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General Equilibrium with Production ISingle-input, Single-output

Exchange Economies

No production only endowments

No consideration of how resources are converted to consumables (tech)

General equilibrium:all markets clear simultaneously

1st and 2ndFundamental Theorems of Welfare Economics

dd !roduction

add" input#output markets

describe"$rm%s technolo&ies' distributions of $rms outputs and pro$ts

2 Kinds o GE with Production

start with sin&leinput production

o input used labour

o model optimal labour supply' demand for &ood

then analy*e production with more than 1 kind of output

+ sets of conditions describin& both e,uilibrium and welfare ma-

o e-chan&e e.ciency (consumers have same /0)

o !roduction E.ciency (producers have same /0T)

o !roduct/i- E.ciency (producer prices consumer prices)

Single-input, Single output !odel

haracteri*e e,uilibrium prices (labour and produced &ood)

E,uilibrium prices separation of consumption decision from production decision

o 3ptimi*ation by consumer is separate from optimi*ation by producer

o llows for decentrali*ed decisionmakin&

"obinson #rusoe$s Econom%& Single input '(abour), single output

3ne a&ent 0 endowed $-ed ,uantity of 1 resource 4 25 hours

6se time for labour (production)#leisure (consumption)

7abour time 7 leisure 25 4 7

What will 0 choose8

0 Technolo&y

Tech" labour produces output (coconuts) accordin& to concave production

function

0%s !references

oconut &od

7eisure &ood

0%s hoice(blue)

!" * !P(#ondition

!"S:rate at which consumer is willin& to sub consumption &oods for leisure and be e,ually well o9

-d#+d" or d#+d( consumerperspective !P(: chan&e in output (d) from usin& little bit more labour

d#+d( producers perspective

t utility ma-imi*in& optimum (hi&hest attainable indi9erence curve touches production function)

o 0ate at which consumer is willin& to trade o9 leisure for coconuts rate at which labour is

transformed into coconuts

:standard; utilityma- problems' at optimum

o rate at which consumer is willin& to sub one &ood for another rate at which she can sub 1

&ood for another while stayin& within her bud&et

marginal product o labour

o rate at which consumer is willin& to sub leisure for &oods rate at which he can sub leisure f

&oods throu&h transformin& hours into &oods

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marginal rate o substitution (of leisure for &oods consumer) * marginal rate o transormation (labour into &oods 4 producter)1)

coconut output level #

0%s wa&e w

"ealwa&e money wa&e# money > of &ood

!ro$t. * # / w( ? @ w7 e,A of isopro$t line lope @ w general (slope @w#!)

Bntercept general (intercept #!)

Proft-Maximization

(2) 6tility/a-imi*ation

Endowed with nonlabour income of >< who can work for >wper hour

0%s most preferred consumption bundle8

Cud&et constraint"

# * 0 w(

Utility-Maximization

6tility/a- = !ro$t/a-

Prot max:

o w * !P((isoproft, prodn. n.)

o ,uantity of output supplied #

o

,uantity of labour demanded ( oconut and labour markets both clear

Pareto E1cienc%(must have /0 /!7)

tilit% max:

o w * !"S(budget, indieren!e)

o ,uantity of output demanded

#

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E-" 0eal Wa&e Bmplementin& !aretoE.cient !lan

!riduction function" D (7) "2#+

Wa&e" 1 !- 5

6tility function of sin&le consumer#labour supplier" 6(0'D) 0D (multiply the 2)

Time endowment GHA

Nonlabour income 1+

(#ststep) $ %ind Proft-Maximizing &ombination o "abour (") and output (')

lope of prodn fn real wa&e at pro$t ma-imi*in& acitivity level

0eal wa&e w#!- 1#5 2#I

Find"

o lope of production fn mar&inal product of labour

D (7) "2#+ /!7 dD(7)#d7o set to real wa&e"

o solve for 7'D

(ndstep) $ %ind Utility-Maximizingemand or ', *upply o "

utility function is obbJou&las with e-ponents

e-pression for demand D and demand for leisure 0

price of leisure w

o total income (m) is w < time endowment @ nonlabour income

o use values of time endowment (GHA)' nonlabour income (1+)' !- (5)' w (1)"

o optimal amount of D consumers and labour supplied are I = 2H

same ,uantities chosen by pro$tma-imi*in&

3irst 3undamental 4heorem o 5elare Economics

competitive market e,uilibrium !areto e.cient if "

o consumers preferences are con6ex

o there are no e-ternalities in consumption production

(public &oods#collective consumption &oods@ve e-ternalities)

Second 3undamental 4heorem o 5elare Economics

any !areto e.cient economic state can be achieved as a competitive market e,uilibrium if"o consumer%s preference are con6ex

o no e-ternalities

o $rms technolo&ies are conve-

7on-#on6ex 4echnologies

do theorems hold if technolo&y is nonconve-8

1sttheorem doesn%t rely on $rms technolo&ies bein& conve-

o competitive e,uilibrium paretooptimal

o with nonconve- technolo&ies there is no &uarantee that a competitive

e,uilibrium e-istso not applicable

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2ndtheorem re,uires $rm technolo&ies to be conve-