EconS 425 - Intertemporal Considerations

EconS 425 - Intertemporal Considerations

Eric Dunaway

Washington State University

[email protected]

Industrial Organization

Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 37

Introduction

Today we�ll look at how time plays a factor in both �rm decisions andmarket e¢ ciency.

Lightboards coming soon, probably a week or two.


Patience

People have di¤erent levels of patience.

We �nd that our patience levels are even dynamic. Are you more orless patient when it is below freezing outside?Some people like the cold and get more patient; others want to getinside as fast as possible.

Firms also vary on their patience levels.

How do pro�ts tomorrow compare with pro�ts today?


Patience

A bit of a thought exercise:

What if I o¤ered you $1,000 today, or $1,100 a year from now. Whichwould you prefer?Why or why not are you willing to wait for that extra $100?


Patience

Now, what if I o¤ered you $1,000 today or $5,000 a year from now?

Has anyone changed their mind?


Patience

People and �rms tend to value the future at less than or equal to thepresent in terms of utility or pro�ts.

We like to have our rewards now, even if it means getting a slightlylarger reward later.

This is actually completely rational behavior, and we can modelpatience into what we have already done in order to see how timeplays a role in our decision making.

We call it discounting.


Discounting

Mathematically, we economists stole the idea of discounting from ourfriends in �nance.

It�s ok. They steal stu¤ from us all the time. And we both steal a lotfrom physics.

We can use the idea of bond interest rates to determine a discountfactor.

The discount factor is by what percent we value the future relative tothe present.


Discounting

Let�s say that I sell you a bond for $100 that has a maturity time ofone period and an interest rate of 5%.

The formula for calculating what a bond is worth in the future is asfollows:

Future Value = (Present Value)(1+ r)

where r is the interest rate.

Once the bond matures, I pay you 100(1+ 0.05) = $105, whichaccounts for both the initial bond price and the amount gainedthrough interest.


Discounting

Future Value = (Present Value)(1+ r)

We�re really not interested in the future value, though. We want toknow what something in the future is valued at in present terms.

Good thing we have algebra.

We can rearrange the above equation to express the present value ofa bond in terms of its future value and the interest rate.

Present Value =Future Value

1+ r


Discounting

Present Value =Future Value

1+ r

From here, we can de�ne the discount factor (The book uses R, I useδ) as

δ � 11+ r

and our above equation becomes

Present Value = δ(Future Value)


Discounting

δ � 11+ r

Lets look at the discount factor a little more.

We aren�t going to deal with negative interest rates.This means that at most, δ could equal 1 (if the interest rate, r , is 0).A discount factor of 1 would imply that the future isn�t discounted atall, and rewards in the future are equally weighed as rewards in thepresent.Most of the time, δ will take a value between 0 and 1. As δ gets closerto 0, the future is discounted more and more and future rewards areweighed less and less relative to present rewards.


Discounting

With this, we can now chain multiple periods together into a singleproblem.

For example, let�s consider a monopolist who needs to make a decisionover two periods. They have inverse demand functions p1 and p2 forperiods 1 and 2, respectively, and need to choose an output level forboth periods, q1 and q2. Their total cost functions are c1 and c2.

The pro�t maximization problem would be

maxq1,q2

p1q1 � c1 + δ (p2q2 � c2)

which we can simplify be de�ning pro�ts as πi � piqi � ci , giving

maxπ1,π2

π1 + δπ2


Discounting

Why stop at 2 periods?

We can extend this out to as many periods as we want. For everyadditional period, we just multiply by δ an additional time.For a third period, we discount it twice, i.e., multiply it by δ2, a fourthperiod is multiplied by δ3, and so on...

The present value for a monopolist who must make their decisionsover n periods can be expressed as

PV = π1 + δπ2 + δ2π3 + ...+ δn�1πn

=n

∑i=1

δi�1πi

Let�s look at an example:


Discounting Example

Headed into a three day weekend, a friend of yours has managed toobtain a 12-pack of the high quality adult beverage Pullman Water.He has nothing else to drink over the weekend and must decide howmuch to drink on each of the three nights.

Their utility function is

log(c1) + δ log(c2) + δ2 log(c3)

where ci is the amount of Pullman Water consumed on night i .

