\dfrac{\partial\pi}{\partial Q_1}=12-2Q_1-2=0,$$ $$\dfrac{\partial\pi}{\partial Q_2}=8-2Q_2-2=0.$$

As we can see this system of equations will satisfy when $Q_1=5$, $Q_2=3$. Then, substituting $Q_1$ and $Q_2$ into the demand functions we get the prices $P_1=7\ Rs$ and $P_2=5\ Rs$.

ii) Let's calculate the marginal revenues in each market:

$MR_1=\dfrac{d}{dQ_1}(TR_1)=\dfrac{d}{dQ_1}(12Q_1-Q_1^2)=12-2Q_1=12-2\cdot5=2\ Rs,$$MR_2=\dfrac{d}{dQ_2}(TR_2)=\dfrac{d}{dQ_2}(8Q_2-Q_2^2)=8-2Q_2=8-2\cdot3=2\ Rs.$

b) (i) Price discrimination occurs when a company takes advantage of its market power to sell the same product in two different markets at two different prices. As we can see above, our company has the following inverse demand functions for the two markets:

$$P_1=12-Q_1,$$ $$P_2=8-Q_2.$$

Let's write the total and marginal revenue for both cases:

$$TR_1=(12-Q_1)Q_1,$$ $$MR_1=12-2Q_1,$$ $$TR_2=(8-Q_2)Q_2,$$ $$MR_2=8-2Q_2.$$

Let's find the price and quantity that maximizes profits in each market separately:

$$MR_1=MC=12-2Q_1=2,$$ $$MR_2=MC=8-2Q_2=2.$$

Solving for $Q_1$ and $Q_2$, we get:

$$Q_1=5, Q_2=3.$$

Substituting $Q_1$ and $Q_2$ into the inverse demand functions, we get:

$$P_1=7\ Rs, P_2=5\ Rs.$$

Finally, let's find the company's total profits:

$$\pi=TR_1+TR_2-TC,$$ $$\pi=(12-5)\cdot5+(8-3)\cdot3-5-2\cdot(5+3)=29\ Rs.$$

(ii) If the company does not engage in price discrimination, we will have the following worldwide demand function:

$$Q=(12-P)+(8-P)=20-2P.$$

Let's find the inverse demand function, TR and MR:

$$P=10-0.5Q,$$ $$TR=(10-0.5Q)Q,$$ $$MR=10-Q.$$

Let's find the price and quantity that maximizes company's profit in the market:

$$MR=MC=10-Q=2,$$ $$Q=8.$$

Substituting $Q$ into the inverse demand function, we get the price:

$$P=6\ Rs.$$

Finally, let's calculate the total profits in the case of absence of price discrimination:

$$\pi=TR-TC,$$ $$\pi=(10-0.5\cdot8)\cdot8-5-2\cdot8=27\ Rs.$$

c) Let's draw the graphs for Market 1 and 2 (price discrimination) and Total Market (absence of price discrimination) and do some conclusions:

Here, the blue curve is the demand function, the orange curve is the marginal revenue and the grey curve is the marginal cost. As we can see, in case of price discrimination to maximize the profits in each markets company produces 5 units and charge price 7 Rs (Market 1) and produces 3 units and charge price 5 Rs (Market 2). In case of worldwide market the company produces 8 units and charge price 6 Rs. Also, if the company will charge the same charge to everyone, its profit will decrease.