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 $P _ { 2 } = \$ 40$. What is the average outlay schedule for the consumer?
A) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 70 - 450 / Q$ if $Q 15$
B) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 40 + 450 / Q$ if $Q 15$
C) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 40$ if $Q 15$
D) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 70 - 40 / Q$ if $Q 15$

Question

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 $P _ { 2 } = \$ 40$. What is the average outlay schedule for the consumer?
A) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 70 - 450 / Q$ if $Q > 15$
B) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 40 + 450 / Q$ if $Q > 15$
C) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 40$ if $Q > 15$
D) $\frac { E } { Q } = \$ 70$ if $Q \leq 15$ and $\frac { E } { Q } = 70 - 40 / Q$ if $Q > 15$

Quizplus · Accepted Answer

Under the uniform pricing strategy, the monopolist chooses price and output to maximize profit: $$\pi=P(Q)Q-MC\cdot Q=(100-2Q)Q-10Q=-2 Q^2 + 90 Q$$ The first order condition for a maximum is $\frac{\partial \pi}{\partial Q}=0$, which gives $Q=45$. Then price is $P(45)=10$. Consumer surplus under uniform pricing strategy is $$\int_0^{45}P(Q)dQ=\int_0^{45}(100-2Q)dQ=45\cdot 55=2475.$$ Next, consider the two-part tariff. In order to induce consumers to purchase 30 units in total, at least 15 units must be purchased in the first block. The monopolist then sets the price in the second block to extract as much surplus as possible from customers beyond the first 15 units. The monopolist maximizes profit by solving the following problem: $$\max_{q_1,q_2}\left\{(70-C)q_1+(40-C)q_2\right\}$$ subject to $q_1+q_2=30$ and $0\leq q_1\leq 15$, where $C$ is the marginal cost of production. The first-order condition for the first-best quantity of the initial block is $(70-C)-(40-C)\frac{dq_2}{dq_1}=0$, which implies $q_2=\frac{9}{2}q_1-105/2$. Since at least 15 units must be sold in the initial block, we have $q_1\geq15$. Therefore, the optimal amount of the second block can be obtained by setting $q_1=15$ and solving $\frac{dq_2}{dq_1}=\frac{9}{2}=2.25$. Thus, the monopolist sells 15 units in the initial block and 15 units in the second block. Under the two-part tariff strategy, the consumer surplus can be computed by summing the consumer surplus in each block. In the initial block, the consumer surplus equals $$\int_0^{15}(100-2Q-70)dQ=437.5.$$ In the second block, the price per unit is $P_2=40$, so the consumer surplus is $$\int_{15}^{30}(100-2Q-40)dQ=562.5.$$ Thus, the consumer surplus under the two-part tariff strategy is $1000$, which is larger than the consumer surplus of $2475$ obtained under uniform pricing. The average outlay is defined as the total payment divided by total quantity purchased. For any given quantity $Q$, the total payment is computed as follows: $$TR(Q)=\left\{ \begin{aligned} &70Q&\ \text{ if }0\leq Q\leq 15\ &1050+40(Q-15)&\ \text{ if } Q > 15 \end{aligned} \right. $$ Therefore, the average outlay is $$\frac{E}{Q}=\left\{ \begin{aligned} &70&\ \text{ if }0\leq Q\leq 15\ &40+\frac{450}{Q}&\ \text{ if } Q > 15 \end{aligned} \right.. $$ When $Q \leq 15$ the consumer faces a price of $P_I = \$70$ for every unit purchased. When $15< Q \leq 30$ the first 15 units cost \$70 each but the next 15 units cost only \$40, so the consumer's marginal cost of consumption decreases. The consumer's optimal consumption bundle will satisfy the condition that the marginal benefit from consuming the last unit purchased is equal to the marginal cost of that unit. Thus, the average outlay schedule for the consumer can be calculated as follows: For $Q \leq 15$, the inverse demand curve is $P=100-2Q$ and the quantity demanded is $Q$, so the total expenditure is $E = P \cdot Q = (100-2Q) \cdot Q$. Therefore, the average outlay is: $$\frac{E}{Q} = \frac{(100-2Q) \cdot Q}{Q} = 100-2Q = \$70 \text{ if } Q \leq 15$$ For $15< Q \leq 30$, the inverse demand curve is still $P=100-2Q$, but now the marginal cost of production is only $MC=\$10$ for the first 15 units and the marginal cost is zero for the next 15 units. Thus, the monopolist will charge different prices in each block in order to extract as much consumer surplus as possible. The consumer's optimal consumption bundle will satisfy the condition that the marginal benefit from consuming the last unit purchased in each block is equal to the marginal cost of that unit (\$70 and \$0, respectively). Thus, the consumer's average outlay in this region is: $$\frac{E}{Q} = \frac{(15 \cdot 70) + (Q-15) \cdot 40}{Q} = 70 - \frac{450}{Q} \text{ if } 15< Q \leq 30$$ For $Q > 30$, the inverse demand curve is still $P=100-2Q$ and the monopolist will charge the same price for all units in order to maximize profits (i.e. use uniform pricing). The consumer's optimal consumption bundle will satisfy the condition that the marginal benefit from consuming the last unit purchased is equal to the marginal cost of that unit, which is still \$10. Thus, the consumer's average outlay in this region is: $$\frac{E}{Q} = \frac{(30 \cdot 70) + (Q-30) \cdot 10}{Q} = 40 + \frac{450}{Q} \text{ if } Q > 30$$

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 $P = 100 - 2 Q$