9. Utility Theory - 9.1 Fundamental Concepts
Let u(x) denote the utility of having x units of a good to a
particular person.
By assumption, 0 is chosen as a reference point, i.e. u(0) = 0. This
should be interpreted as the utility of having no possessions.
u is an increasing function (u 0 (x) > 0), i.e. the more one has of a
particular good, the better.
The marginal utility function is given by u 0 (x).
It is assumed that u is a concave function (u 00 (x) ¬ 0), this means
that doubling the amount of a good at most doubles the utility of
an individual (equivalently, the marginal utility of purchasing extra
units is decreasing).
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A Utility Function
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Utility Functions
When there is no randomness, then maximization of one’s utility,
u(x), is equivalent to maximizing x.
When there is randomness, one may well maximize expected utility,
E [u(X )].
However, this is not equivalent to maximizing the expected value
of the amount of a good possessed, E (X ).
It should be noted that multiplying an individual’s utility function
by a positive constant would not change their decision (the
expected utility from a decision would be multiplied by that
constant, which would not change the ordering of expected utilities
in any way).
Hence, we may assume, for example, that u(1) = 1.
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Example 9.1
Suppose an individual can make one of two decisions:
Decision A : guarantees $36.
Decision B : if a coin toss results in tails, the individual gains
$100, otherwise the individual does not gain
anything.
a) Which decision maximizes the individual’s expected payoff?
√
b) Suppose the individual’s utility from gaining $x is x. Which
decision maximizes the individual’s expected utility?
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Example 9.1
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Example 9.1
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Aversion to Risk
If an individual is risk neutral, then he/she is indifferent between a
sure payoff and a lottery which gives the same expected payoff.
If an individual’s utility function is linear, i.e. u 00 (x) = 0, then such
an individual is risk neutral.
If an individual is risk averse, then he/she will prefer a sure payoff
to a random lottery which gives the same expected payoff.
If an individual’s utility function is strictly concave, u 00 (x) < 0,
√
then he/she is risk averse (e.g. when u(x) = x in Example 9.1,
the individual preferred the sure payoff to a lottery which gives a
higher expected reward).
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Risk Seeking Behaviour
If an individual is risk seeking, then he/she will prefer a lottery to a
sure payoff equal to the expected payoff from the lottery.
Classical utility theory predicts that individuals will either be risk
averse or risk neutral.
However, many people clearly show risk seeking behaviour (playing
the lottery, betting on sports).
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Risk Seeking Behaviour
The assumption that u(2x) ¬ 2u(x), i.e. doubling one’s
possessions at most doubles one’s utility seems reasonable in
normal circumstances.
Hence, the seemingly high frequency of risk seeking behaviour
must result some other factor(s) that utility theory does not take
into account.
One possible reason for this may lie in the fact that individuals
gain utility from winning a bet (or added excitement from
watching a match).
Another possibility is that individuals ”overestimate” small
probabilities (it could though be argued that in the case of the
lottery this effect is associated with the ”excitement” effect).
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Multivariate Utility Functions
The properties of multivariate utility functions are very similar to
those of univariate utility functions.
We will just consider functions of two variables u(x, y ).
u(x, y ) is increasing in each of its arguments (given the other is
fixed), i.e. ux (x, y ) > 0 and uy (x, y ) > 0.
u(x, y ) is concave in both of its arguments (given the other is
fixed), i.e. uxx (x, y ) ¬ 0 and uyy (x, y ) ¬ 0.
This is equivalent to the marginal utility from purchasing extra
units of a good is decreasing, given the amounts of the other
goods purchased are fixed.
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Multivariate Utility Functions
We assume that u(0, 0) = 0.
Since multiplying utility functions by a constant does not affect
decision making, we can scale utility functions in any way we want,
e.g. define u(1, 1) = 1.
It should be noted that, in the case of a univariate utility function,
the above properties were sufficient to ensure that returns to scale
are negative, i.e. doubling one’s possessions at most doubles one’s
utility.
This is not the case for multivariate utility functions. Hence, one
should specifically add the assumption that returns to scale are
negative, i.e. that for k > 1, u(kx, ky ) ¬ ku(x, y ).
