Homework Assignment 7

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Instructions:

You will scan your homework into a single PDF file to be upload.
Only legible assignments will be graded.
Late assignments will not be accepted.

Only two randomly-chosen questions will be graded.

Question 1:

Suppose that the monopolist faces a linear demand curve, $P(Q) = A - BQ$. Further suppose that the monopolist has the marginal cost function: $MC = Q$.

  1. Find the revenue as a function of Q.
  2. Find the marginal revenue as a function of Q.
  3. Find the quantity that maximizes the monopolist's profit as a function of A and B.
  4. Find the equilibrium price as a function of A and B.
  5. Let's use some numbers. Suppose $A=10$ and $B=2$. Solve for the profit-maximizing quantity and price.
  6. Using $A=10$ and $B=2$, draw a demand curve and a marginal revenue function as well as marginal cost. Shade the deadweight loss. Also label clearly the profit-maximizing quantity and price chosen by the monopolist.
  7. What would have been the competitive equilibrium price and quantity $($hint: equate MC and the demand function$)$? Label the competitive equilbrium point. Also compute the size of the deadweight loss due to inefficiency casued by the monopolist behavior.

Solutions 1:

  1. $R(Q) = P(Q)\cdot Q = (A - B \cdot Q)\cdot Q = AQ - BQ^2$
  2. $MR(Q) = A - 2BQ$
  3. Setting $MR = MC$ gives $A-2BQ = Q$. By solving for $Q$, we get $Q_M = \frac{A}{1+2B}$.
  4. Plug $Q_M$ into the demand curve. $$\begin{aligned} P_M &= A - B \cdot Q_M \\ P_M &= A - B \cdot \frac{A}{1+2B} = A - \frac{AB}{1+2B} = \frac{A+AB}{1+2B}\\ \end{aligned}$$
  5. $Q_M = \frac{10}{1+2\cdot 2} = 2$ and $P_M =\frac{10+10\cdot 2}{1+2\cdot 2} = 6$.
  6. Green is marginal cost $($supply curve$)$, red is linear demand, and blue is marginal revenue function. The equilibrium quantity and price is $(2,6)$.
7. The competitive equilibrium price can be found by $Q = 10-2Q$, which gives $Q = P = 10/3$. The deadweight loss is $\frac{1}{2} \cdot 4\cdot (\frac{10}{3} - 2)= \frac{8}{3}$.

Question 2:

Suppose a monopolist faces a market demand of $Q^D=500-P$ and has a total cost function of $TC\left(Q\right)=4Q^{2}$.

  1. What is the equilibrium price and quantity decided by the monopolist?

  2. What is the average cost at the equilibrium quantity?

  3. How much profit does the monopolist make at the equilibrium price and quantity?

Solutions 2:

  1. Recall that the monopolist chooses a price on the basis of $MR=MC$. Note that we need the demand function re-written to get $P$ as a function of $Q$. $$\begin{aligned} Q & = 500-P\\ P & = 500-Q\end{aligned}$$ In order to find $MR$: $$\begin{aligned} MR & = \frac{\partial\left(PQ\right)}{\partial Q}=\frac{\partial}{\partial Q}\cdot\left[\left(500-Q\right)Q\right]=\frac{\partial}{\partial Q}\cdot\left[500Q-Q^{2}\right]\\ & = 500-2Q\end{aligned}$$ We can find $MC$ by taking the derivative of total cost: $$MC=\frac{\partial TC}{\partial Q}=8Q$$ Setting $MR$ and $MC$ equal: $$\begin{aligned} 500-2Q & = 8Q\\ 500 & = 10Q\\ Q_{M} & = 50\end{aligned}$$ To find the price, recall that: $$\begin{aligned} P_{M} & = 500-Q_{M}\\ & = 500-50\\ & = 450\end{aligned}$$

  2. $AC = \frac{TC}{Q} = \frac{4Q^2}{Q} = 4Q = 200$.

  3. For each unit, the monopolist makes a profit of 250 $(=P - AC = 450 - 200)$. Hence, the total profit is $400 \cdot Q_M = 250 \cdot 50 = 12,500$.

Question 3:

Suppose a monopolist faces a demand curve $Q^d = 200 - P $ and that the monopolist has a constant marginal cost of $c$ where $ 0 < c <200$. Find the monopolist’s profit-maximizing quantity and price; and describe how they vary with the marginal cost $c$.

Solution 3:

Demand Curve: The linear demand curve is given by:

$ Q_d = 200 - P $

Inverse demand: $ P = 200 - Q_d $

Revenue: The total revenue (TR) for the monopolist is:

$ TR = (200 - Q_d) \times Q_d $

$ TR = 200 Q_d - Q_d^2 $

Marginal Revenue (MR): Differentiate $ TR $ with respect to $ Q $ to get $ MR $.

Cost Structure: The total cost (TC) function is:

$ TC = c \times Q $

Marginal Cost (MC): Differentiate $ TC $ with respect to $ Q $ to get $ MC $.

Profit Maximization: The monopolist will choose the quantity where $ MR = MC $ to maximize profit. Setting the MR and MC equations equal to each other and solving for $ Q $ will give the profit-maximizing quantity. The corresponding price can be found by plugging this quantity into the inverse demand curve.

$ TR = MC$

$ 200 - 2 Q_d = c$

The profit-maximizing quantity $ Q $ for the monopolist is: $ Q^M = 100 - \frac{ c}{2} $

The corresponding profit-maximizing price $ P $ is: $ P^M = 100 + \frac{ c}{2} $

The higher the MC, the lower the optimal quantity and the higher the price.

Question 4:

A profit-maximizing monopolist faces a downward-sloping demand curve that has a constant elasticity of -2. The firm finds it optimal to charge a price of $8 for its output. What is its marginal cost at this level of output?

Solution 4:

We have a special demand function that gives us constant price elasiticity. The demand function is given by $Q = kP^r$ where k is some constant and r is the price elasticity. So in this case, we have $P = (\frac{Q}{k})^{\frac{1}{r}}$ $$\begin{align} \implies TR = (\frac{Q}{k})^{\frac{1}{r}}*Q \\ \implies TR = (\frac{1}{k})^{\frac{1}{r}} * Q^{\frac{1}{r} + 1} \\ \implies MR = (\frac{1}{k})^{\frac{1}{r}} (\frac{1}{r} + 1)Q^{\frac{1}{r}} \\ \implies MR = P(\frac{1}{r} + 1) \end{align}$$
We are given that P = 8 and we know that r = -2. So substituting P and r we get $$\begin{align} MR = 8(\frac{-1}{2} + 1) \implies MR = 8(\frac{1}{2}) = 4 \\ \text{And, at profit maximisation we have MR = MC} \implies MR = MC = 4. \\ \text{So MC} = 4\end{align}$$