Answer to Question #177613 in Microeconomics for Ishita Srivastava

Question

In western Texas, in a small town called Shallowater, just nothwest of Lubbock, two stores sell tires, Store FlatsAreUs and Store We'reNotGood. Both have zero cost because they just cheat customers. Demand for tires in the town is Q = 550 - P, where P is price. Say that We'reNotGood produces Q = 110. If FlatsAreUs is trying to maximize revenue, then the store's optimal choice for Q would be _____ and they would then earn revenue _____.  

Answer 1

Let's first write the inverse demand function:

$$P=550-Q.$$

Let's write the total revenue:

$$TR=QP=Q_1P+Q_2P=(Q_1+Q_2)P,$$ $$TR=(Q_1+Q_2)(550-(Q_1+Q_2)).$$

Then, we can find $TR_2$:

$$TR_2=Q_2P=Q_2(550-Q_1-Q_2).$$

The quantity that maximizes the revenue can be found as follows:

$$\dfrac{\partial TR_2}{\partial Q_2}=0,$$ $$\dfrac{\partial TR_2}{\partial Q_2}=550-Q_1-2Q_2=0,$$ $$550-110-2Q_2=0,$$ $$Q_2=220.$$

Finally, we can find how much revenue they earn:

$$TR_2=Q_2(550-Q_1-Q_2),$$ $$TR_2=220\cdot(550-110-220)=\$48400.$$