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ๅ ้ซ ็ถ ๆฟ ๅญธ ไธ M i c r o e c o n o m i c s (I) ๏ผUntility function Total utility of consuming (x, y), denoted as u(x, y), is the total level of total satisfaction of consuming(x, y).
๏ฟฝCardinal utility analysis (number itself is meaningful)
Ordinal utility analysis (number is used to rank bundles)๏ฟฝ
In Ordinal utility analysis, higher value of TU is associated with higher level of satisfaction.
utility function is a function from bundles to a real number
such that ๏ฟฝ ๐ข(๐ฅ1,๐ฆ1) > ๐ข(๐ฅ2,๐ฆ2) (๐ฅ1,๐ฆ1) > (๐ฅ2,๐ฆ2)๐ข(๐ฅ1,๐ฆ1) = ๐ข(๐ฅ2,๐ฆ2) (๐ฅ1,๐ฆ1) ~ (๐ฅ2,๐ฆ2)
๏ฟฝ
u : (x, y) โ โ
EX: (2, 3) ๏ผ 100 u(x, y) total utility (3, 7) ๏ผ 130 (5, 10) ๏ผ 200
๏ผMarginal utility (of X, of Y) u = u(x, y)
MUx = โ๐ข(๐ฅ,๐ฆ)โ๐ฅ
(๐๐ข(๐ฅ,๐ฆ)๐๐ฅ
)
MUy = โ๐ข(๐ฅ,๐ฆ)โ๐ฆ
(๐๐ข(๐ฅ,๐ฆ)๐๐ฆ
)
The law of Diminishing marginal utility
x โ MUXโ ( ๐2๐ข(๐ฅ,๐ฆ)๐๐ฅ2
< 0 )
y โ MUYโ ( ๐2๐ข(๐ฅ,๐ฆ)๐๐ฆ2
< 0 )
CH2 The analysis of consumer behavior
Any positively monotonic transformation of a utility function is also a utility function representing the same preference. ๏ผCardinal v.s. Ordinal utility analysis
EX: x : # of toast y: # of ham
u(x, y) = min ๏ฟฝx3
, y2๏ฟฝ positively monotonic
transformationโ min{x
3, y2}2 + 100
=> same preference. ๏ผMarginal utility
MUx = โu(x,y)โx
MUy = โu(x,y)โy
or โu(x,y)โx
or โu(x,y)โy
or Ux(x, y) or Uy(x, y) (or simple Ux) (or simple Uy) ๏ผMarginal utility is the slope of the utility function
If more is better (assumption of nonsatiation) MUx > 0 , MUy > 0
But โLaw of Diminishing marginal utilityโ xโ, MUx โ yโ, MUy โ
๏ผฆigure1 8:
{(x, y)|(x, y)~ (x1, y1)for all (x, y) โ S} Given (x1, y1) โ S, we have total utility
u(x1, y1) : {(x, y)|u(x, y) = u(x1, y1)for all (x, y) โ S}
๏ผฆigure19 : Indifference curve
MRSxy= โ๐ฅ๐ฆ๐ฅ๐ฅ
(given an indifference curve)
MRSxy depends on (x, y)
note that โ๐ข(๐ฅ, ๐ฆ) = โ๐ข(๐ฅ,๐ฆ)โ๐ฅ
โ โ๐ฅ + โ๐ข(๐ฅ,๐ฆ)โ๐ฆ
โ โ๐ฆ
(๐๐ข(๐ฅ,๐ฆ) = ๐๐ข(๐ฅ,๐ฆ)๐๐ฅ
โ ๐๐ฅ + ๐๐ข(๐ฅ,๐ฆ)๐๐ฆ
โ ๐๐ฆ)
๐๐ข(๐ฅ,๐ฆ) = ๐๐ข๐ฅ โ ๐๐ฅ + ๐๐ข๐ฆ โ ๐๐ฆ On an indifference curve, a change in X and Y must satisfy
๐๐ข๐ฅ โ ๐๐ฅ + ๐๐ข๐ฆ โ ๐๐ฆ = 0
โ๐๐ฆ๐๐ฅ
= ๐๐ข๐ฅ๐๐ข๐ฆ
๏ผฆigure20 :
(note: MRSxy= โ๐ฅ๐ฆ๐ฅ๐ฅ
of an indifference curve)
MRSxy(x, y) = ๐๐ข๐ฅ(๐ฅ,๐ฆ)๐๐ข๐ฆ(๐ฅ,๐ฆ)
suppose๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐๐ข๐ฅ(๐ฅ,๐ฆ) is diminishing on ๐
๐๐ข๐ฆ(๐ฅ,๐ฆ) is diminishing on ๐ฆ } Diminishing marginal utility.
