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Chapter 4: Utility
Utility way to describe
consumer preferences
doesn’t matter how much higher doesn’t matter.
Utility function
assigning numbers to cons. bundles, more
preferred= higher number.
Ordinal utility
ranking cons bundles.
Monotonic transformation
transform one set of numbers into another set of numbers, preserves order of
numbers (e.g. multiplying all numbers by 2). Rate of change of f(u) as u changes
∆
f/
∆
u= (f(u
2
)f(u
1
))/(u
2
u
1
).
Monotonic function always pos. rate of change
(pos slope)
f(u
2
)
f(u
1
) always has same sign as u
2
u
1
.
If f(u) is a mon trans of a utility func that reps some partic pref, then f(u(x
1
,x
2
)) reps same pref. Utility func= way of assigning # to the diff
indiff curves so higher curves get larger #s.
Cardinal utility
size of utility difference between 2 bundles supposed to have significance (really no way to determine, almost
useless).
Level set
set of all (x
1
,x
2
) such that u(x
1
,x
2
)= a constant.
UTILITY FUNCTION= CONSTANT ALONG INDIFF CURVES and ASSIGN HIGHER LABEL TO MORE
PREFERRED BUNDLES
.
Perfect substitutes
u(x
1
,x
2
)= ax
1
+bx
2
(a and b are positive # that measure value of goods 1 and 2 to customer,
slope= a/b
perfect complements (left
and right shoes)
u(x
1
,x
2
)= min{x
1
,x
2
} in proportions other than onetoone: (e.g 2:1) min{x
1
,1/2x
2
} can multiply to get rid of fraction: min{2x
1
,x
2
}. Basically:
u(x
1
,x
2
)=
min{ax
1
,bx
2
}.
Qusilinear “partly linear” preferences:
indiff curves vertical translates of one another (vertically shifted) x
2
=kv(x
1
) k= diff constant for each indiff curve
height
of each indiff curve is some function of x
1
, v(x
1
)+constant k: higher value k give higher indiff curves
.
u(x
1
,x
2
)=k=v(x
1
)+x
2
.
(e.g.: sqrt(x
1
)+x
2
or lnx
1
+x
2
).
CobbDouglas utility
function:
u(x
1
,x
2
)= (x
1
c
x
2
d
) c and d are positive # that describe pref of consumer
cobbdouglas indifference curves look just like the convex, monotonic indiff curves referred to as
“well behaved indiff curves”
examples of mon transformation of cobbdoug util func:
u(x
1
,x
2
)=ln(x
1
c
x
2
d
)=clnx
1
+dlnx
2
and: u(x
1
,x
2
)= x
1
c
x
2
d
then raise utility to 1/(c+d) power we
get: x
1
c/c+d
x
2
d/c+d
then define a number a=c/(c+d) rewrite as: u(x
1
,x
2
)= x
1
a
x
2
1a
.
We can always take a monotonic transformation of the cobbdouglas utility function that make the
exponents sum to 1.
Marginal utility
rate of change of consumer’s utility as we give him more of good 1:
MU
1
=
∆
U/
∆
x
1
=(u(x
1
+
∆
x
1
,x
2
)u(x
1
,x
2
))/
∆
x
1
∆
U= rate of change in
utility,
∆
x
1
= change in amount of good 1
.
to calculate change in utility associated with a small change in consump of good 1 multiply change in cons by marg utility of the good:
∆
U=MU
1
∆
x
1
,
marginal utility for
good 2: MU
2
=
∆
U/
∆
x
2
= (u(x
1
,x
2
+
∆
x
2
)u(x
1
,x
2
))/
∆
x
2
to calc change in utility associated with change in consump of good 2:
∆
U=MU
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This note was uploaded on 09/23/2010 for the course BMGT 381 taught by Professor Dawson during the Spring '10 term at University of Maryland Baltimore.
 Spring '10
 DAWSON
 Utility

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