Econ 439
Notes on Utility Possibilities Frontier
Source: Schotter, Microeconomics, ch 4
What is UPF?
- plots combinations of utilities
- maximum attainable utilities, given
resources
- requires efficiency in
1. production: combination of goods
produced is on ppf (slope of
ppf=MRT)
2. consumption/distribution: MRS's
of individuals equated
3. overall: MRT=MRS
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Econ 439
To derive efficiency in consumption: use
Edgeworth box
- analyzes process of trade between two
parties, given quantities of goods
- size of box determined by quantities
Ex: goods are apples and raspberries;
10 apples (height of box), 8 raspberries
(length of box) (both in pounds)
- individual endowments measured from
bottom-left and top-right corners
Ex: Geoffrey from A, Elizabeth from C
- any point in box indicates an allocation
of goods to individuals exhausting
resources
- individuals' evaluations of allocations
indicated by indifference curves
from relevant origins (figure 4.2)
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Econ 439
Efficient allocations: not possible to find
another feasible allocation which
increases utility of at least one,leaving
other no worse off
(Pareto optimal allocations)
In box, efficient allocations points where
indifference curves are tangent.
Contract curve: curve joining all Pareto optimal
allocations.
Each point on contract curve corresponds to pair
of utilities; can plot these utilities
(negatively sloped curve)
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Econ 439
Utility trade-off above derived given a particular
combination of goods available for
distribution.
Take step back:
- goods produced by resources
- society can produce many efficient
combinations of goods, given set of
resources
Each point on ppf generates different
Edgeworth box
Each Edgeworth box generates different
utility tradeoff
Utility possibility frontier is envelope of these
curves
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Alternative derivation of UPF:
Given resources and production technologies, have PPF
- fix utility of individual A at U A = U A
o if put A’s IC map on PPF diagram, keep A on particular
IC
- for each product mix on PPF, find distribution of products
which max B’s utility while keeping A on designated IC
- find U (U ) = max U over all product mixes, given A on
designated IC
B*
A
B
A
B*
A
- then ( U , U (U ) is a point on UPF for that economy
- mathematically: given resources ( z1 , z2 ) and production
technologies x1 = f ( z1 , z2 ) and x2 = g ( z1, z2 ) , UPF is derived
from
B
B
B
U
(
x
,
x
max
1
2 ) subject to
z ,z ,z ,z
1
1
1
2
2
1
2
2
i) U A ( x1A , x2A ) ≥ U A
ii) x1A + x1B ≤ f ( z11 , z12 )
iii) x2A + x2B ≤ g ( z12 , z22 )
iv) z11 + z12 ≤ z1
v) z12 + z22 ≤ z2
(how would this differ if good two a public good?)