Microeconomics I. Antonio Zabalza. University of Valencia 1
Micro I. Lesson 4. Utility
In the previous lesson we have developed a method to rank consistently all bundles in the (x,y) space and we have introduced a concept –the indifference curve- to help us in this analysis. Now we introduce a related concept to rank bundles –the utility function- that will be useful to solve the equilibrium of the consumer in terms of calculus.
4.1 Ordinal versus cardinal utility
In the past, utility was conceived as a quantitative measure of a person’s welfare out of consuming goods. Now it is recognised that utility cannot be quantified because of the impossibility of interpersonal comparisons. So we do now as we did in Lesson 3, we only rank bundles.
Utility function: A way of assigning a number to every possible consumption bundle such that more- preferred bundles get assigned larger numbers than less-preferred bundles.
(,x¢y¢)ff(x¢¢,y¢¢) if and only if Ux(¢,y¢)U(xy¢¢,)¢¢
The only property about the numbers the utility function generates which is important is how it orders the bundles; how it ranks them. The size of the difference between the numbers assigned to each Microeconomics I. Antonio Zabalza. University of Valencia 2 bundle does not matter. Thus we talk about ordinal utility.
Since only the ranking matters, there can be no unique way to assign utility to bundles. If U(x,y) represents one way of ranking goods, 2U(x,y) is equally acceptable: it ranks bundles in the same manner. Multiplying by 2 is an example of a monotonic transformation.
Example: U=xy
Bundle x y U=xy
A 1 4 4 B 2 3 6 C 1 2 2
Ranking: B>A>C
U=2xy
Bundle x y U=2xy
A 1 4 8 B 2 3 12 C 1 2 4
Ranking: B>A>C
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A monotonic transformation is a way of transforming one set of numbers into another set of numbers in a way that the order of the numbers is preserved.
If the original utility function is U(x,y), we represent a monotonic transformation by fUxy[ (,)]. The property the function f[.] has to have is that
If U1>U21 Þ> f(U)fU()
Examples of monotonic transformations: f(U) = 2U f(U) = 3U f(U) = U+2 f(U) = U+10 f(U) = 5+3U f(U) = U3 (What about f(U) = U2?)
See that to preserve the order, f(U) must be a strictly increasing function of U.
Utility functions have indifference curves too; they are the level curves in the space (x,y) of the three dimensional function U=f(x,y). The indifference curves of a monotonic transformation of a utility function are the same as the indifference curves of the original utility function, only that the numbers attached to each indifference curve are different.
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4.2 From utility functions to indifference curves
If you are given a utility function U(x,y), it is easy to derive a given indifference curve from it: simply plot all points (x,y) such that U(x,y) equals a constant.
Examples: U(x,y)=xy k=xy y=k/x
y Rectangular hyperbola
k=3
k=2 k=1 x
Ux(,)y= xy22
Notice that since xy cannot be negative (we are in the positive quadrant), x2y22= ()xy preserves the same order. So this utility function is a monotonic transformation of the previous function. The formula for the indifference curve is: Microeconomics I. Antonio Zabalza. University of Valencia 5
f = xy22 f12 = xy yx= f12/
This is the same indifference curve map as before, only that the levels of the indifference curves are the squared of the previous levels
y Rectangular hyperbola
f=9
f=4
f=1
x
Perfect substitutes (blue pencils, red pencils) Ux(,)y=+xy kxy=+ ykx=-
Slope -1
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Perfect substitutes but at differents proportions: for example, suppose for the consumer x is twice as valuable as y. Ux(,y)2=+xy k=+2xy y=-kx2
Slope -2
In general, Ux(,)y=+axby k=+axby ka yx=- bb This is a utility function in which the consumer values x as much as a/b units of y.
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Perfect complements (left shoe, right shoe)
Ux(,y)= min,{xy}
If I have 2x (two right shoes) and 1y (one left shoe) it is like if I had only one pair of shoes: I get the same utility as with 1x and 1y.
Slope 1
2
1
1 2
The proportion need not be 1 to 1. Say, a consumer uses always 1x (cup of cofee) with 2y (two sugars), then ìü1 ux(,y)= min,íýxy îþ2
Slope 2
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In general Ux(,y)= min,{axby} The slope of the axis is then a/b. Check you understand the function min{×}.
Cobb-Douglas utility function
Ux(,y)=xaby; ab>>0; and 0
This is a well known function which generates well behaved indifference curves (smooth, negative and convex). y y
a>b a
Good x is relatively preferred to good y x Good y is relatively preferred to good x x
Cobb-Douglas functions are frequently used in production theory, where instead of utility we talk of output, and instead of goods we talk of inputs.
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4.3 Marginal Utility and the MRS
Consider a consumer that consumes the bundle (x,y). How does this consumer’s utility change when we maintain the amount of y and give him a little more of x? The change in utility per unit of change in x is the marginal utility of x (MU x ). DUU(x+D-x,y)Uxy(,) MU == x DDxx
MU x measures by how much utility changes when we change x by a small amount holding y constant. In the limit, if the change in x is infinitesimal, dU MU = x d x From the definition of marginal utility it follows that the change in utility that results from a small increase in x, holding y constant is:
DU=MUxx ×D
We have the same sort of definitions for good y; marginal utility of y (MU y ).
DUUx(,y+D-y)Uxy(,) MU == y DDyy dU MU = y d y
DU=MUyy ×D
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The utility function can be used to measure the MRS defined in the previous lesson.
Suppose that, for a given utility function, both x and y change. In general, when we change the quantities consumed of x and y, the level of utility will change. The total change in utility will be the sum of the change in utility generated by the change in x plus the change in utility generated by the change in y.
DU=MUxy×Dx+MUy×D
Suppose additionally that this change in x and y is a movement along a given indifference curve. This means that after the movement, the level of utility must be the same, and that D=U 0. Therefore,
0 =MUxy×Dx+MUy×D
Or Dy MU =- x DxMU y But DDyx, measured along a given indifference curve, is (minus) the MRS. Therefore,
Dy MU MRS =-= x DxMU y
The MRS can be measured by the ratio of the respective marginal utilities of the two goods. Microeconomics I. Antonio Zabalza. University of Valencia 11
Annex (Some calculus)
U=Uxy(,) (1) ddUUxy(,) MU == x ddxx ddUUxy(,) MU == y ddyy
Totally differentiating (1) we find
ddUU dU=+dxdy ddxy
dU=+MUxydxMUdy
Along an indifference curve dU = 0. Therefore,
0 =+MUxydxMUdy dy MU =- x dxMU y dy MU MRS =-= x dxMU y