Psurplus Handout

Producers Surplus

Philip A. Viton

October 24, 2011

Philip A. Viton ()CRP 763 Producers Surplus October 24, 2011 1 / 22

Introduction

We have seen how to evaluate consumers wtp for a market-inducedprice change: we measure this (with appropriate cautions, primarilythat we treat it as an approximation) by the change in consumerssurplus, the area under the demand function between the two pricehorizontals.

We now address to the remaining question in market-induced (price)changes: how to evaluate the benets to producers or rms.

Fortunately this is easy, and has none of the complications ofconsumers surplus.

The obviously correct measure of benets to rms is prots.

If a project changes the level of the rms prots, we measure theimpact by the change in its prots.


Prots

Consider a rm selling x1 units of output at price p1.

Its Total Revenues (TR) are p1 x1.Write its Total Costs as TC .

Then prots are Total Revenues minus Total Costs or:

= TR TC .


Total Costs: Fixed and Variable

Typically a rm will have both xed and variable costs.

So write total costs as:

TC = TFC + TVC

Then we have: = TR (TFC + TVC )


Change in Prots

Suppose a project changes revenues from TRa to TRb and costs fromTC a to TC b

Using our decomposition of costs, we have

= b a= (TRb TC b) (TRa TC a)= (TRb (TFC + TVC b)) (TRa (TFC + TVC a))= (TRb TVC b) (TRa TVC a)

since by denition xed costs are unchanged by the project.The quantity

PS = TR TVCis called the rms Producer Surplus.So we see that the change in the rms prots () is the same asthe change in producers surplus:

= PSb PSa = PSPhilip A. Viton ()CRP 763 Producers Surplus October 24, 2011 5 / 22

Producers Surplus I

It is important to understand that while the change in prots equalsthe change in producers surplus, prots and producers surplusthemselves are not the same.

They dier by the quantity of xed costs:

= TR (TVC + TFC )= TR TVC TFC= PS TFC

But since we will always be looking at the change in producerssurplus, this wont matter.

Finally, remember that unlike the case of consumers surplus,producers surplus isnt an approximation to anything.


Producers Surplus II

Using the change in producers surplus has two advantages over thechange in prots:

1. It does not require us to understand xed costs; and empirically xedcosts (ie the value of the constant or intercept in a regressionequation) is often imprecisely estimated.

2. If we know something about the rms marginal cost schedule, it iseasy to calculate total variable costs: they are simply the area underthe marginal cost curve from x1 = 0 (zero output) to xa1 (observedoutput).


Total Variable Costs

x1

$

p1a

x1a

MC

0

The gure shows a rmsupward-sloping marginal costcurve ( = supply function)

If it is a prot-maximizingprice-taker, then, if it facesmarket price pa1 its output willbe xa1Total revenue is pa1 xa1(hatched)

Total variable costs are the areaunder MC from x1 = 0 tox1 = xa1 ( dark shade)

So producers surplus is the areaabove MC but below the pricehorizontal (light shade)


Project

x1

$

p1b

x1b

MC

0

Suppose the rm is a price-takerin the output market, and thatour project has the eect ofraising the rms market priceto pb1 . The rm responds byraising its output to xb1 .

The new total revenue is thehatched area, total variablecosts are darkly shaded andproducers surplus is lightlyshaded.


Change in Producers Surplus

x1

$

p1a

x1a

p1b

x1b

MC

0

Comparing the previouspictures, we can see that thechange in producers surplus isthe area to the left of themarginal cost curve and betweenthe two price horizontals.

Note that this is valid only for acompetitive rm that sees itsmarket price change with noshift in its cost curves.


Multiproduct Producers Surplus

When a rm produces several products and the project changes theirprices, we naturally measure the full impact as the change inproducers surplus in each of the product markets. So:

PS = PS1 + PS2 + . . .

(just as we did for consumers surplus).

