Question

Suppose there are two beer companies. One produces a beer that has a high alcohol content (ABV 10%), while the other company produces a light beer that has a low ABV (5%). Assume there are 50 consumers whose preferences for alcohol content (ABV) are uniformly distributed between 5% and 10%. Consumers all value drinking a beer their ideal beer at $10 but dislike a beer with a different ABV than their ideal ABV by $1 per percentage point. That is, if I prefer a beer with 6% ABV and I drink the light beer, my utility will be $1 lower. If I prefer a beer with 5.5% and I drink the light beer my utility will be $0.50 lower. Marginal cost is the same for both companies and is equal to $1. The two companies compete by choosing prices simultaneously.

1. What is the utility of purchasing the low ABV beer for a consumer whose preferred beer contains x ABV?

2. What is the utility of purchasing the high ABV beer for a consumer whose preferred beer contains x ABV?

3. Find an expression for the “location” of marginal consumer given pl and ph. In other words, given prices, what is the ABV preference for a consumer who is indifferent between consuming the light beer and the heavy beer. Call this function x m(pl , ph).

4. What happens to the “location” of the marginal consumer as the price of the heavy beer increases?

5. Using this expression, what is the demand curve for the two beers?

6. What is the profit function for each firm?

7. What is the best response function of each firm?

8. Solve for the pure strategy Nash Equilibrium in prices. What are profits in this equilibrium?

Answer 1

Solution:-

(1)We are asked to find out the utility function for the low beer and the high beer.

The utility can be defined as of purchasing the low ABV beer is

U xl = 10 - p l - 1 * ( x - 5) - (I)

where pI is the price of the light beer.

(2)Ono observing the utility of of purchasing the high ABV beer as per the question, the conclusion comes out to be

Uxh= 10-ph-1*(10-x) - (ii)

where ph is the price of the heavy beer.

Observing that the coefficient of Ph in (I) is positive,

There will be an increase in x​​​​​​m​​​​​ and the it will be followed as ph increases along with it.  

Also, holding pl as a constant, we have preferred that the ABV must be closer to 10 as (given in question) to prefer the heavy beer as there is an increase in it's price.

(4).Those consumers whose location is lower that x​​​​​​m​​​​​ has the preference of light beer,

the demand for the same would be the total number of consumers which in this case is 50 times the share of consumers who preferred ABV lower than x​​​m​​​.

Therefore the demand for the low beer will be

= Dl (ph,pl) will be equal to 50* x​​​​​​m​​​​​-5 /10-5

= 10* 7.5+ 1/2 (ph - pl) -5

= 25+ 5 (ph-pl)

Similarly , the demand for the heavy beer will be,

Dh(ph,pl) = 50 (10- x​​​​​m)/10-5

= 25-5 (ph-pl)