Examples for equilibrium and optimization problems

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ELTE Faculty of Social Sciences, Department of Economics

Microeconomics I.

week 3

WORKING TOOLS, PART 1 Authors:

Gergely K®hegyi, Dániel Horn, Klára Major Supervised by

Gergely K®hegyi

June 2010

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Equilibrium Demand and supply analysis Changing external factors Eects of government intervention

The course was prepaerd by Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer and David Hirshleifer (2009) Mikroökonómia. Budapest: Osiris Kiadó, ELTECON-books (henceforth HGH), and Gábor Kertesi (ed.) (2004) Mikroökonómia el®adásvázlatok.

http://econ.core.hu/ kertesi/kertesimikro/ (henceforth KG).

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Equilibrium and optimization Equilibrium

Demand and supply analysis Changing external factors Eects of government intervention

1 Equilibrium and optimization

2 Equilibrium

Demand and supply analysis Changing external factors

Eects of government intervention

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Microeconomic problems always come in two forms:

equilibrium problems optimization problems

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Examples for equilibrium and optimization problems

Optimization problems

Should we buy a new car of keep the old one for a while?

Are we happier if we work or if we stop this drudgery and live on aid?

Should we buy or rent a at?

Should drug policy be stricter or lighter?

Equilibrium problems

Is it possible that prices of new cars will be lower next year?

Would unemployment assistance increase unemployment rate?

What aects the rent and the price of a at?

Would the number of drug users increase if drug use was legal?

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Optimization problems

Should a trade union go on strike or accept the

management oer?

Should the wings of a plane be made out of titanium or stainless steel?

Should the doctor prescribe antibiotics or placebo?

Should the general attack or retreat?

Should we rent or buy an oce?

Equilibrium problems

Why is the male/female birth ratio always around 1:1?

Why do some cultures prefer equality while others favor a more hierarchic structure?

What aects the index of the stock exchange?

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Equilibrium and optimization

Typical methods for dealing with the two problem types

Equilibrium problems: we are looking for the equality of the demand and the supply, (in a broad sense). [necessary mathematical tool: to solve a(n) (system of) equation]

Optimization problems: we are looking for the minimum or the maximum value by comparing the marginal values with each other. [necessary mathematical tool: (conditional) minimization/maximization]

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Denition

A reservation price

consumers: the highest price they are willing to pay for a unit of the good.

sellers: the lowest price they are willing to sell a unit of a good.

Note

Reservation price depends on the actor, the good, the situation and the amount of the good (how many units have been sold/consumed before).

Let's look at the market for ats to rent.

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Demand and supply analysis (cont.)

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Demand and supply analysis (cont.)

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Demand and supply analysis (cont.)

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Demand and supply analysis

Let's look at the market for memory chips.

(endogenous) variables:

Quantity (in natural units e.g. pieces) within a given time (e.g. 1 month)

Price (dollar/pc)

Relationship between variables (functions):

The demand curve shows the quantity the consumers are willing to buy on a given P price.

The supply curve, similarly, shows the quantity the sellers are willing to sell at a given price.

External factors (exogenous variables): technology, taste, resources, market characteristics of other goods, incomes, legal framework, etc.

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Note

We talk about price in terms of money, but in reality it is the exchange rate of two goods. For instance if 1 MB memory costs 1 dollar, and a cartridge 15 dollars, then the price of 1 MB memory in cartridge is 1/15. Or we can consider money as a good with unit price (this could be problematic).

Note

The decreasing slope of the demand curve shows the law of demand, so (intuitively) the fact that if the price of memory chips or phone calls, or shoes drops then usually consumers want to buy more from them. The increasing slope of the supply curve

indicates that the greater the price the greater the quantity oered.

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Demand and supply analysis

Equilibrium

P^∗: equilibrium price Q^∗: equilibrium quantity At the equilibrium point E, the quantity that consumers wish to purchase equals the quantity that sellers want to sell.

