Answer to Question #157866 in Microeconomics for Rosey

Question

1. The monopolist’s demand is represented by p=20-q, and the marginal cost function is MC(q)=2q, in which q is the quantity produced/sold. What is the production quantity and the price which can maximize the monopolist’s profit?

2. Show in a figure the profit and deadweight loss for a monopolistic firm.

3. The demand curve is p=10-q. The supply curve is p=2+q. Calculate the social welfare (consumer surplus + producer surplus) for the market equilibrium. If the government impose a price cap of 5, then how much is the deadweight loss? You may draw figures to assist your calculation.

4. For a monopolistic firm, the marginal cost is 20. The demand elasticity is -2. What is the optimal price for the firm to set?

5. Two identical firms(firm 1 and firm 2) face the following market demand curve p=40-(q1+q2), in which q1 and q2 are the production quantity of firm 1 and firm 2. Their marginal cost is a constant 4 per unit of product. What is the equilibrium price and quantity in the Cournot competition?

Answer 1

1) Let's first find the total revenue:

$$TR=PQ=(20-q)q=20q-q^2.$$

By the definition of the marginal revenue, we have:

$$MR=\dfrac{d}{dQ}(20q-q^2)=20-2q.$$

The level of output that maximizes a monopoly's profit is when the marginal cost equals the marginal revenue (MC=MR):

$$2q=20-2q,$$ $$q=\dfrac{20}{4}=5 units.$$

Finally, we can find the price that maximize the monopolist’s profit:

$$p=20-5=\$15.$$

2) Let's show in a figure the profit and deadweight loss for a monopolistic firm

Here, $P_C$ and $Q_C$ is the price and quantity at competitive market; $P_M$ and $Q_M$ is the price and quantity at monopolistic market; $D$ is demand curve; $MC$ is the marginal cost; $MR$ is the marginal revenue.

3) Let's plot both demand and supply curves in Excel:

As we can see in graph, the market equilibrium is at point $P_E=\$6$, $Q_E=4$. Let's calculate the social welfare (CS+PS):

$$SW=\dfrac{1}{2}\cdot(\$10-\$2)\cdot4=\$32.$$

If the overnment impose a price cap of $5 (grey line), the DWL will be:

$$DWL=\dfrac{1}{2}\cdot(\$7-\$5)\cdot(4-3)=\$1.$$

4) By the definition of the total revenue, we get:

$$TR=P(Q)Q.$$

Since $MR=\dfrac{\partial TR}{\partial Q}$, we get:

$$\dfrac{\partial TR}{\partial Q}=(\dfrac{\partial P}{\partial Q})Q+(\dfrac{\partial Q}{\partial Q})P,$$ $$\dfrac{\partial TR}{\partial Q}=(\dfrac{\partial P}{\partial Q})Q+P.$$

Dividing and multiplying by $P$, we get:

$$MR=(\dfrac{Q\dfrac{\partial P}{\partial Q}}{P})P + P.$$

Let's reacall the equation for the elasticity of demand:

$$E_d=\dfrac{\dfrac{\partial Q}{\partial P}P}{Q}.$$

Then, we get for MR:

$$MR=(\dfrac{1}{E_{d}})P+P,$$ $$MR=P(1+\dfrac{1}{E_d}).$$

The optimal price for the firm to set is when MC=MR:

$$MC=P+\dfrac{P}{E_d}.$$

Let's substitute the numbers and solve for P:

$$2=P+\dfrac{P}{-2},$$ $$P=\$4.$$

5) Let's look at the demand facing firm 1:

$$P=(40-q_2)-q_1.$$

By the definition of the total revenue (TR=PQ), we get:

$$Pq_1=(40-q_2)q_1-q_1^2.$$

Let's write MR:

$$MR=(40-q_2)-2q_1.$$

Since MR=MC and MC=4, we get:

$$(40-q_2)-2q_1=4,$$ $$q_1=18-0.5q_2.$$

The equation above is the firm 1 best response function. The firm 2 best response function is perfectly symmetric to firm 1:

$$q_2=18-0.5q_1.$$

Substituting $q_2$ into $q_1$we get the equilibrium price:

$$q_1=18-0.5\cdot(18-0.5q_1),$$ $$q_1=9+0.25q_1,$$ $$q_1=q_2=12.$$

Substituting $q_1$and $q_2$ into the demand function we get the equilibrium price:

$$P=40-(12+12)=\$16.$$