# COMM 295 MT Review Session

By:
Ubc Tutoring
09-12-2011

**Comm 295 REVIEW SESSION Solutions**

Telus Telecommunications is a monopolist supplier of telephone services in the Fraser Valley. Total market demand Telus faces is: Q^{d} = 400 – 4P, where Q = number of hours of calls per day and P = price per hour of call (in dollars). The cost of providing the service is: C(Q) = 2000 + 20Q.

a) If Telus could perfectly price discriminate (i.e., use first-degree price discrimination), how many hours of calls would its customers make per day? Illustrate your results on a well-labeled diagram. **4 points**

Rearranging the demand function gives P = 100 – Q/4.

A perfect price discriminating monopolist supplies the quantity where

P = MC.

100 – Q/4 = 20 **Q = 320 (hr/day)**

b) Suppose aggregate consumer demand is comprised of 20 identical consumers. Derive the demand curve for each individual consumer. **2 points**

Aggregate consumer demand: Q^{d} = 400 – 4P

Individual consumer demand: q = Q^{d}/20 = 20 – P/5

c) If Telus decided to use a two-part tariff instead of perfect price discrimination, what fixed/entrance fee would each customer be charged if customers are identical, as described in part (b)? Without using math, is it possible to compare profits for Telus with this optimal two-part tariff scheme versus the perfect price discrimination scheme? **4 points**

If consumers are identical, the profit maximizing usage fee P^{*} = MC = 20.

At this usage fee the individual q demanded q^{*} = 20 – 20/5 = 16 hr/day

The fixed entrance fee T_{i}^{*} per consumer

= CS of an individual consumer (before the fee is paid)

= 0.5(100 – 20)*16 = **$640**

**Alternative Solution:**

The optimal total entry fee T equals total consumer surplus. We know that the usage fee P equals marginal costs for a two-part tariff with homogenous consumers. Thus, P = MC = 100 – Q/4. Furthermore, quantity supplied is identical to the quantity we could observe in perfectly competitive market:

Q = 320 **P = $20**

P(Q=0) = 100

CS = (320)(100 – 20)/2 = 12,800

Thus, the individual entry fee T_{i} = CS/20 = **$640**

Since all individuals have identical demand (i.e. they are homogenous) the profit for Telus using a two-part tariff is the same as for applying a first-degree (perfect) price discrimination. (You can easily calculate that Profits = 10,800 and PS = 12,800 under both the schemes.)

d) Suppose perfect price discrimination is not possible, and a two-part tariff is deemed illegal. However, Telus discovers that it can divide the customers into two groups: business and residential. It therefore decides to adopt a third-degree price discrimination strategy. There are 10 residential customers, each with demand Q_{R}^{d} = 15 – 0.2P and there 10 business customers, each with demand Q_{B}^{d} = 25 – 0.2P. What price will Telus charge to each group in order to maximize profits? How many hours of calls will each group make per day? **5 points**

Aggregate demand residential: Q_{R}^{D} = 10 * Q_{R}^{d} = 150 – 2P

Aggregate demand business: Q_{B}^{D} = 10 * Q_{B}^{d} = 250 – 2P

For each market: MR = MC

** Residential: **

P = 75 – Q/2 MR_{R} = 75 – Q

For profit maximization, the third-degree price discriminating monopolist sets:

**MR _{R} (Q_{R}) = MC(Q)**

75 – Q_{R} = 20 **Q _{R} = 55 (hr/day)**

P_{R} = 75 – 55/2 = $**47.5**

** Business: **

P = 125 – Q/2 MR_{B} = 125 –Q

For profit maximization,

**MR _{B} (Q_{B}) = MC(Q)**

125 – Q_{B} = 20 **Q _{B} = 105 (hr/day)**

P_{B}= 125 – 105/2 = $**72.5**

** QUESTION 2 **

Suppose that firms A and B are the only two producers of hockey sticks in Canada. A’s cost of production is given by C_{A} = 200 + 20Q and B’s cost is given by C_{B} = 600 + 20Q. The market demand function is Q = 200 – P. Assume that the hockey sticks produced by the two firms are identical.