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Oversupply

If P⁰>P^∗then sellers would oer Q_S⁰ quantity, while buyers would only wish to buy Q_d⁰ quantity.

The size of oversupply:

Q_S⁰ −Q_d⁰ At P⁰ there is a downward pressure on price.

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Demand and supply analysis

Overdemand

If P”<P^∗then sellers would oer Q”_S quantity, while buyers would wish to buy Q”_d) quantity.

The size of overdemand:

Q”_d−Q”_S At P”there is an upward pressure on price.

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If the price diers from P^∗ then one of the above pressures (downward or upward) will start.

Consequence

The point where demand curve intersects the supply curve determines the equilibrium price P^∗ and quantity Q^∗.

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Demand and supply analysis (cont.)

Let Q_d =D(P)be the demand function which denes the demanded quantity at every price.

Let QS =S(P)be the supply function which sets the oered quantity at each price.

In equilibrium Q_d =Q_S, that is D(P^∗) =S(P^∗)

Note

Conventionally we depict the inverted demand and supply functions (D⁻¹(Q)and S⁻¹(Q))

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Supply function: S(P) =P In equilibrium

D(P^∗) =S(P^∗) 100−P^∗=P^∗

100=2P^∗ P^∗=50

Q^∗=D(P^∗) =S(P^∗) =D(50) =100−50=S(50) =50 P^∗=50 (equilibrium price) Q^∗=50 (equilibrium quantity) To invert the demand and supply functions let's equal them with Q and rearrange the equation to P

D(P) =100−P[=Q] D⁻¹(Q) =100−Q[=P]

S(P) =P[=Q]

S⁻¹(Q) =Q[=P]

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Demand and supply analysis (cont.)

2 Exercise (nding the equilibrium and graphical depiction) Demand function: D(P) =150−3P

Supply function: S(P) =2P−20 In equilibrium

D(P^∗) =S(P^∗) 150−3P^∗=2P^∗−20

170=5P^∗ P^∗=34

Q^∗=D(P^∗) =S(P^∗) =D(34) =150−3∗34=S(34) =

=2∗34−20=48

P^∗=34 (equilibrium price) Q^∗=48 (equilibrium quantity) To invert the demand and supply functions let's equal them with Q and rearrange the equation to P

D(P) =150−3P[=Q] D⁻¹(Q) =50−Q/3[=P]

S(P) =2P−20[=Q] S⁻¹(Q) =Q/2+10[=P]

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Equilibrium Demand and supply analysis Changing external factors Eects of government intervention 3 Exercise: general linear case (nding the equilibrium and

graphical depiction)

Demand function: D(P) =a+bP Supply function: S(P) =cP−d Equilibrium:

P^∗=a+d

c+b,Q^∗= ac−bd a+b

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Demand and supply analysis (cont.)

4 Exercise: non-linear case (nding the equilibrium and graphical depiction)

Demand function: D(P) = ²⁰⁰_P Supply function: S(P) =2P

D(P^∗) =S(P^∗) 200/P^∗=2P^∗

200=2(P^∗)² P^∗=10

Q^∗=D(P^∗) =S(P^∗) =D(10) =20

P^∗=10 (equilibrium price) Q^∗=20 (equilibrium quantity)

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Equilibrium Demand and supply analysis Changing external factors Eects of government intervention 5 Exercise: non-linear case (nding the equilibrium and

graphical depiction)

Demand function: D(P) =a−bP Supply function: S(P) =√

P−c D(P^∗) =S(P^∗) a−bP=√

p−c p^∗=2ab+1−√

1+4ab−4b²c 2b²

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Demand and supply analysis (cont.)

The state of equilibrium is a special state, but we do NOT claim it to be either GOOD or BAD.

Note

Remember the dierence between normative and positive analysis.

"Good" things, such as a house, can have equilibrium price and quantity just as much as "bad" things , such as heroin.

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Pareto-improvement is when an economic actor is better o while no-one else is worse o.

Denition

A state of allocation is Pareto-ecient (or Pareto-optimal), if no Pareto-improvement can be made.

