Discussions of Modeling [1 ed.] 9780740932786

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Purpose and Practices in Econo111ic Theory

Brennan

C. Platt

B1.t.T .l\. types of· cons,11ne1·s etc.)) and must compute For a continuous

the average outcome 01·a r·andom dravv. dist1·ibution)

this is done using integr·als.

These appeai· less fi·equently in om· courses do, I cover· them

3. Algebra -

so "vhen they

imple distr·ibutions.

To get to a final ans,ve1· 1 we have to do a lot

of· substitution equations

lowl)' and use

1

and/or

eli1ni11ation to get t·1·om perhaps

and 5 ,mlmowns to a specific solution.

It·onicall;>r,

this mundane vvrap-up of the p1·oblem is vvhere 90% of math e1·1·orsoccU1·.

5

1. A Conve1·sation Virith a Math Atbei t

O'-V,

that

all

3

I don t 1nean. to ay that these at.·e t1·i,,ial, but at least )'OU

have to deal with. I, for 011e,a.in t,hankful that \Ve

don>t use t1·igonomet1·y 01· pai·tial difl'ei·ential equations in Econ

3 2. All the calculus we use i11class can be done with a handful of 1·ules about de1·i,rati,,es and integi·als. The algeb1·a is usually a matte1· of sol,,ing a couple eq,1ations and u11l{now11s: which have si1nila1·algo1·ithms. I once 1·ecei,reda student evaluation fo1·Econ 3 2 that claimed

I was dema.nding graduate-level 1nath of them. Ho\A.co1nfo1·table g1·ad school ,vould ha,,e been had \Vebeen confined to deri,,atives 1

a.nd algeb1·a! No ... 1·eal anal)rsis, topolog-y, and measure theory ai:e the tools .of graduate economics. At the inte1·mediate le,,el) "ve only bring out tl1e es ential tools. 1

But why use it at all? At this point, the math atheist ma)' 1·eto1·t 'Slll·e~ )'OU re not using the hai:dest ma.tl1 in existence - but \ivhy use any 1nath at all?''

This ai·g,1rnent aga.in.st teaching mathematically-1·igo1·ous

economics generally tal{es t'\vo angles: fu·st, math is hard, and second, math distracls

tlS

1·1·omthe ideas we 1·eall)' \ivant to study.

I concede the fi.1·stpoint. l\llath is difficult. It requi1·es abst1·act tbi11kir1g.It is easy to make little 1nistakes that tht·ow the solution off. It take

some pla1ming and perhap

ome false stai·ts a.1Jd

erased pages to figure out 110\.Vto tackle a p1·oblem. To make matters wo1·se, 1·1·omgrade

chool on, math is taught \iViih vef)'

few motivating examples.

tudents fail t•o see the usefulne s 01·

the algebra 01· calculus techniques,

o they put no effo1·t into

it. Jviost of us ( and I include my elf.) only lear·n math wl1en we absolute!)' must~ a.1Jdare not anxious to pull Olll' matl1ematical 1

For those who see an Economics Ph.D. in yolu· fut1u·e, Econ 5 0 \Vill gi,,e you an importa11t i11trod11ction to the application of gradt1ate le,rel matl1ematics to economics.

1. A Conve1:sation «rith a Ivlath Atheist

4

kno,vledge out of . to1·age. Howeve1·1 this onl)' establishes that using math is costl,, that it doe n t come natU1·ally. Slll·el)' '\.Vecan't dismiss the use of· math in econo1nics Oill)' on this basi ! We tnust compar·e the co ts with the benefitsi vvhich I spell out belotiv. Eve1·ything comes at a cost (i1nplicit or explicit), and usuall)', things of a g1·eate1·,,alue come at a g1·eate1·cost.

Lost in translation Then what are the benefits? the gi·ad-school-bound

F\u·the1·mo1·e, ,vill they onl)' help

student, 01· a1·e these benefits •equall)' dis-

tributecl? The ftu1da.1nental 1·eason why we use 1nathematics

in eco-

nomic is because it fo1·ces us to be logical. The Engli h language is slippe1·3r. Wor·d have multiple definitions a.nd nuances~ o ii is ha1·d to be ure that you ai:e talking about the

ame thing. Fur-

the1:mo1·e,comn1U11icating in English is sloppy. In the coUI·seo.fa11 ar·g,1ment, it i entirely po . ible to cont1·adict one'

self' \\1 ithout

noticit1g it, or 1nake a conclusion that does 11ot follow 1·rom tl1e pr·eceding con,,e1·sation. Someone could also make a t1·ue cla.im but express it in a way that is diffict1lt to follow. Ce1·ta.i11l)'you have been in an a:r·gument or debate vvhere )'OUI'opponent keeps changing the topic to avoid getting pinned down 01· 1·edefines his pre,,ious statements

as he goes, or tvvist

the meani11g of your \vords. The fi.·ustration of· Sl1ch e.>...1)eriences i enough to mal{e yotr 11ot v.1 a11t to talk to the othei· per' on at

all, and it ce1·ta.in]y do•es not facilitate any exchange of ideas. Fo1· a moment~ think of Mathernat;ics as a language of· it O\Vn. Each equa,tion makes a clai1n or· state1nent lation into English.