Can we have diminishing MRSxy? (MRSxyโ as xโ or yโ)
xโ => ๐๐ข๐ฅโ(Diminishing ๐๐ข๐ฅ) yโ=> ๐๐ข๐ฆโ(stays on the same IC)
Since MRSxyโ=๐๐ข๐ฅโ๐๐ข๐ฆโ
โ as xโ
๏ผฆigure21:Diminishing MRSxy
๐๐๐ ๐๐ฅ๐ฆ๐๐ฅ
>< 0 ?
We hope ๐๐๐ ๐๐ฅ๐ฆ๐๐ฅ
< 0.
Mux= ux Muy= uy
โMux(x,y)โx
or โuxโx
= uxx
โMux(x,y)โy
or โuxโy
= uxy
โMuy(x,y)โx
or โuyโx
= uyx = uxy
โMuy(x,y)โy
or โuyโy
= uyy
๐MRSxy๐x
=๐(Mux
Muy)
๐x=๐(ux
uy)
๐x=
uy๐๐ข๐ฅ๐x โ ux
๐๐ข๐ฆ๐x
uy2
๏ฟฝnote that duxdx
โ โuxโx๏ฟฝ =
uy๏ฟฝโuxโx +
โuxโy โ
dydx๏ฟฝโux๏ฟฝ
โuyโy +
โuxโy โ
dydx๏ฟฝ
uy2
aโc
โuxโx
(given y)
cโb โuxโx
and dxdy
๏ผฆigure22 :Diminishing MRSxy
(note that โdydx
= uxuy
(= MuxMuy
))
uy๏ฟฝuxxโuxyโ
uxuy๏ฟฝโux๏ฟฝuyxโuyyโ
uxuy๏ฟฝ
uy2
uy2๏ฟฝuxxโuxyโ
uxuy๏ฟฝโuxuy๏ฟฝuyxโuyyโ
uxuy๏ฟฝ
uy3
uy2uxxโ2uxuyuxy +ux2uyy
uy3< 0
none satisfaction ux๏ผuy > 0 uxx<0uyy<0
} Diminishing Mux, Muy
Diminishing marginal utility is not sufficient to imply diminishing MRSxy, we need to know uxy >0<0
๏ผBudget Constraint X, Y Quantity:x, y price: Px, Py income: m Expenditure of(x, y) = ๐๐ฅ โ ๐ฅ + ๐๐ฆ โ ๐ฆ Budget constraint: ๏ฟฝ(๐ฅ, ๐ฆ)๏ฟฝ๐๐ฅ โ ๐ฅ + ๐๐ฆ โ ๐ฆ โค ๐๏ฟฝ Budget line: ๏ฟฝ(๐ฅ,๐ฆ)๏ฟฝ๐๐ฅ โ ๐ฅ + ๐๐ฆ โ ๐ฆ = ๐๏ฟฝ
๐๐ฅ โ ๐ฅ + ๐๐ฆ โ ๐ฆ = ๐ ๐๐ฆ โ ๐ฆ = ๐ โ ๐๐ฅ โ ๐ฅ
๐ฆ =๐๐๐ฆโ๐๐ฅ โ ๐ฅ๐๐ฆ
๏ผฆigure23 :Budget constraint
๐๐ ๐๐ฅ๐ฆ = โ๐ฅ๐ฆ๐ฅ๐ฅ
|on an indifference curve = subject exchange rate.