It can be shown that PS is not path-dependent: that is, for amultiproduct rm you will get the same answer in whatever order youdo the calculation (unlike consumers surplus).


Market Impacts Full Analysis

We are now in a position to evaluate the full impact of a policy thatchanges a market price from its pre-project level pa to its post project levelpb

For the impact on consumers we use the change in consumerssurplus:

CS = CS(pb) CS(pa)For the impact on rms we use the change in producers surplus(aggregating over rms, if necessary):

PS = PS(pb) PS(pa)And then the full impact is:

W = CS + PS


Example Monopolization

We have seen that monopolization has two impacts: it reduced thequantity produced and increases the price consumers pay for theproduct.

Intuitively, this is a bad thing ; but you might wonder if it really is:what if the price increase leads to an increase in rm prots mightnot this outweigh the loss to consumers?

By using our new tools, we can see that this is not so.


Monopolization Setup

S MC=S

DMR

$

xxaxb

papb

The gure shows the basicposition: under competition,demand = supply, where supplyis the sum of the rmsmarginalcosts ; the result is price pa andoutput xa.

Under monopolization,prot-maximizing output isgoverned by the condition MC= MR, leading to output xb .The monopolist prices to sell alloutput, resulting in a price pb .


Computational Aside I

Suppose you want to calculate the pre-project equilibrium (pa, xa).Conceptually, you want to equate supply (= MC) and demand.

However, note that the marginal cost function is MC (x) : that is, ittakes a quantity and delivers its marginal cost, in dollars.

While the demand function is x(p) : that is, it takes a price anddelivers the demand (quantity) at that price.

So you cant just equate x(p) and MC (x) : that would bemeaningless.

Instead, you must rst invert (solve) either the demand function orthe marginal cost function, so they both deliver the same thing(output or money). Then you can equate the two equations.


Computational Aside II

For example, suppose in the linear case you have:

Marginal cost: c = 0 + 1x (1 > 0)Demand: x = 0 + 1p (1 < 0)The in order to nd the equilibrium you can invert the demandfunction, giving p = (x 0)/1 and then equate supply anddemand:

(x 0)/1 = 0 + 1xwhich is an equation in quantity only.Alternatively, you can just feed in the price (cost) equation into thedemand function:

x = 0 + 1p

= 0 + 1(0 + 1x

which is also an equation in x only; and solve.


Monopolization Impact on Consumers

S MC=S

DMR

$

xxaxb

papb

As usual, the impact onconsumers is CS = signedarea under the demand functionbetween the two pricehorizontals.

This is the shaded area, and it isimportant to note that it is anegative quantity.


Producers Surplus under Competition

S MC=S

DMR

$

xxaxb

papb

Pre-project producers surplusPSa is TR ( = pa xa) TVC(= area under MC from x = 0to x = xa).

In the gure, this is shaded.


Producers Surplus under Monopoly

S MC=S

DMR

$

xxaxb

papb

Similarly, relative to themonopoly equilibrium (xb , pb)post-project producers surplusPSb is the shaded area.


Change in Producers Surplus

S MC=S

DMR

$

xxaxb

papb

+-

Comparing the two previousgures, PS is the shaded areaNote that this has two parts: arectangle, representing wherePSb is more than PSa andhence has a positive sign ; and alittle triangle representing partof PSa not covered by PSb : sothis bit has a negative sign.

Obviously the net eect ispositive : as wed expect,monopoly increases rm prots.


Net Impact

S MC=S

DMR

$

xxaxb

papb

For the overall net impact, wecompare PS to CS . We seethat CS more than covers thearea of the rectangle part ofPS . The left-over area is partof CS and hence is negative.There is also the little triangleof PS , which is also negative.So all net impacts work out to adeterminately negative area,shown in the gure as theshaded area.


Conclusion

We conclude that going from a situation of competition to one ofmonopoly has an unambiguously negative impact.

In the literature, this is referred to as the Deadweight Loss tomonopolization.


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