which has s01ne tra.n.s-

The beautiful thing about mathematics

that it is p1·ecise and il1controve1·tible. Mathematical

is

co11cept

1. A Conve1·sation Virith a Math Atbei t

ai·e well defined,

5

o tl1at eve1·yone t1·ained in Nlathematic

common tinder tanding

has a

as to the meaning of a particular· equa-

tion. 11Io1·ei1npo1·tantly, Math is relentles ly logical. If t,vo equations contradict

one ano·ther· a

p,ei•

on tr·ying to olve the problem

will find out that the1·e is no solution· the,y ca11not simultaneously be Lr·ue. Thi

make

it impo ..ible to hide logical inconsistencies.

If' someones equation ( tatement)

only holru t1·t1esome of the

time ( e.g. under· certa.in as timptions) be

tated or el e the equation is false.

f1·om sneaking in 11evvassumptions Thi

tho e a s11mptions must Thi

p1·event s01neo11e

i11the middle of an ai·gument.

also expo es the false analogy, where a true concept for· one

ituation is u ed to justify a f"alse conclusion for distil1ct ci1·cumstances.

It i also easier to find common grou11d while peaking Mathematics.

u·someone

clairns that these tvvo equations can be com-

bined to form a third equation,

anyone else ~·ith mathematical

t1·aining can ,,erif) 1 for· themsel\res if thi impossible

is true.

This mal this hard wo1·k \vith math will gi,,e you relative!)' rare talent -

which; as

higher· f'uture

alar·y. How is math co1111.ectedto analytical

)'OU

know; allo\.v you to command

a

thinl 'Is

include the ti 1ne and

1·equit·ed or· 1·eader· of the model, which incomplexity,

and specialized n1ath used in the

The benefits depend on hovv novel 01· stu:p1·ising the 1·e-

st1lts are, a11d l1ow illu1ninati11g tl1e inte1·11almecl1anic;;1nsa1·e.

In ho1·t, no one get

high ma1·k for proving

at the same time, elaborate Balanced

the obviou ·

con1ple:xity i not an end in it elf·.

omevvhe1·e be·tvvee11these, theor)' has a 1.1nique 1·ole to

pla)'· If' e,,e1· that 1·ole eems t1·i,rial or it1·elevant,. pe1·hap we just sent tl1e \v1·011gmodel to ca1-ve out that 1·ole.

Theory in Pmctice 1. List tht·ee standards

a theo1·etical model.

by ,vhich

\.\re

judge the lL.~ef'ulnessof

Explain each (in a entence or two),

1·efe1·ring to one of the models f1·om class as an exa.mple. 2. A friend fr·om anothe1· discipline signed a.n eq·uation that describe ho,v doe

that mathematical

the 1·eal -vvo1·ld?' Give

)'Olli'

once told me

{You de-

a result you vvanted; but

model pr·ove anytlling

about

1·esponse.

3. Suppo e congre s is contemplating

a dt·amatic policy change

fo1·,vhich the1·e is little data. D•esc1·ibe ho,v economist ate a counte1·factual p1·ecliction of its efl·ect.

cr·e-

CHAPTER7-------------------

Assume

Responsibly

An)' theo1·etical claim boils dovvn to an if-then tateme11t. If specified assumptions

1

hold, then a specified 1·esult always occurs.

The cla.i1n must be justified in a mathen1atical

pr·oof de1non-

trating ho\v those assumptions inevitabl)' lead to that co11clusion. But notice that the proof only co1mect the path bet,vee11 assumptions

and 1·esult. It does not (and cannot) p1·ove that

the assumptions ai·e co1·1·ect. As fai· as the math is concerned, the stai·ting poi11t is unquestioned

and

Olli)'

it consequences ar·e

exa.rni11ed. Tllis pr·ocedure i excellent at establishing the logic behind one's thi11king. Done properly~ the proof

"'ill be

iI1cont1·overtible

easil)' ve1·ified b)' anyone mathe1natical1)' tr·ained. If a11y cont1·0,,e1·s)' re1nai11S,it ca11 only 1·est in the a..ssumptions used. And ho,v do vve adjudicate tlia,t contest? What makes an as.c;111nption acceptable? 1

A natliral inclination is to ask '·Is the assumption 1·ealistic?

