Question

Demand for alcohol has been present for much of human history, but supply has faced many forms of restriction for a variety of reasons. Consider a competitive market for wine, where the costs of a typical supplier can be represented by the function ?(?)=175+7?2, where ? represents bottles of wine produced.

What is the long-run equilibrium price per bottle of wine?  

We know that competitive markets lead to the lowest possible price and thus the largest quantity demanded, which is often a good economic outcome. In the case of alcohol, however, consumption of large quantities can lead to negative effects.

Suppose that many citizens of a wine-producing (and -consuming) region are frustrated with public drunkenness and lobby their representatives to somehow limit the consumption of alcohol. The local government thus decides to require licenses for any firms that wish to produce wine in order to restrict production and also ensure quality. The price of a license is set at $77

In the short run, existing firms have to pay for the license, but the price of wine does not change. What are the short-run profits for a typical firm if the price of wine is the same as in Part 1 and firms must also purchase a license?

Answer 1

The total cost function of the alcohol producing firms in a competitive market is given as c(y)=175+7y^2 where y represents the bottle of wine produced. Therefore, the average cost function of the alcohol producing firms or ac(y), in this case would be=c(y)/y=(175+7y^2)/y=175/y+7y. The marginal cost function faced by the alcohol producing firms or mc(y)=dc(y)/dy=14y. Now, based on the long-run profit-maximizing condition or principle of a competitive firm, the firm would maximize profit at the level or point which corresponds to the equality between the average cost function and the marginal cost function.

Therefore, based on the long-run profit maxizing condition of a competitive firm, we can state:-

ac(y)=mc(y)

175/y+7y=14y

175/y+7y-14y=0

175/y-7y=0

(175-7y^2)/y=0

175-7y^2=0

-7y^2=-175

y^2=-175/-7

y^2=25

y=5

Hence, the long-run profit-maximizing equilibrium number of bottles produced by the competitive alcohol-producing firm would be 5 bottles.

Now, plugging the value of the profit-maximizing equilibrium number of bottles into the mc(y), we obtain:-

14y

=14*(5)

=70

Thus, the equilibrium price of per bottle of wine would be $70.

Now, considering that the firms have to a pay an additional licence fee of $77 the total cost function of the firms would now become=(175+7y^2)+77=175+7y^2+77=252+7y^2. The equilibrium price price of per bottle of wine remains $70 which is also the long-run equilibrium price of per bottle of wine. Therefore, the total revenue generated by the firms in the short-run at the profit-maximizing equilibrium level=$70*5 bottles=$350 and the total cost incurred by the firms at the profit-maximizing equilibrium level=252+7*(5)^2=252+175=$427. Therefore, the short-run profit of the each competitive firm in the short-run=($350-$427)=-$77 or a total loss of $77.