Note: in economics, if we write log, we mean the natural logarithm (ln)


Discounting Example

log(c1) + δ log(c2) + δ2 log(c3)

c1 + c2 + c3 = 12

We can write their lagrangian as

L = log(c1) + δ log(c2) + δ2 log(c3) + λ(12� c1 � c2 � c3)and take �rst-order conditions to obtain

∂L∂c1

=1c1� λ = 0

∂L∂c2

=δ

c2� λ = 0

∂L∂c3

=δ2

c3� λ = 0

∂L∂λ

= 12� c1 � c2 � c3 = 0


Discounting Example

1c1� λ = 0 δ

c2� λ = 0

δ2

c3� λ = 0 12� c1 � c2 � c3 = 0

From the �rst two equations, we can set them equal to one anotherto obtain

1c1� λ =

δ

c2� λ

c2 = δc1

Likewise, with the �rst and third equations,

1c1� λ =

δ2

c3� λ

c3 = δ2c1


Discounting Example

c2 = δc1c3 = δ2c1

Plugging both of these into the last equation,

12� c1 � c2 � c3 = 0

12� c1 � δc1 � δ2c1 = 0

Rearranging,

c1 + δc1 + δ2c1 = 12

c1(1+ δ+ δ2) = 12

c�1 =12

1+ δ+ δ2


Discounting Example

c2 = δc1c3 = δ2c1

c�1 =12

1+ δ+ δ2

From here, we can solve for c2 and c3,

c�2 =12δ

1+ δ+ δ2c�3 =

12δ2

1+ δ+ δ2

Let�s take a look at what happens.


Discounting Example

Comparing a few values of δ

δ c�1 =12

1+δ+δ2c�2 =

12δ1+δ+δ2

c�3 =12δ2

1+δ+δ2

0 12 0 00.25 9.14 2.29 0.570.5 6.86 3.43 1.710.75 5.19 3.89 2.921 4 4 4

As δ increases, your friend lowers their consumption on the �rst nightand instead consumes more on nights two and three.

At the extreme case, when δ = 1, they actually value all of the nightsthe same and consume an equal amount each night.


Discounting

As people get more patient, they consume less in earlier periods andshift that consumption to later periods.

This is analogous for �rms; as they get more patient, they may makedecisions that bene�t later periods.

We do need to be able to link the periods, though.

Let�s brie�y look at another model.


Learning by Doing

In the learning by doing model, a monopolist produces for twoperiods. In the �rst period, they select an output quanitity, q1.

We say that they learn how to more e¢ ciently produce the product dueto their experience in period 1. In period 2, the total cost functionactually decreases the more that the �rm produces in q1.Their experience from the �rst period allows them to make the productmore e¢ ciently in the second period.This could cause a monopolist to deviate from the monopoly outputlevel.

You�ll calculate this equilibrium for homework.


Time as a Constraint

There are other ways time can enter into our models, speci�cally inthe nature of goods.

If a good lasts for more than a single period, we call it a durable good.

Durable goods, if valued appropriately, can cause contraints onMonopoly power.

This was proposed by Ronald Coase in 1972.


Coase Conjecture

Consider a monopolist that has two units of a durable good that willlast for two periods.

The monopolist already has them, so the cost is 0.

There are two consumers, and they value the good at 50 and 30,respectively.

If the market were perfectly competitive, the price of the good wouldbe 0 (Bear with me).

Thus, the consumer surplus for each consumer would be:

High Type : 50+ δ50 = 50(1+ δ)

Low Type : 30+ δ30 = 30(1+ δ)

with a total surplus of 50(1+ δ) + 30(1+ δ) = 80(1+ δ).


Coase Conjecture

Starting with the monopolist,

If for some reason he doesn�t sell to either consumer in the �rst period,he can either charge 50 in the second period and sell 1 unit, or charge30 and sell both units. Obviously, he�ll charge 30 and sell both since60 > 50.

Since the consumers can enjoy the good for more than one period,they might be willing to pay more than their actual valuation in the�rst period, since they get the bene�t in both periods.

In fact, the high type consumer might be willing to pay up to50(1+ δ) in the �rst period, while the low type might be willing to payup to 30(1+ δ) (Those are their surpluses).


Coase Conjecture

Why so much?

This will make more sense once we get to game theory, but you canthink of the consumers�consumer surplus as how much bene�t theyget from buying the product.If they don�t buy it, their consumer surplus is 0 (Not necessarily bad,just status quo).If, in the case of the high type consumer, they pay something just lessthan 50(1+ δ) for the good, their consumer surplus will be greaterthan 0, which implies that they are better o¤ than not purchasing thegood at all.


Coase Conjecture

Back to the monopolist; they have three real options in the �rstperiod.

If they charge more than 50(1+ δ) in the �rst period, nobody willbuy the good, and then the monopolist will sell both goods at a priceof 30 in the second period. The surpluses would be:

High Type : 0+ δ(50� 30) = 20δ

Low Type : 0+ δ(30� 30) = 0Monopolist : 2 � δ30 = 60δ

Total Welfare : 20δ+ 0+ 60δ = 80δ

DWL : 80(1+ δ)� 80δ = 80


Coase Conjecture

If they charge a price of 30(1+ δ) in the �rst period, both consumerswill buy the good in the �rst period, and there will be nothing left tosell in the second. The surpluses would be:

High Type: [50� 30(1+ δ)] + δ50 = 20(1+ δ)

Low Type: [30� 30(1+ δ)] + δ30 = 0

Monopolist: 2 � 30(1+ δ) = 60(1+ δ)

Total Welfare: 20(1+ δ) + 0+ 60(1+ δ) = 80(1+ δ)

DWL: 80(1+ δ)� 80(1+ δ) = 0

Note: It would be irrational for a monopolist to charge less than30(1+ δ) in the �rst period. They would not gain anything for thelower price.


