Econ 302: Solutions for Practice Questions 9


Question 1 Because the demand for the product is y(p)=100-p, the price elasticity of demand is given by

ep=-p/(100-p).

The cost functions are c(yi)=10yi. Therefore MC=10. Let si=yi/y be firm i's market share. Then

MCi=p(1+si/ep).

Therefore

10=p(1+s1/ep);
10=p(1+s2/ep).

Adding these two equations yields 10=2p+p/ep. Therefore 10=2p+p-100. Thus, 3p=110 which implies p=36.67.

Inserting p=36.67 in the demand function yields y=63.33.

If we insert p=36.67 in

10=p(1+si/ep)=p+si(p-100)

then we get si=0.5, i.e., each firm has 50% of the market share and therefore produces 31.665 units of output.


Question 2 The demand for the product is y(p)=40-p. Therefore the price elasticity of demand is given by

ep=-p/(40-p).

The cost functions including transportation costs are c(y1)=5y1, and c(y2)=6y2. Thus, MC1=5 and MC1=6. Again, let si=yi/y be firm i's market share. Then

MCi=p(1+si/ep)=p+si(p-40).

Therefore

5=p(1+s1(p-40);
6=p(1+s1(p-40).

Adding these two equations yields 11=2p+(p-40). Thus, 3p=51 which implies p=17.

Inserting p=17 in the demand function yields y=23.

If we insert p=17 in

5=p+s1(p-40)

we get s1=12/23. As a consequence, s2=11/23.

Because total output is 23, it follows that firm 1 produces 12 units, and that firm 2 produces 11 units.

Question 3 As in Question 1 we can conclude that

MCi=p+si(p-100).

Therefore

10=p+s1(p-100);
10=p+s2(p-100);
10=p+s3(p-100).

Adding these three equations yields 30=3p+(p-100). Thus, 4p=130 which implies p=32.5.

Inserting p=32.5 in the demand function yields y=67.5.

If we insert p=67.5 in

10=p+si(p-100)

then we get si=1/3, i.e., each firm has 1/3 of the market share and therefore produces 22.5 units of output.

Now assume there are nine firms. Then we get nine equations of the form

10=p+si(p-100).

Adding these equations yields 90=9p+(p-100). Therefore p=19. This implies y=7.29 . Each firm now has 1/9th of the market share and therefore produces 9 units of output.