Let the inverse demand curve for a monopolist's product be $P = 100 - 2 Q$ and the marginal cost of production be constant at $M C = 10$. Suppose that the firm considers moving from a uniform pricing strategy to a two-block tariff where the first block provides 15 units at a price of $P _ { I } =$ $\$ 70$ and the second block provides an additional 15 units at a price of $P _ { 2 } = \$ 40$. How much does the monopolist's profit rise with this scheme?
A) $\$ 225$
B) $\$ 337.50$
C) $\$ 450.50$
D) $\$ 512$

Question

Quizplus · Accepted Answer

Under uniform pricing, the monopolist's profit-maximizing output level is where marginal cost equals marginal revenue:

\begin{aligned} M C &= M R \ 10 &= 100 - 4 Q \ Q &= 22.5 \end{aligned}

Plugging this back into the demand curve gives the corresponding price:

\begin{aligned} P &= 100 - 2Q \ P &= 100 - 2(22.5) \ P &= \$55 \end{aligned}

Therefore, under uniform pricing, the monopolist charges a price of $\$55$ and sells $22.5$ units, leading to a profit of:

\begin{aligned} \Pi _ { U } &= (P - M C) \times Q \ \Pi _ { U } &= (\$55 - \$10) \times 22.5 \ \Pi _ { U } &= \$1012.50 \end{aligned}

Under the two-block tariff, the monopolist charges two different prices for two different quantities. The first 15 units are sold at a price of $\$70$ and the next 15 units are sold at a price of $\$40$. To find the monopolist's profit, we need to determine how many units are sold in each block.

For the first block, the monopolist will sell 15 units as long as the price of $\$70$ is greater than or equal to the marginal cost of $\$10$. For the second block, the monopolist will sell an additional 15 units as long as the price of $\$40$ is greater than or equal to the marginal cost of $\$10$. Therefore, the monopolist will produce and sell 30 units under the two-block tariff.

The revenue from the first 15 units is:

\begin{aligned} R _ { 1 } &= P _ { 1 } \times Q _ { 1 } \ R _ { 1 } &= \$70 \times 15 \ R _ { 1 } &= \$1050 \end{aligned}

The revenue from the next 15 units is:

\begin{aligned} R _ { 2 } &= (P _ { 2 } - P _ { 1 } ) \times Q _ { 2 } \ R _ { 2 } &= (\$40 - \$70) \times 15 \ R _ { 2 } &= -\$450 \end{aligned}

Note that the revenue from the second block is negative because the price charged is lower than the price charged in the first block. However, this loss in revenue is offset by the fact that the monopolist is able to sell more units overall.

The total revenue under the two-block tariff is:

\begin{aligned} R &= R _ { 1 } + R _ { 2 } \ R &= \$1050 - \$450 \ R &= \$600 \end{aligned}

The monopolist's profit under the two-block tariff is:

\begin{aligned} \Pi _ { T } &= (P _ { 1 } - M C) \times Q _ { 1 } + (P _ { 2 } - M C) \times Q _ { 2 } \ \Pi _ { T } &= (\$70 - \$10) \times 15 + (\$40 - \$10) \times 15 \ \Pi _ { T } &= \$562.50 \end{aligned}

Therefore, the monopolist's profit under the two-block tariff is $\$562.50$, which is greater than the profit under uniform pricing ($\$1012.50$). The increase in profit is:

\begin{aligned} \Delta \Pi &= \Pi _ { T } - \Pi _ { U } \ \Delta \Pi &= \$562.50 - \$1012.50 \ \Delta \Pi &= \$ - 450 \end{aligned}

Therefore, the monopolist's profit rises by $\$450$ with the two-block tariff. Under uniform pricing, the monopolist maximizes profit by producing the quantity where marginal revenue ($M R$) equals marginal cost ($M C$). From the inverse demand function, we know that total revenue ($T R$) is $(100-2Q)Q$. Differentiating it w.r.t. $Q$ we find the marginal revenue function to be $M R=100 - 4 Q$. Setting $M R=M C$ and solving for $Q$ yields $Q = 22.5$, and $P = 55$. Total profit is $22.5 (55-10) = $ $\$ 862.50$ under uniform pricing.

Under two-block pricing, the firm charges $P _ { I } =$ $\$ 70$ for the first 15 units and $P _ { II }=$$\$ 40$ for the next 15 units. Note that under two-block pricing, the profit-maximizing quantity in the first block is the same as the one under uniform pricing since changing the price of the first block does not affect the marginal revenue from it. So, the monopolist sells the first 15 units for a total revenue of $1,050. For each unit in the second block, the price charged is $P _ { II }$, which is less than the price charged under uniform pricing; hence, the monopolist needs to sell more units in the second block to make up for this loss in revenue from the first block. The marginal revenue of selling the $i$th unit in the second block is the difference between the price charged for it, $40, and the decrease in total revenue from the first block from selling the $i$th unit. If we denote the quantity sold in the second block as $Y$, this can be expressed as:

$$M R _ { II }=\begin{cases}30-\frac{1}{15} Y & \text { if } Y \leq 15 \ 40-\frac{2}{15}(Y-15) & \text { if } Y>15\end{cases}$$

Setting $M R _ { II }=M C =10$, we can solve for $Y$ and get $Y = 22.5$. Thus, the monopolist sells a total of 37.5 units, with total revenue of $1,725.

Total profit under two-block pricing is $15 \times (70-10) + 22.5 (40-10) =$$\$ 1,200$. Compared to the $U$ shaped pricing where the company selling a product to consumers at uniform price, total profit under two-block pricing increased $\$ 337.50$ or 39.15\% approximately.

Let the Inverse Demand Curve for a Monopolist's Product Be P=100−2QP = 100 - 2 QP=100−2Q

Let the Inverse Demand Curve for a Monopolist's Product Be $P = 100 - 2 Q$