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Multivariate Utility Functions
For example, u(x, y ) = x 0.75 y 0.75 satisfies the monotonicity and
concavity properties.
However,
u(kx, ky )=(kx)0.75 (ky )0.75 = k 1.5 x 0.75 y 0.75
=k 1.5 u(x, y ) > ku(x, y ), when k > 1.
On the other hand, when a, b > 0 and a + b ¬ 1, then
u(x, y ) = x a y b satisfies the conditions required to be a utility
function.
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9.2 Maximizing Utility under Budget Constraints
Utility functions with multiple arguments describe to some degree
the preferences of individuals.
For example, if x denotes the number of apples and y the number
of bananas and let 0 < α ¬ 1:
1. u(x, y ) = x α + y α - The individual shows no
preference for either apples or bananas (the utility
function is symmetric with respect to x and y ).
2. u(x, y ) = x α - The individual simply does not like
bananas (gains no utility from them).
3. u(x, y ) = 3x α + y α - The individual prefers apples,
(ascribes greater weight to them in the utility
function), but values variety.
The larger α, the ”greater the appetite” of an individual (the
utility curve is less curved downwards).
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Maximizing Utility under Budget Constraints
Obviously, if an individual has a choice between various bundles of
goods, she would choose the one which gives her the greatest
utility.
In practice, an individual should maximize her utility for a given
budget.
Let the unit cost of good 1 be k1 , the unit cost of good 2 be k2
and the individual’s budget be c.
An individual buys x units of good 1 and y units of good 2, the
budget constraint states that the individual can afford the bundle
of goods purchased, i.e.
k1 x + k2 y ¬ c
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Maximizing Utility under Budget Constraints
First we assume that x and y can take any real non-negative
values. The utility maximization problem is given by
max u(x, y )
subject to
k1 x + k2 y ¬ c
In graphical terms, the boundary of the region described by the
budget constraint is given by a downwards sloping line.
If the utility function u(x, y ) is strictly concave in its arguments,
i.e. uxx (x, y ) < 0 and uyy (x, y ) < 0, then any contour of the utility
function, i.e. any curve of the form u(x, y ) = K , is convex.
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Maximizing Utility under Budget Constraints
The solution of the utility maximization function is found by
determining the highest contour of the utility function which is
tangent to the boundary given by the budget constraint (see next
slide).
The bundle (x ∗ , y ∗ ) which should be bought is given by the
intersection point of this highest contour and the boundary given
by the budget constraint.
Any bundle giving a higher utility cannot be afforded.
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Maximizing Utility under Budget Constraints - Graphical
Interpretation
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Maximizing Utility under Budget Constraints - Analytic
Solution
It is intuitively clear that when x and y can take any non-negative
real values, then the individual spends all of her budget.
It follows that k1 x + k2 y = c and hence y =
c−k1 x
k2 .
Using this equation, we can reduce the problem to one in which we
maximize a function of one variable, namely
max u x,
c − k1 x
k2
.
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When only an Integer Number of Units of each Good can
be Purchased
Suppose that good 1 is more expensive and that given the budget
constraint
j k a maximum of n units of good 1 can be bought, where
n = kc1 and bc denotes the integer part.
We then compare the utility obtained by purchasing the following
n + 1 bundles of goods:
(x, y ) ∈ {(0, m[0]), (1, m[1]), (2, m[2]), . . . , (n, m[n])},
where m[x] is the number of units of good 2 which can be
purchased
j when
k x units of good 1 are purchased, i.e.
1x
m[x] = c−k
.
k2
Note that it is assumed that an individual does not gain any utility
from the change which might remain.
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Example 9.2
Suppose the utility of an individual from having x apples and y
√
√
oranges is u(x, y ) = 3 x + y .
i) Would this person prefer a bundle of four apples and one
oranges, or a bundle containing one apple and nine oranges?
ii) Suppose an apple costs $1 and an orange costs 50 cents. Given
than an individual has $4, what combination of apples and oranges
maximize his utility (assume only an integer number of apples and
oranges can be bought andignore the utility from any change she
might obtain).
iii) Assuming that fractions of apples and oranges can be
purchased, find the optimal bundle.
