ECON 306 — Microeconomic Analysis
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4.1 — Modeling Firms With Market Power — Appendix

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    • 4.1 — Modeling Firms with Market Power
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On this page

  • Monopolists Only Produce Where Demand is Elastic: Proof
  • Derivation of the Lerner Index
  • Firms With Market Power vs. Competitive Firms’ Responses to Market Changes

4.1 — Modeling Firms With Market Power — Appendix

Monopolists Only Produce Where Demand is Elastic: Proof

Let’s first show the relationship between MR(q) and price elasticity of demand, ϵD.

MR(q)=p+(ΔpΔq)qDefinition of MR(q)MR(q)p=pp+(ΔpΔq)qpDividing both sides by pMR(q)p=1+(ΔpΔq×qp)⏟1ϵSimplifyingMR(q)p=1+1ϵDRecognize price elasticity ϵD=ΔqΔp×pqMR(q)=p(1+1ϵD)Multiplying both sides by p

Remember, we’ve simplified ϵD=1slope×pq, where 1slope=ΔqΔp because on a demand curve, slope=ΔpΔq.

Now that we have this alternate expression for MR(q), lets assume MC(q)≥0 and set them equal to one another to maximize profits:

MR(q)=MC(q)p(1+1ϵD)=MC(q)p(1−1|ϵD|)=MC(q)

I rearrange the last line only to remind us that ϵD is always negative!

Now note the following:

  • If |ϵD|<1, then MR(q) is negative. Since MC(q) is assumed to be positive, it cannot equal a negative MR(q), hence this is not profit-maximizing.
  • If |ϵD|=1, then MR(q) is 0. Only if MC(q) is also 0 is this profit-maximizing.
  • If |ϵD|>1, then MR(q) is positive. It can equal a positive MC(q) to be profit-maximizing.

Hence, a monopolist will never produce in the inelastic region of the demand curve (where MR(q)<0), and will only produce at the unit elastic part of the demand curve (where MR(q)=0) if MC(q)=0. Thus, it generally produces in the elastic region where MR(q)>0.

See the graphs on slide 33.

Derivation of the Lerner Index

Marginal revenue is strongly related to the price elasticity of demand, which is ED=ΔqΔp×pq1

We derived marginal revenue (in the slides) as: MR(q)=p+ΔpΔqq

Firms will always maximize profits where:

MR(q)=MC(q)Profit-max output conditionp+(ΔpΔq)q=MC(q)Definition of MR(q)p+(ΔpΔq)q×pp=MC(q)Multiplying the left by pp (i.e. 1)p+(ΔpΔq×qp)⏟1ϵ×p=MC(q)Rearranging the leftp+1ϵ×p=MC(q)Recognize price elasticity ϵ=ΔqΔp×pqp=MC(q)−1ϵpSubtract 1ϵp from both sidesp−MC(q)=−1ϵpSubtract MC(q) from both sidesp−MC(q)p=−1ϵDivide both sides by p

The left side gives us the fraction of price that is markup (p−MC(q)p), and the right side shows this is inversely related to price elasticity of demand.

Firms With Market Power vs. Competitive Firms’ Responses to Market Changes

Consider a firm in a competitive market (left) and a firm with market power (right):

An Increase in Firms’ Marginal Cost

A competitive firm responds by only changing its output q⋆ (since it cannot control price), whereas the firm with market power changes both its p⋆ and q⋆.

A Shift in Market Demand

Both firms change p⋆ and q⋆, but there is a much smaller change in q⋆ for the monopolist.

A Change in Price Elasticity of Demand

For the competitive market on the left, there is no change in q⋆ or p⋆ for the industry! On the right, the monopolist will lower (raise) p⋆ and raise (lower) q⋆ as demand becomes more (less) elastic!

Footnotes

  1. I sometimes simplify it as ED=1slope×pq, where “slope” is the slope of the inverse demand curve (graph), since the slope is ΔpΔq=riserun.↩︎

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