(a) If the two firms compete in quantities, find the Cournot-Nash profit-maximizing quantities and the equilibrium market price. (**3 points**)

Inverse demand function à P = 200 – Q^{d} = 200 – Q_{A} – Q_{B}

Firm A:

R_{A} = P*Q_{A} = (200 – Q_{A} – Q_{B}) Q_{A} = 200Q_{A} – Q_{A}^{2} – Q_{B}Q_{A}

MR_{A} = 200 – 2Q_{A} – Q_{B}

Firm A’s RF is given by

MR_{A} = MC_{A}

200 – 2Q_{A} – Q_{B} = 20

**Q _{A} = 90 – ½Q_{B}**

Firm B

R_{B} = P*Q_{B} = (200 – Q_{A} – Q_{B})Q_{B} = 200Q_{B} – Q_{B}^{2} – Q_{B}Q_{A}

MR_{B} = 200 – 2Q_{B} – Q_{B}

Set MR_{B} = MC_{B}

200 – 2Q_{B} – Q_{A} = 20

**Q _{B} = 90 – ½Q_{A}**

(Note that since A and B are identical you can get RF of B by symmetry)

Solve the two RFs for Q_{A} and Q_{B}. For example, plug RF of B into RF of A

Q_{A} = 90 – ½Q_{B } = 90 – ½(90 – ½Q_{A})

3Q_{A}**/4 = 45**

Q_{A} = 60.

Plug the value of and Q_{A}** = **60 into RF of B to get Q_{B} = 60

so **Q = 120**

From demand function, P = 200 – Q = 200 – 120 = $80

**QUESTION 3**

a) MP_{K} = 5L/K^{0.5} MP_{L} = 10K^{0.5}

The production function exhibits the property of diminishing marginal returns in K. However, MP_{L} is independent of L.

b) AP_{L} =Q/L = 10K^{0.5}. Therefore AP_{L} (K = 9) = 30. AP_{L} is independent of L.

c) Assume that both the inputs are increased by t (>1) times, then the new production Q_{n} = 10(tK)^{0.5} (tL) = t^{1.5} 10K^{0.5} L = t^{1.5} Q. As the inputs are increased by t times, production increases by more than t times, implying that the production function exhibits IRS.

d) At cost-minimizing levels of K and L,

MRTS =

Therefore, at K = L = 4, MRTS = 2.

Q= 80 can be obtained by employing the following combinations of K and L:

K= | 2 | 3.9 | 4 | 4.1 | 9 | 16 |

L= | 5.66 | 4.05 | 4 | 3.95 | 2.66 | 2 |

Approximate slope at K=L=4 is (approximately), which is equal to the negative of MRTS.

(Note: an alternative mathematical way to find the slope at any point of an isoquant (i.e., MRTS itself) is by taking total derivatives. Taking total derivatives of the Q function: . Substituting dQ = 0 (since Q is fixed) and after some manipulation slope of the isoquant dK/dL = – 2K/L, which is equal to – 2 (at K = L =4))

** Question 4 **

** Solution 4 **

(a) At equilibrium, Q^{D} = Q^{S}.

6 – 4.5P = – 1 + 3.5P

P^{*} = $ 7/8 and

Q^{*} = 2.062

(c) Consumer Surplus, CS = ½ * (P^{D}(Q^{D} = 0) – P*) * Q*

= 0.5(1.333 – 0.875)2.062 = 0.472

Producer Surplus, PS = ½ * (P* – P^{S}(Q^{S} = 0)) * Q*

= 0.5(0.875 – 0.285)2.062 = 0.608

(d) The price ceiling of P = 5/4 is higher than the prevailing market equilibrium price. So this price ceiling is non-binding and will have no effect on market outcomes, including the consumer and producer surpluses.

(e) At P^{C} = 0.8, the new Q^{S} = 1.8 and Q^{D} = 2.4.

So the there is a shortage = 2.4 – 1.8 = 0.6.

(g) At Q = 1.8, P^{D} = 0.933

Dead weight loss, DWL =0.5(0.933 – 0.8)(2.062 – 1.8) = 0.0174.

**Question 5**

The pricing rule is: P_{1}/P_{2} = (1 + 1/E_{2})/(1 + 1/E_{1}). If E_{1} is smaller in absolute value than E_{2} (i.e., less elastic), then to maximize profits we require **P _{1} > P_{2}**. Since BP have a less elastic demand than GS, the BP group should be charged the higher price.

The best way to show this result on a diagram is to graph these two schedules along with MC and then show that the profit-maximizing price is higher for BP than for GS.

If Bios can set a different administration fee F for GS and BP, then it can independently set P = MC = 20 and then set F(GS) = CS of one member from GS group and F(BP) = CS of one member from BP group. This way the monopolist can extract all consumer surpluses from both groups. Thus CS for both groups will be zero and PS will be maximized.

If Bios wants to keep both groups as customers, then the common F = lower CS of an individual group member. With P = 30, Q(GS) = 100 – 2(30) = 40 and Q(BP) = 55 – 0.5(30) = 40.

Therefore,

CS(GS group) = 0.5(50 – 30)*40 = 400 and

CS(GS group) = 0.5(110 – 30)*40 = 1600

If there are 10 students in each group, then F = lower CS/10 = 400/10 = $40.

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