4. consumer optimum

7/25/2019 4. Consumer Optimum

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4. Consumer OptimumMax utility on budget set constrained optimization

Most preferred aordable consumption bundle

• Highest feasible utility

Economic Rationality

• Decision maker chooses most preferred alternative from availables

• Optimal choice: where indierence curve is tangent to budget line

• Choice set: available choiceso Most preferred bundle in choice set located

Rational constrained choice:



• oundary optimum: where indierence curve isn’t tangent to budget

line & meets at the endo Consuming 0 units of good !not tangent to budget ling

• !nterior optimum: where indi"curve is tangent to budgetline in between

• More thank # tangency$ only above optimal

• Convex preference any point satis%es tangency conditionmust be an optimal point

• Strict convex only # optimal choice

Rational Constrained Choice: Ordinary "emands

• ordinary demand: Most preferred aordable bundle at given prices and budget

o denoted by

• contrast with compensated demand utility constant

Rational Constrained Choice

x#$ 0 and x%$ 0 interior !not boundary'

• if$

o buying !(#)*()' costs +m

budget is exhausted



• when$

o (#) 0 and () 0 ,-./1-213

o !(#)*()' exhausts the budget

o indierence curves have 4no kinks’

ordinary demands obtained by solving$

a' p#x#$ & p%x%$ ' yb' slopes of budget constraint !5p#6p' and of indi"curve through

!(#)*()'are 7 at !(#)*()'

(x#$)x%$* satis%es conditions

a' budget is exhausted8p#(#) 9 p() 7 m

b' slope of budget constraint!5p#6p' and of indi"curve through !(#)*()'are 7 at !(#)*()'

Computing Ordinary "emands

info used to calculate !(#)*()' for given p#*p* and m:

Cobb5Douglas ;reference

MR+

<t !(#)*()'* M1= 7 5p#6p

(,*

!(#)*()' e(hausts the budget

(* p#(#) 9 p() 7 m

=ub !<' into !>' & simplify$ (#)

sub (#) in !>' & simplify$ ()



cobb5Douglas used in e(amples$

o buying6selling

o intertemporal choice

o e(change e?uilibrium

Rational Constrained Choice

• if (#) 7 0 8 or () 7 0

o if either 7 0 ordinary demand is at a corner solution to the problem of

ma(imi@ing utility subAect to constraint

Corner =olution$ ;erfect =ub

B!(#*(' 7 (# 9 ( most preferred !(#)*()'

• 2ptimal choice usually on boundary

Example:

− 2ptimal bundle when p# 7 p:

o hen slope of straight indierence curve 7 slope of budget constraint'

Corner =olutions$ .on5Conve( ;references



ink =olutions$ ;erfect Complements

• =ub !ii' into !i'

o # commodity * # unit and a commodity

units costs$ p# & ap% <EE21D<>F

4. consumer optimum

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