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Example 9.2
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Example 9.2
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Example 9.2
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Example 9.2
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Example 9.2
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9.3 Production Functions
A firm’s level of production can depend on a number of factors,
e.g.
1. The number of employees.
2. Fixed capital (machinery).
3. Raw materials.
We assume that there are at most two factors of production, which
will be understood as human capital and fixed capital.
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Production Functions
A production function, f (x1 , x2 ) gives the value of a good
produced using x1 and x2 units of factors 1 and 2, respectively.
Production functions satisfy the following conditions:
1. Production is non-decreasing in xi , i = 1, 2, given
that the other variable is fixed (e.g. if more
individuals are employed, then production cannot
fall).
2. Production is a concave function of xi , i = 1, 2, given
that the other variable is fixed (e.g. the marginal gain
from employing an additional employee decreases as
the number of employees increases).
It is assumed that in the short term at least one of the two factors
is fixed.
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Production Function with a Fixed Level of Human Capital
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Production Functions and Returns to Scale
It should be noted that these are the same as the first two
assumptions regarding multi-dimensional utility functions.
However, it is not assumed that e.g. doubling each input leads to
output increasing by a factor of at most two.
There are constant returns to scale when f (kx, ky ) = kf (x, y ), i.e.
increasing the levels of human and fixed capital by a factor k
increases production by a factor k.
There are increasing returns to scale when for k > 1,
f (kx, ky ) > kf (x, y ), i.e. increasing the levels of human and fixed
capital by a factor k increases production by a greater factor.
There are decreasing returns to scale when for k > 1,
f (kx, ky ) < kf (x, y ), i.e. increasing the levels of human and fixed
capital by a factor k increases production by a smaller factor.
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Marginal Productivity of Capital
Suppose the production function is f (x, y ), where x is human
capital and y is fixed capital.
The marginal productivity of human capital is given by the partial
derivative fx (x, y ). This is the rate of increase in production when
human capital is increased by a small amount.
The marginal productivity fixed capital is given by the partial
derivative fy (x, y ). This is the rate of increase in production when
fixed capital is increased by a small amount.
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Optimizing Production Levels
It is assumed that cost of each unit of capital (human or fixed) is
constant.
The cost of a unit of human capital is c1 and the cost of a unit of
fixed capital is c2 .
There are two problems which seem natural within this framework:
1. Maximizing the value of production given a total
budget for labour and capital.
2. Maximizing the value added by production (i.e. value
of production minus production costs) when the
budget is unlimited .
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Optimizing Production Levels
The first problem is analogous to the problem of maximizing utility
on a fixed budget and can be solved using the same methods.
The second problem requires a slightly different approach.
The solution to the first problem can be thought of as defining a
short-term goal.
The solution to the second problem can be thought of as defining
a long-term goal.
It will be assumed that the returns to scale are negative, otherwise
e.g. doubling each factor of production will at least double the
value added by production.
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Optimizing Production Levels - Unlimited Budget
Hence, a firm should maximize s(x, y ), where
s(x, y ) = f (x, y ) − c1 x − c2 y ,
i.e. s(x, y ) is the value of production minus the production costs.
Differentiating with respect to x and y individually, the optimality
conditions are given by
fx (x, y ) = c1 ,
fy (x, y ) = c2 ,
i.e. the marginal gains from each form of capital are equal to the
marginal costs of employing that form of capital.
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Example 9.3
Suppose the production function of a certain firm is given by
f (x, y ) = 16x 0.25 y 0.5 , where x is human capital and y is fixed
capital.
Suppose the unit cost of human capital is 1 and the unit cost of
fixed capital is 8.
Find the levels of human capital and fixed capital which maximize
the value added by production.
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Example 9.3
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Example 9.3
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Example 9.3
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Weaknesses of Production Functions
In the real world, the returns to scale are increasing at low levels of
capital and decreasing at high levels of capital.
This property is not reflected using a simple production function.
In addition, the level of technology is developing and it is thus
difficult to ”measure” fixed capital.
Also, the state of fixed capital will deteriorate over time unless it is
replaced.
It is a reasonable approximation to assume that the costs of human
capital are linear in x. However, the costs of fixed capital are split
into running costs and purchase costs.
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