๐๐ฅ๐๐ฆ
= โ๐ฅ๐ฆ๐ฅ๐ฅ
|on an budget line = object exchange rate.
1. ๆๅพ(m)ๆน่ฎ, mโ
๏ผฆigure24 :Budget constraint when income change
2. Px, Py chang
Pxโ, Py fixed, m fixed
๏ผฆigure25 :Budget constraint when price of X change(increase)
*Special case 1.Quantity Discount on X
๏ฟฝ๐ โค ๐0 ,๐๐ฅ
๐ > ๐0 ,๐๐ฅ2๏ฟฝ
m: incomePy:Price of gppd Y} as usual
๏ผฆigure26 :Budget constraint with special case
๐ฆ0 =๐ โ ๐๐ฅ๐0
๐๐ฆ
(x0, y0) is on D, B
Price of ๐ =0.5 ๐๐ฅ =๐๐ฆ
}on D, B
income = mโ
0.5๐๐ฅ๐0 + ๐๐ฆ๐ โ ๐๐ฅ๐0
๐๐ฆ= ๐โฒ
0.5๐๐ฅ๐0 + ๐ โ ๐๐ฅ๐0 = ๐โฒ ๐ โ 0.5๐๐ฅ๐0 = ๐โฒ ๐ โ๐โฒ = 0.5๐๐ฅ๐0
Distance between A.D=๐โ0.5๐๐ฅ๐0๐๐ฆ
D.B budget line = ๏ฟฝ(๐ฅ,๐ฆ)๏ฟฝ0.5๐๐ฅ๐ + ๐๐ฆ๐ = ๐ โ 0.5๐๐ฅ๐0๏ฟฝ X > X0
2.Quota on X
๏ผฆigure27 :Budget constraint with quota
3. WIC(Woman.infant.children) Food coupon X0:good X of food coupon
๏ผฆigure28 : Budget constraint with food coupon
4. Endowment ็จ่ณฆ (x0, y0)~m= Pxx0+Pyy0
Pxโ, PxโPxโ, Pxโ>Px mโ = PxโX0+PyY0
๏ผฆigure29 :Budget constraint with endowment
๏ผฆigure30 :Interior solution of consumer equilibrium
๏ผConsumer equilibrium A consumer maximizes his or her utility subjected to his/her budget constraint. (X*, Y*) is a consumer equilibrium where indifference curve is tangent to the budget line(indifference curve touches the budget line at e) or slope of the indifference curve at e = slope of the budget line.
orโMRSxy(X*, Y*) = ๐๐ฅ
๐๐ฆ
โone more condition: Pxx*+Pyy* = m
corner solution ่ง่งฃ e: x*>0, y*=0
or x* = 0, y* = 0 Pxx*+Pyy* = m
Pxx*= m, x*=๐๐๐ฅ
x*= ๐๐๐ฅ
, y* = 0, corner solution at e.
MRSxy โฅ๐๐ฅ๐๐ฆ
๏ผฆigure31 :Corner solution of consumer equilibrium
another corner solution:
x*= 0, y* = ๐๐๐ฆ
, MRSxy โค๐๐ฅ๐๐ฆ
๏ผIn the interior solution case:
if MRSxy > ๐๐ฅ๐๐ฆ
subject change object exchange rate (ไธป่ง)ๅไบบๅๅฅฝ (ๅฎข่ง)ๅน้ขๆฏ
xโ, yโ
๏ผฆigure32 :
aโc c has more X => c is better than a. cโe
MRSxy = ๐๐ฅ๐๐ฆ
ๅ็, MRSxy < ๐๐ฅ๐๐ฆ
=> xโ, yโ
๏ผWIC case
๏ผฆigure33 :Consumer equilibrium with food couponโs type budget constraint
๏ผX and Y are perfect complements
๏ผฆigure34 :Interior solution under the preference is perfect complements
๏ผTo max u(x, y) s.t ๐ท๐ โ ๐ + ๐ท๐ โ ๐ = ๐ u(x, y) is some continuously differentiable function How to solve the consumerโs problem? General problem: max f(x. y) s.t. g(x, y) = 0
(min) โ(๐ฅ,๐ฆ, ๐) = ๐ข(๐ฅ,๐ฆ) + ๐(๐โ ๐๐ฅ๐ โ ๐๐ฆ๐) โฆLagrange multiplier method = ๐(๐ฅ,๐ฆ) + ๐๐(๐ฅ,๐ฆ) constrained problem โnon-constrained problem and apply FOC to the
non- constrained problem.