Consequence

A state of allocation is Pareto-ecient if no actor can be better o without making some other actor worse o.

Note

Pareto-eciency has nothing to do with justice!

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Equilibrium and eciency (cont.)

Examples:

If a parent allocates 10 apples amond 5 kids, and one receives all, while the others none, that is a Pareto-ecient allocation (If each gets 2, that is also Pareto-ecient),

A state of allocation, where some small percentage of the population holds most of the national income, can be Pareto-ecient, even if there are many poor people in the country.

If I do not use my textbook and lend it to someone else, that is a Pareto-improvement (assuming that I am not too envious).

If a seller gives a Christmas-tree to someone for free after Christmas eve, that could be a Pareto-improvement. If the seller gets some money for it that is also Pareto-improvement.

The price can go as high as the reservation price of the consumer, and as low as the reservation price of the seller (which is zero, in this case).

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Statement

Market equilibrium is a Pareto-ecient state.

Denition

The (net) individual consumer surplus is the dierence between the reservation price and the market price of one good. This is the net monetary yield of buying a product. The consumer surplus for the whole market is the sum of the individual consumer surpluses, that is the area above the market price and under the inverted demand curve.

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Equilibrium and eciency (cont.)

Denition

The individual producer surplus is the dierence between the market price and the reservation price of one good. That is the net monetary yield of selling a product. The consumer surplus for the whole market is the sum of the individual consumer surpluses, that is the area below the market price and over the inverted supply curve.

Note

The sum of the consumer and producer surpluses is the wealth (in money) of the actors of the given economic system. I market equilibrium this amount is maximal.

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If the external factors change demand and/or supply will change as well. Such a change in external factors can be:

change in tastes (news about the dangers of high cholesterol could make people not to consume butter);

technological change (the integrated circuit has increased the supply of electrical home devices);

changes in resources (a new oil-led could increase the supply of raw oil.);

changes in regulation (legalizing marihuana could change both its supply and demand.);

changes in the price or quantity of other goods that aect the demand of the given good (lowering the price of printers will increase the demand for paper);

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How changes in supply and demand aect equilibrium? (cont.)

changes in the price or quantity of other goods that aect the demand of the given good (increase in beef supply will increase the supply of leather.);

increase in income (the increase in the income of brokers has increased the demand for luxury cars);

Note

By limiting the economic system, the system under study we dene what is external and what is internal factor.

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Note

By limiting the economic system, the system under study we dene what is external and what is internal factor.

Examples: Internal factors could be the

change in the price of another good, if we examine the interaction of the two markets.

change in income, if comes from the change in salaries and we look at the interaction of the labor market and the luxury goods.

change in tastes, if marketing is the part of our investigation.

Commercial costs change supply and advertisements change tastes.

change in technology and change in resources if increasing supply side research and development (R+D) is under investigation.

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How changes in supply and demand aect equilibrium? (cont.)

change in regulation, when we study the eects of corruption on the costs of highway construction. Or the eects of government programs on voting behavior. p

etc.

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If consumer preferences change so that they want to buy more from a product at each price then demand curve will shift to the right, from D1 to D2. In such cases both equilibrium price and quantity will increase.

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Change in supply

If changes in circumstances will make sellers oer more at each price then supply curve will shift to the right from S1to S2. In such cases equilibrium quantity will increase while equilibrium price will decrease.

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Statement

Increase in demand will increase both P^∗ equilibrium price and Q^∗ equilibrium quantity. Increase in supply will increase P^∗but decrease Q^∗. A simultaneous increase in demand and supply will increase Q^∗but can increase as well as decrease P^∗ (or leave it unchanged).

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A methodological guideline

Above we were concentrating on comparing equilibrium situations.

So we have NOT dealt with the shift from one equilibrium to another (we considered it as being outside our model). So we have not dealt with the DYNAMICS of the economic system.

Denition

The principle of COMPARATIVE STATICS means that we compare equilibrium situations; the original with another with changed conditions.

