Answer to Question #162803 in Microeconomics for krysty

Question

Market surveys show that there are two types of consumers for frozen yogurt. The first type like frozen yogurt and have an inverse demand curve of P =5 − 21Q. The second type is crazy about frozen yogurt and has an inverse demand curve of P = 20 − Q. In the town of Smallville, there are only 2 consumers: one of them likes frozen yogurt and the other is crazy about frozen yogurt.

Using the individual demand curves above, derive the market demand for frozen yogurt in Smallville. Plot the market demand curves.

2. Suppose that the market supply for frozen yogurt in Smallville is given by QS = 2+P. Find the equilibrium price and quantity. How much does each consumer buy at the equilibrium price?

Answer 1

(a) Let's first find the direct demand functions for 2 consumers:

$$Q_1=10-2P,$$ $$Q_2=20-P.$$

Then, the total direct demand function will be:

$$Q=Q_1+Q_2=30-3P.$$

Let's plot the market demand curves:

Here, the blue curve is the demand curve for the consumer that like frozen yogurt, orange curve is the demand curve for the consumer that crazy about frozen yogurt, the gray curve is the total market demand curve for two consumers.

(b) Let the market supply for frozen yogurt in Smallville is given by $Q_s=2+P.$ Let's first find the direct demand functions for 2 consumers:

$$Q_1=10-2P,$$ $$Q_2=20-P.$$

Then, the total direct demand function will be:

$$Q=30-3P.$$

At the market equilibrium $Q_d=Q_s$, therefore, the equilibrium price can be found as follows:

$$30-3P=2+P,$$ $$28=4P,$$ $$P=\$7.$$

Then, we can find the equilibrium quantity:

$$Q=30-3\cdot\$7=9\ units\ of\ yogurt.$$

c) Let's first find the direct demand functions for 2 consumers:

$$Q_1=10-2P,$$ $$Q_2=20-P.$$

Then, we can find how much each consumer buy at the equilibrium price. For the first consumer, we get:

$$Q=10-2P+20-P=30-3P,$$ $$Q=30-3\cdot\$7=9\ units\ of\ yogurt.$$

For the second consumer, we get:

$$Q=20-\$7=13\ units\ of\ yogurt.$$