But by that sta11dai·d, eve1·,ymodel fails. Even the most v,rellaccepted models make

i1nplil'ying as umptions that are laugh-

able ( e.g. fu·ms know p1·ecisely 1nar·ket demand or one another· cost function, corh~lllner ii1stantaneously know all prices and utilit)' from 1

8nj'

combination of goods etc). As put by the

obel

We give the e claims different names: theorem or proposition a.re \1Sed interchangeabl)' for the important clain1S, ,vhile lemma is used for an intermediate (but other1,vise 11ninteresting) step.

62

7. A -st1n1eResponsibly

LaUI·eate Finn Kydland and Ed Prescott,

~To c1·iticize

01·

1·eject

a model because it is a.n ab t1·action is foolish: all models ai·e necessa1·il)' abst,1·actions.

In othe1· \.vords, the model is a model -

a si1nplification -

not a literal depiction of tl1e real ,vo1·ld. Without tho e simplifications, we \Vould be 1111ableto make any conclusions about beha,,ior. A the old adage goes) '' o ass11mptions, no theo1·ems.~' We even if acrificing realism,

need some context in which to operate 01· else vve can

a)' nothing about that envit·onment.

J\1Io1·eo,,e1·1

as \ve obse1·ved in Chapte1· 6, sometimes a simpler model is mo1·e info1·mative than a complex one, because it eleg-antly highlights the mecha.nisms at work ,vhich would othe1·wise be obscured by othe1· bells and whistles.

The ''As If'' Analogy Econo1nists ar,e tjrpically \re1·y upf"ront abo11t the assumptions their 1nodels, but one

i11

hould not 1nistal;, I a1n 1·eal1)'sum1nai·izing my intro pection about my •O\vn expe1·ience, as in) I can t thitlk of a time that I pla.yed the game ,vhen my patience 1·an out. 1 That effective!)' r·eturns to the empu·ical ai·g11ment: does the model Alte1·nati,,ely my statement

p1·edictions beai· out il1 practice?

could be t1·anslaled a , I don t ee

hot\l disco11nti11gcould make a diffe1·ence il1 playing the game. 1'

7. A -st1n1eResponsibly

64

-.. . • ,,. .,•'

1

~ ~ --..._

(I,)

0 --..._

tv

0

0 ., I).

Howe,rer·, if Px = I

and tl1e ove1·all claim is still tr·ue.

the11

A. Logic dictates that ...

5

Tl1us to p1·ove an or claim you only have to p1·0·\1eone of its subpart

to be t1·t1e;)'OU need not \Vor·r·yabout whether both

hold 01· not. To disp1·0,,e an or clai1n

)'OU

would have to shovv

that both su bpai·ts ai·e false.

If-then • Disp1·ove an if- then claim by showing that the condition holds and the consequence fails. • P1·ove an if- then clai 1n by howing that either the condition fails or· the conseque11ce holds.

For those ,vho ha,,en t studied logic, the conditional operator,

if-then, i a bit tricky because 01·theu· 11arrow scope. In addition, the sa.1neclai111can be stated using a number of difl'e1·ent English wo1·ds. But fi1·st, let us set up an exa1nple Lo use thi·oughot1t.

• Claim P: The outcome of a model is Pai:eto efficient. 1 • Claim T: The outcome

•Of a

model is technicall)' efficient. 2

Jote that these claims may be t1·ue 01·false, depending on which model "ve conside1·. Suppose I make the cla.i1n: !If a models outcome is Par·eto efficient 1 the11 it is also technically efficient 1' (that is, :11·P, then T.; ). The scope of this clai1n is limited to times when P is t1·ue; it has no bite othei·wise. For i11Stance if the model is a competitive market with exte1·nalities 1 the outcome will never be Pai·eto efficient. Still, the outcome may or 1nay not be technically ef'ficient 1

tl1at depends 011,vhethe1· the externali ty only affects

1tleaning, it is not possible to increase the utilit)' of one person without reducing the t1tilit)r of another. 2 :£\lleaning, it is not possible to increase production of one good ,,rithout reducing proclt1ction of another.

A. Logic dictates lhat ...

6 utility of diJ:ectly impact

the p1·oduction technolog)'· But he1·e~

the kicker: when Pis 1·a1se,it doesn't matter what happens to T.

In that case we ay the claim 'If P then T i vacuo11.slyt1·ue. The conditional clai1n actually has some bite when its co11clition (P) is true. For e:xan1ple, in a pe1·fecily co1npetiti,,e 1na1.-l