Coase Conjecture

If the monopolist charges a price between these two in the �rstperiod, he could only attract the high type consumer to the good.But what price should he charge?

What if he charged the maximum the high type was willing to pay,50(1+ δ)?

If the high type accepted this price, then their consumer surplus wouldbe 0 (He would be indi¤erent about buying the good).If the high type rejected this price, they could buy the good in thesecond stage for a price of 30, and their consumer surplus would be20δ.Since 20δ > 0 as long as δ > 0, the high type would just wait it outrather than pay the high price in the �rst period.


Coase Conjecture

Thus, the price has to be less than 50(1+ δ) in the �rst period.

We can �gure out the correct price. It has to be such that the hightype is indi¤erent between buying the good in the �rst period andwaiting out for the cheaper price in the second period.

[50� p] + δ50| {z }CS from buying in the �rst period

= 20δ|{z}CS from waiting for the second period

Solving for p,p = 50+ 30δ


Coase Conjecture

Using p = 50+ 30δ, we can calculate our surpluses,

High Type: [50� (50+ 30δ)] + δ50 = 20δ

Low Type: δ(30� 30) = 0Monopolist: [50+ 30δ] + δ30 = 50+ 60δ

Total Welfare: 20δ+ 0+ 60δ = 80δ

DWL: 80(1+ δ)� 80δ = 80


Coase Conjecture

What should the monopolist do? Let�s compare the surplus for themonopolist for each option in the �rst period,

30(1+ δ) 50+ 30δ No saleSurplus 60(1+ δ) 50+ 60δ 60δDWL 0 80 80

Since charging a price of 30(1+ δ) yields the highest surplus level forthe monopolist, that�s the price they want to charge.

Interestingly, this price leads to no deadweight loss, i.e., it�seconomically e¢ cient.


Coase Conjecture

What�s going on here?

The high type consumer can use the threat of waiting it out until thesecond period to force the monopolist to lower its price in the �rstperiod.It becomes better for the monopolist to just charge an even lower priceand sell both units right away, leading to an economically e¢ cientoutcome.

The key is the threat. If the high type consumer couldn�t threaten towait out for a lower price, the monopolist would be able to leverageits market power.


Coase Conjecture

What if the low type consumer was only willing to pay 20 for thegood?

This changes the monopolist�s behavior in the second period. Now,they could charge 50 and sell 1 unit, or charge 20 and sell two units.Naturally, now the monopolist would want to charge the higher priceand sell one unit. This removes the high type consumer�s ability tothreaten the monopolist with waiting out for the lower price.

In this situation, the monopolist would prefer to charge a price of50(1+ δ) in the �rst period. In the second period, the monopolistcharges 50 if the high type didn�t purchase in the �rst period, and 20if only the low type remains.

There ends up being no consumer surplus, and deadweight loss of 20.


Coase Conjecture

In order for durable goods to mitigate a monopolist, the willingness topay for both the high and low type consumers must be fairly close toone another.

If the low type is willing to pay less than half of what the high type iswilling to pay, the outcome will not be economically e¢ cient.

As a note: this is just one of the ways that durability plays a role ineconomic e¢ ciency. There are many others; most of which arebeyond the scope of our class.


Summary

Consumers and producers use discounting when dealing with rewardsthat occur in the future.

More patient consumers are willing to give up present consumption forfuture bene�t.

Durable goods, under certain conditions, can reduce the power of amonopolist.


Next Time

Technology and Cost Relationships.

How do cost structures in�uence a �rm�s decision?What happens when we add time as a factor to our models?

Reading: Sections 3.1and 3.2.

Reminder: No class Monday, Holiday!


Assignment 1-3

Consider a market served by a monopolist that faces the followinginverse demand and total cost curves,

P = 100�QTC = 50+ 10Q + 0.5Q2

1. Calculate the equilibrium price and quantity.2. Now, suppose the monopolist produced over two periods, and was ableto bene�t from learning by doing. In the second period, the total costfunction becomes

TC2 = 50+ 10Q2 + 0.5Q22 � 9Q1

where Qi is the monopolist�s output in period i . The �rst period totalcost function remains unchanged. Calculate the equilibrium price andquantity for both periods.

3. Compare your results from parts 1 and 2. Intuitively, what is themonopolist doing?


EconS 425 - Intertemporal Considerations

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