SM 06 L Capacity Constraints

Motivation Rationing of Demand Capacity Constrained Price Competition Capacities Discussion

Capacity Constraints

Franz Hubert


Motivation

Bertrand price competition:

1 Prices are set.

2 For given prices quantities are supplied until the demand issatisfied.

Two (related) objections:

1 Often it is easier to change prices than to change quantities.

2 Usually a single firm cannot serve the whole market at shortnotice (capacity constraints).


Effect of Capacity Constraints

Intuition:

With capacity constraints the incentive to undercut the rival isweakened.

Suppose he undercuts, but can serve only half of the customers,then I am still a monopolist for the residual demand.

Do we observe this often?


Alternative Approach

Prices are fast, capacity is slow.

Dynamic game:

1 Capacities are set.

2 Given capacities, prices are set.

3 Demand is served, subject to capacity constraints.

Previous decisions cannot be changed (commitment).

Agents rationally anticipate the future.

Solution concept: subgame perfect Nash-equilibrium.


Think forward, solve backward

We find the subgame perfect equilibrium by backward induction.

3 How is the demand served for all possible prices andcapacities.

2 Nashequilibrium in prices for any given pair of capacities:pA(kA, kB), p

B(kA, kB).

1 Nashequilibrium in capacities (kA, kB) given that price

competition is correctly anticipated.


Two Strategic Decisions

We have to understand two types of strategic decisions:

Choosing prices when capacity is constrained (anticipating howdemand is rationed)

Choosing capacities in anticipation of price competition.


Capacity Constraints

For simplicity: zero marginal cost of quantity up to the capacitylimit, which is strictly binding. Installing capacity is costly.

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Rationing of Demand

Suppose pA < pB but D(pA) > kA.

random or proportional rationing:Faced with excess demand buyers are selected at random.

parallel or efficient rationing:Buyers with the largest willingness to pay are served first.

What does this imply for the residual demand of high price firm B?


Random rationing

Residual demand for high price firm:

DR(pB) = D(pB)(1 kA

D(pA)

)= D(pB) kAD(pB)

D(pA).

pA

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Efficient rationing

Residual demand for high price firm:

DR(pB) = D(pB) kA, or P (kB + kA)

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D(p)

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Residual demand is independent of pA.

Rationing is efficient: there is no mutually beneficial trade possiblebetween those who obtained the good cheap and those who whererationed out.

Residual demand with efficient rationing is lower than underrandom rationing. Hence, efficient rationing is worse for the highprice rival.


Analogy to Monopoly

In a sense both firms resemble a monopolist:

low price firm faces D(p),high price firm faces DR(p)

When I know which firm is cheaper, it does not matter whether Iconsider price or quantity as instrument.

Define accommodating (market clearing) price p0:

D(p0) = kA + kB, or p0 = P (kA + kB)

If both firms charge p0 then available capacities will be fully used.


Capacity Constrained Price Competition

We have to find the Nashequilibrium in prices for all possible(kA, kB) [0, D(0)]2.We do so in several steps.

First we divide the parameter space into different regions, todistinguish cases, in which capacities are small, intermediate, andlarge.


Dividing the Parameter Space

Let QA(qb) be firm As best reply in quantities (ignoring capacityconstraints):

QA = argmaxqA P (qA + qB)qA,

(and likewise QB).

small: Both firms would like to increase their quantities (reducethe prices).

intermediate: If one firm sells all its capacity, the rival uses onlypart of its own capacity.

large: If one firm sells all its capacity, the rival will not sellanything.


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kA, qA

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Small capacities

Small capacities: Ks = {kA, kB : kA QA(kB), kB QB(kA)}Result 1: For small capacities, a Nash equilibrium in purestrategies exist, in which both firms charge the market clearingprice p0.

Proof:Undercutting p0 does not pay because additional demand couldnot be served.

Overbidding p0 does not pay by definition of small capacity.

Note: Firms set prices as an auctioneer would do.


Small capacities

kA

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Large capacities

Result 2: If both firms can serve the market at zero price, then weobtain the Bertrand result pA = pB = 0.


Intermediate capacities

Result 3: There exists no equilibrium in pure strategies(Edgeworth 1897)

Intuition: for small prices, there is excess demand and one firmwould deviate by charging the monopolist price for the residualdemand (becoming thereby the high price firm). If prices areunequal, the low price firm (selling up to capacity) increases itsprice just below the high price. But then the high price firm cangain by undercutting the low price firm, and prices will fall again(Edgeworth cycle).

Result 4: There exists an equilibrium in mixed strategies. (Kreps& Scheinkmann 1983)

Result 5: In the mixed strategy equilibrium the profit of the largerfirm is smaller than in the unconstrained Cournotequilibrium.


Selection of capacities

How will the firms select their capacities if they anticipate this kindof price competition?

Installing capacity is expensive!

Suppose my rival installs a small capacity. Within the range ofsmall capacities my optimal response function would be of theCournot form.

Since installing capacity is costly, the reaction functions of the twofirms shift inwards.


Cournot solution

kcA

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kB(kA)

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Cournot solution

It does not pay to deviate to large capacities because thisdecreases profit for the deviating firm (Results 4 and 5).

Therefore:

Result 6: In equilibrium both firms will choose Cournot capacities.


Cournot solution

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piD < pic

piB < pic

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Discussion

Competition in capacities followed by price-competition yieldsCournot result.

This explains, why we rarely see high and low prices (for the samegood!) and quantity rationing. Firms will not choose the capacitieswhich lead to this outcome.

This is true only for efficient rationing of demand. Other forms ofrationing leave more residual demand to the high-price firm andcreate incentives to raise the price at the Cournot Solution.

MotivationRationing of DemandCapacity Constrained Price Competition CapacitiesDiscussion

Documents

SM 06 L Capacity Constraints