Note

Microeconomics follow the principle of comparative statics (with exceptions). We do that too!

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values of the (endogenous) variables.

Example:

Let the demand function for university sublets be:

D(P) =80000−2P

Let supply function be: S(P) =P−10000

How does the equilibrium price and quantity change university students receive 30000 HUF subsidy.

Original equilibrium:

D(P^∗) =S(P^∗) 80000−2P^∗=P^∗−10000

P^∗=30000,Q^∗=20000 D(P) =80000−2P[=Q] D⁻¹(Q) =40000−Q/2[=P]

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Changing external factors (cont.)

S(P) =P−10000[=Q] S⁻¹(Q) =Q+10000[=P]

Let's assume that each student is willing to add the amount of subsidy amount s/he wanted to spend on the sublet.

Then the new inverted demand function is:

D⁻¹(Q) =40000+30000−Q/2=70000−Q/2 The new demand function:

Dˆ(P) =80000+2×30000−P=140000−2P The new equilibrium:

Dˆ( ˆP^∗) =S( ˆP^∗) 14000−2Pˆ^∗= ˆP^∗−10000

Pˆ^∗=50000,Qˆ^∗=40000

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Supply function: S(P) =cP−d (Endogenous) variables: P,Q

Parameters (exogenous variables): a,b,c,d

Let's assume that the value of d parameter changes to d⁰, where (d⁰ >d)

Original equilibrium:

P^∗= a+d

c+b,Q^∗= ac−bd a+b New equilibrium:

P^∗= a+d⁰

c+b,Q^∗= ac−bd⁰ a+b

Equilibrium price increases, equilibrium quantity decreases.

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Another methodological guideline

So far we have assumed that only one of the external factors change, while others remain unchanged (and the changing factor aects only one parameter).

We assumed, for instance, that the university students will spend all their subsidy on their sublet and thus aecting their demand on sublets, although they might as well spend it on food (or beer...)

When we make a model we have to decide which factors are unchanged when we look at the eects of another changing factor.

Denition

The principle of CETERIS PARIBUS means that we look at the eects of one changing factor, while everything else is unchanged.

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Without import the E0

equilibrium is the intersection of the D demand and the S^h domestic supply curve The aggregated supply curve is PS, which is the

horizontal sum of S^h and Sⁱ curves. The new equilibrium is E₁.

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Taxing

When a tax is issued there will be two prices on the market, the dierence between the two is the size of tax

what buyers pay (demand price, or gross price): P⁺ what sellers pay (supply price, or net price): P⁻

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unit tax: a xed amount of money for each unit of the sold good (e.g.: Alcohol, petro, cigarettes)

The sellers are obligated to pay: P⁺=P⁻+T The buyers are obligated to pay: P⁺−T =P⁻

proportionate tax: a xed percent of the price (pl.: VAT) The sellers are obligated to pay: P⁺= (1+t)P⁻

The buyers are obligated to pay: (1−τ)P⁺=P⁻;τ=₁₊^t_t In equilibrium:

D(P⁺) =S(P⁻)

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Eect of unit tax

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Welfare eects of taxing

Note

When a good is taxed, both consumer and producer surplus will decrease, since price paid by buyers will increase while price received by sellers decrease.

The new consumer surplus is the area: A− −F⁰− −P⁺. The new producers surplus is the area: C− −G⁰− −P⁻. The amount of tax collected: (P⁺−P⁻)∗Q⁰.

The tax burden on consumers: (P⁺−P^∗)∗Q⁰. The tax burden on producers: (P^∗−P⁻)∗Q⁰.

The ratio of consumer and producer tax burden depends on the ratio of the elasticity of demand and supply.