โ1 ๐โ(๐ฅ,๐ฆ,๐)๐๐ฅ
=๐๏ฟฝ๐ข(๐ฅ,๐ฆ)+๐๏ฟฝ๐โ๐๐ฅ๐โ๐๐ฆ๐๏ฟฝ๏ฟฝ
๐๐ฅ= ๐๐ข(๐ฅ,๐ฆ)
๐๐ฅโ ๐๐๐ฅ = 0
๐๐ข(๐ฅ,๐ฆ)๐๐ฅ
= ๐๐๐ฅ --โ1 โ
โ2 ๐โ(๐ฅ,๐ฆ,๐)๐๐ฆ
=๐๏ฟฝ๐ข(๐ฅ,๐ฆ)+๐๏ฟฝ๐โ๐๐ฅ๐โ๐๐ฆ๐๏ฟฝ๏ฟฝ
๐๐ฆ= ๐๐ข(๐ฅ,๐ฆ)
๐๐ฆโ ๐๐๐ฆ = 0
๐๐ข(๐ฅ,๐ฆ)๐๐ฆ
= ๐๐๐ฅ --โ2 โ
โ3 ๐โ(๐ฅ,๐ฆ,๐)๐๐
=๐๏ฟฝ๐ข(๐ฅ,๐ฆ)+๐๏ฟฝ๐โ๐๐ฅ๐โ๐๐ฆ๐๏ฟฝ๏ฟฝ
๐๐= ๐โ ๐๐ฅ๐ฅ โ ๐๐ฆ๐ฆ = 0 --โ3 โ
budget constraint
โ1 '/โ2 '=๐๐ข๐ฅ๐๐ข๐ฆ
= ๐๐ฅ๐๐ฆ
--โ4
MRSxy = ๐๐ข๐ฅ๐๐ข๐ฆ
โ4 MRSxy = ๐๐ข๐ฅ๐๐ข๐ฆ
=๐๐ฅ๐๐ฆ
ๅจๅๆจฃ็$1 ไธๆฏ่ผ
๐๐ข๐ฅ๐๐ฅ
= ๐๐ข๐ฆ๐๐ฆ
โ ๐๐ข๐ฅ๐๐ฅ
= โ๐ข(๐ฅ,๐ฆ)โ๐ฅ
โ๐๐ฅ๐๐๐๐๐๐ก๐ข๐๐โ๐ฅ
๏ผSpecial case 1. X and Y are perfect complement. 3 units of X is always consumed with 2
units of Y in fixed proportion. Px=$2, Py=$5, m=$80.
๐ข(๐ฅ,๐ฆ) = ๐๐๐ ๏ฟฝ๐ฅ3
,๐ฆ2๏ฟฝ
Optimal combination of X&Y: ๐ฅ3
= ๐ฆ2โ ๐ฆ
๐ฅ= 2
3
2๐ฅ +103๐ฅ = 80
163๐ฅ = 80
๐ฅโ = 15,๐ฆโ = 10
Generalized to min ๏ฟฝ๐ฅ๐
, ๐ฆ๐๏ฟฝ โๆไธๆตช่ฒป็ๆฏไพ
s.t. Pxx+Pyy=m Ex. X and Y are perfect complements. A consumer always uses 3 units of X
with 2 units of Y in the fixed proportion.
