A firm faces the demand for its product, $P = 100 - 0.5 Q$ ,as shown in the figure below.It produces under conditions of constant costs in the long run,and LMC = LAC = $12 per unit. If the firm can practice first-degree price discrimination,it will be able to collect $________ in total revenue under perfect price discrimination.
A)$1,872
B)$1,936
C)$7,744
D)$9,856
E)none of the above

Question

A firm faces the demand for its product, $P = 100 - 0.5 Q$ ,as shown in the figure below.It produces under conditions of constant costs in the long run,and LMC = LAC = $12 per unit.  If the firm can practice first-degree price discrimination,it will be able to collect $________ in total revenue under perfect price discrimination.
A)$1,872 
B)$1,936 
C)$7,744 
D)$9,856 
E)none of the above

Quizplus · Accepted Answer

Under first-degree price discrimination, the firm charges each customer their maximum willingness to pay for the product. This means that each customer will be charged a different price, depending on their willingness to pay.

To find the total revenue under first-degree price discrimination, we need to find the maximum price that each individual customer is willing to pay. Since the demand function is linear, we can find the willingness to pay for each customer by setting the marginal revenue equal to their willingness to pay:

$MR = \frac{dTR}{dQ} = P + \frac{dP}{dQ}Q = 100 - Q$

Setting MR equal to the LMC of $12, we get

$12 = 100 - Q $$\Rightarrow Q = 88$

So the firm can sell 88 units at a price of $12 per unit.

To find the total revenue, we need to add up the revenue from each customer:

$TR = P_1 + P_2 + ...+ P_{88}$

where $P_1, P_2, ..., P_{88}$ are the prices charged to each individual customer.

Since the demand function is $P = 100 - 0.5Q$, we can solve for the price for each quantity demanded:

$Q = 100 - 2P \Rightarrow P = 50 - 0.5Q$

Substituting Q = 1, 2, 3, ..., 88, we can find the price charged to each customer:

For Q = 1, P = 49.5

For Q = 2, P = 49

For Q = 3, P = 48.5

and so on, until

For Q = 88, P = 6

Adding up all the prices, we get

$TR = 49.5 + 49 + 48.5 + ... + 6 = \frac{1}{2}[50 + 49 + 48 + ... + 3] = \frac{1}{2}[1275] = 637.5$

Therefore, the firm can collect $7,744 in total revenue under perfect price discrimination (88 units sold at $12 per unit).

Note: The formula for the sum of an arithmetic series is used here to add up the individual prices (from $49.5$ to $6$).

A Firm Faces the Demand for Its Product P=100−0.5QP = 100 - 0.5 QP=100−0.5Q

A Firm Faces the Demand for Its Product $P = 100 - 0.5 Q$