Denition

Deadweight loss is the welfare loss in monetary terms created by taxes (the (F⁰− −G⁰− −E area)

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Demand: D(P) =220−P Supply: S(P) =P

Equilibrium: P^∗=110,Q^∗=110

1 How does the equilibrium price and quantity change when the unit tax is T =30?

2 How does the equilibrium price and quantity change when the proportionate tax is t=20?

3 How can we get to the equilibrium of the 20%-os proportionate tax with a unit tax?

4 How can we compensate consumers after the proportionate tax so that they can buy the product on the same price as before the tax. (How big should the subsidy be?)

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Taxes (cont.)

1 Unit tax:

D(P⁺) =S(P⁻);P⁺=P⁻+T 220−P⁻−30=P⁻

P⁻=95,P⁺=125,Q =95,Q×T =2850

2 Proportionate tax:

D(P⁺) =S(P⁻);P⁺= (1+t)P⁻ 220−1,2P⁻=P⁻

P⁻=100,P⁺=120,Q =100,Q×T =2000

3 The eect of the proportionate tax, with unit tax:

220−100−t=100;t =20

4 Compensation:

220+K−(1,2)×110=110;K =22 amount of subsidy: K/1,2=22/1,2'18,33

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Beer taxes and drinking by high school students, 1989 (percentage) Category (of drinking Actual Estimate after occasions in past year) distribution tax adjustment

Abstainers (none) 15,3 18,6

Infrequent (19) 44,4 46,1

Fairly frequent (1030) 27,1 24,7

Frequent (more than 30) 13,2 10,6

Total 100,0 100,0

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Price ceiling and price oor

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Note

Price ceiling and price oor, as well as other government interventions that distort market prices, will create deadweight loss.

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Price ceiling and price oor (cont.)

E.g.:

Demand: D(P) =220−P Supply: S(P) =P

Equilibrium: P^∗=110,Q^∗=110

1 How does the equilibrium price and quantity change when the price ceiling isP¯ =80?

2 How does the equilibrium price and quantity change when the price ceiling isP¯ =120?

3 How does the equilibrium price and quantity change when a quota of Q¯ =50 is introduced, so that no more than this amount can be sold?

4 How does the equilibrium price and quantity change when a quota of Q¯ =140 is introduced, so that no more than this amount can be sold?

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Equilibrium Demand and supply analysis Changing external factors Eects of government intervention 1 P¯ =80<110=P^∗, soP¯ =80 is equilibrium price,

S( ¯P=80)is equilibrium quantity, and

D( ¯P)−S( ¯P) =220−80−80=60 is the size of over demand.

2 P¯ =120>110=P^∗, so P^∗=110,Q^∗=110.

3 Q¯ =50<110, so D( ¯Q) =220−50=170=P^∗.

4 Q¯ =140>110, so P^∗=110,Q^∗=110.

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Interdiction of supply

The government intercepts the fraction i=50%of the amounts supplied. The new supply curve S⁰S⁰ is twice as high as the original. Since the delivered quantity Q⁻ = (1−i)Q⁺is just half the produced quantity, the eective supply curve is S”S”.

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Yearly acquisitions of three supported crops by the commodity credit corporation, selected years

Sorghum Corn Wheat

1953 40,9 422,3 486,1

1954 110,1 250,6 391,6

1955 92,6 408,9 276,7

1956 32,5 477,4 148,4

1957 279,5 268,1 193,5

1958 258,0 266,6 511,0

1963 125,1 17,9 85,1

1964 66,8 29,1 86,9

1965 85,0 11,2 17,4

1966 0,3 12,4 12,4

1967 9,1 191,0 90,0

1968 13,7 34,4 182,9

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Dening the demand and supply

Dening the relevant market:

What are the boundaries of a market? What eects should we considers and what to neglect?

Dening the demand and supply functions:

Problem: what sort of data do we have?

In theory there are only one equilibrium price (which changes in time, but the reasons are important . . . )

Time series regressions

Tests 1. ("Would you be willing to pay 10 000 HUF for the internet at home? And 11 000 HUF?")

Tests 2. ("If gas would be 20% more expensive, would you switch to oil heating?")

Note

The above list brings up several methodological diculties. These will be discussed in later subjects (e.g. econometrics).