Chapter 15, Problem 15.5P

a)

To determine

To calculate: A’s profit-maximizing strategy and profit earned along with the amount of surplus obtained by the consumer.

Explanation of Solution

The profit-maximizing strategy for A would be to sell so much of the a small cup and big cup at which its marginal cost of coffee production is equal to the marginal market surplus of each form of a coffee drinker. A will therefore offer 8 ounces of a a small cup at 80 cents, and 10 ounces of big cups at $1.50 to maximize profit.

The market gets no net surplus since the price paid for each form of item is equal to each market's total profit surplus.

The profits earned by A will be the sum of profit earned by selling small cups and profit earned by selling a large cup. Assuming that profit from a small cup and a large cup is denoted as ${\pi }_S$ and ${\pi }_L$ . A’s gross profit from selling a small and a large cup is

  $\begin{array}{c}{\pi }_S=\left(50\right)\left[0.80-\left(8\right)\left(0.05\right)\right]\\ =\left(50\right)\left[0.80-0.40\right]\\ =\left(50\right)\left(0.40\right)\\ =\$20\end{array}$

  $\begin{array}{c}{\pi }_L=\left(100\right)\left[1.50-\left(10\right)\left(0.05\right)\right]\\ =\left(100\right)\left[1.50-0.50\right]\\ =100\left(1\right)\\ =\$100\end{array}$

Total profit,

  $\begin{array}{c}\pi ={\pi }_S+{\pi }_L\\ =\$20+\$100\\ =\$120\end{array}$

Hence, total profit is $120.

b)

To determine

To evaluate: The surplus from the a small cup is more than the surplus from a large cup.

Explanation of Solution

The above strategy of A would work only if there is no information asymmetry concerning the consumer’s type. However, given the situation of asymmetric information, the full information-based strategy of A would not be incentive compatible. This is because if A continues with the above-given strategy then it would incentivize the consumers buying a large cup to purchase the quantity of a small cup (8 ounces) at a price of $0.15 and earn positive consumer surplus.

When consumers with a large cup were buying 10 ounces of coffee at $0.15 per ounce, they were paying $\$1.50\left(10\times \$0.15\right)$ and the net consumer surplus was zero.

Now, when consumers with a large cup coffee started buying 8 ounces of coffee at $0.15 per ounce, they are paying $\$1.20\left(8\times \$0.15\right)$

The net surplus obtained by big consumers is equal to the difference between the gross consumer surplus of big consumers consuming a small quantity of coffee and the gross consumer surplus of a small consumer. That is, $1.20-0.80=0.40$ and $0.40\succ 0$ .

Hence, when big consumers buy a small cup they end up gaining positive consumer surplus.

c)

To determine

To calculate: the profit A can, earn from charging 10-ounce cup and 8-ounce cup and still have some customers buy each sized cup.

Explanation of Solution

As we saw in part (b) that if big consumers bought small cups, they gain a positive consumer surplus reducing the profits of A. Now if A has to earn a profit and still have some consumers buy each sized cup, then he should reduce the price of big cups equal to the gain in consumer surplus for big consumers. That is he should reduce the price of big cups by 0.40.

Hence, the price charged by A for a big cup will be, $\$1.50-\$0.40=\$1.10$ .

A’s profit from such a strategy will be

  $\begin{array}{c}{\pi }_S=\left(50\right)\left[0.80-\left(8\right)\left(0.05\right)\right]\\ =\left(50\right)\left[0.80-0.40\right]\\ =\left(50\right)\left(0.40\right)\\ =\$20\end{array}$

  $\begin{array}{c}{\pi }_L=\left(100\right)\left[1.10-\left(10\right)\left(0.05\right)\right]\\ =\left(100\right)\left[1.10-0.50\right]\\ =\left(100\right)\left(0.60\right)\\ =\$60\end{array}$

Total profit,

  $\begin{array}{c}\pi ={\pi }_S+{\pi }_L\\ =\$20+\$60\\ =\$80\end{array}$

Hence, total profit from such a strategy for A will be $80.

d)

To determine

To show: The increase in profit when price fall reduces from 8 to 6 ounces.

Explanation of Solution

An even better strategy for A will be to reduce the quantity of small cups. Although it would reduce the profits of A which he will earn from small consumers, it would also lead to gain from the “bundle effect” and the gain will greater than the loss.

If A reduced the quantity of a small cup from 8 ounces to 6 ounces, then assuming that both types are buying a quantity of small cups, the gross consumer surplus of small consumers and big consumers will be $0.60\left(6\times \$0.10\right)$ and $0.90\left(6\times \$0.15\right)$ .

Thus, the gain in the consumer surplus of big consumers will be, $0.90-0.60=0.30$ , which is less than 0.40.

Hence, the total profit of A will be with such a strategy is

  $\begin{array}{c}{\pi }_S=\left(50\right)\left[0.60-\left(6\right)\left(0.05\right)\right]\\ =\left(50\right)\left[0.60-0.30\right]\\ =\left(50\right)\left(0.30\right)\\ =\$15\end{array}$

  $\begin{array}{c}{\pi }_L=\left(100\right)\left[1.20-\left(10\right)\left(0.05\right)\right]\\ =\left(100\right)\left[1.20-0.50\right]\\ =\left(100\right)\left(0.70\right)\\ =\$75\end{array}$

Total profit,

  $\begin{array}{c}\pi ={\pi }_S+{\pi }_L\\ =\$15+\$70\\ =\$85\end{array}$

Hence, A will earn a total profit of $85 which is higher than the profit of $80 with a price reduction strategy. A will be even better off with a quantity reduction strategy.

e)

To determine

To calculate A’s profit if it ignores small customers and just sells one size of cups, which big customers end up buying.

Explanation of Solution

It might be that A may altogether stop producing small cups and will only produce big cups. This strategy will be even better than quantity reduction strategy. Because with such a strategy, A will earn a total profit of $100 equal to the profit earned from big consumers when there was full information. This is shown below:

  $\begin{array}{c}{\pi }_L=\left(100\right)\left[1.50-\left(10\right)\left(0.05\right)\right]\\ =\left(100\right)\left[1.50-0.50\right]\\ =\left(100\right)\left(1\right)\\ =100\end{array}$

Hence, A will even better with such strategy as the profit of $100\succ \$85\succ \$80$