u(๐ฅ,๐ฆ) = min ๏ฟฝ๐ฅ3
,๐ฆ2๏ฟฝ โ u(๐ฅ,๐ฆ) = min ๏ฟฝ
๐ฅa
,๐ฆb๏ฟฝ
max u(๐ฅ,๐ฆ) = min ๏ฟฝ๐ฅ3
,๐ฆ2๏ฟฝ
s.t. Pxx+Pyy=m ๐ฅa
=๐ฆbโ ๐ฆ =
ba๐ฅ
budget constraint: Pxx+Pyy=m
๐๐ฅ๐ฅ + ๐๐ฆba๐ฅ = ๐
๏ฟฝ๐๐ฅ + ๐๐ฆba๏ฟฝ ๐ฅ = ๐
๐ฅ =๐
๐๐ฅ + ๐๐ฆba
Px=$2, Py=$5 a=3, b=2, m=80
๐ฅโ =a๐
a๐๐ฅ + b๐๐ฆ= 15
๐ฆโ =ba
a๐a๐๐ฅ + b๐๐ฆ
=b๐
a๐๐ฅ + b๐๐ฆ= 10
Ex. Cobb-Douglas utility function ๐ข(๐ฅ,๐ฆ)total utility = ๐ฅฮฑ๐ฆฮฒ, ฮฑ, ฮฒ > 0
๐๐ข๐ฅ = โ๐ข(๐ฅ,๐ฆ)โ๐ฅ
= โ๐ฅฮฑ๐ฆฮฒ
โ๐ฅ= ฮฑ๐ฅฮฑโ1๐ฆฮฒ > 0 no satiation point (no bliss point)
Muy = โu(x,y)โy
= โxฮฑyฮฒ
โy= ฮฒxฮฑyฮฒโ1 > 0
โ๐๐ข๐ฅโ๐ฅ
= ฮฑ(ฮฑโ 1)๐ฅฮฑโ2๐ฆฮฒ โ ฮฑ < 1
๐๐๐ข๐ฆ๐๐ฆ
= ๐ฝ(๐ฝ โ 1)๐ฅ๐ผ๐ฆ๐ฝโ2 โ ๐ฝ < 1 โ ๐ท๐๐๐๐๐๐ โ๐๐๐ ๐๐๐๐๐๐๐๐ ๐ข๐ก๐๐๐๐ก๐ฆ
๐๐ ๐๐ฅ๐ฆ =๐๐ข๐ฅ๐๐ข๐ฆ
=๐ผ๐ฅ๐ผโ1๐ฆ๐ฝ
๐ฝ๐ฅ๐ผ๐ฆ๐ฝโ1=๐ผ๐ฝ๐ฆ๐ฅโ ๏ฟฝ๐ฅ่ถๅคง,๐ฆ่ถๅฐ๏ฟฝ โ
๐๐๐ ๐๐ฅ๐ฆ๐๐ฅ
< 0
For any ฮฑ, ฮฒ>0 and for all x and y ๐๐๐ ๐๐ฅ๐ฆ๐๐ฅ
< 0
i.e. MRSxy is diminishing (with x) i.e. indifference curve is always convex
In equilibrium, ๐๐ ๐๐ฅ๐ฆ = ๐๐ฅ๐๐ฆ
๐ผ๐ฝ๐ฆ๐ฅ
=๐๐ฅ๐๐ฆ
๐๐ฆ๐ฆ(expenditure on Y) =๐ฝ ๐ผ๐๐ฅ๐ฅ(expenditure on ๐)
๐๐ฅ๐ฅ + ๐๐ฆ๐ฆ = ๐
๐๐ฅ๐ฅ +๐ฝ ๐ผ๐๐ฅ๐ฅ = ๐
๐ผ + ๐ฝ ๐ผ
๐๐ฅ๐ฅ = ๐โน ๐๐ฅ(expenditure of x) = ๐๐ฅ๐ฅ =๐ผ
๐ผ + ๐ฝ ๐(short of ๐๐ฅ)
๐๐ฆ(expenditure of y) = ๐๐ฆ๐ฆ =๐ฝ
๐ผ + ๐ฝ ๐(short of ๐๐ฆ)
๐ฅโ =๐ผ
๐ผ + ๐ฝ๐๐๐ฅ
๐ฆโ =๐ฝ
๐ผ + ๐ฝ๐๐๐ฆ