1. 12. Consumer Theory
Econ 494
Spring 2013

2. Agenda Shifting gears…Focus on the consumer, rather than the firm
Axioms of rational choice
Primal: Utility maximization  Marshallian demands
Dual: Expenditure minimization  Hicksian demands
Link between utility max and expenditure min.
Welfare measures
Readings
Silb. Ch 10; also p 53-55

3. Introduction Consumers purchase goods, x1…xn, at prices, p1…pn.
Consumers have a budget or income (M) with which to purchase these goods/services
How do consumers decide what to buy?
Buy goods to make them happy
Derive satisfaction from consumption
This satisfaction can be described by a utility function
Consumers will maximize utility s.t. budget constraint

4. Axioms of rational choice
These are behavioral postulates  no need to prove
Budget constraint is straightforward, but notion of a utility function is not.
We assert that consumer preferences must exhibit the following characteristics:

5. 1. Completeness
Individuals are always able to choose between two bundles
Consumers can rank bundles
Binary preference comparison
One of the following must be true:
A is preferred to B
B is preferred to A
A and B are equally preferred

6. 2. Transitivity
The rankings consumers assign to bundles must be consistent or transitive
We only require that consumers can compare 2 bundles at a time, but the pairwise comparisons must be linked
Transitivity:
If A is preferred to B,
And if B is preferred to C,
Then A must be preferred to C.

7. Comment on transitivity
Assuming transitivity is somewhat controversial
There is some evidence that indicates that choices are not always transitive
3. Reflexivity
Bundle A is at least as preferred as itself
Almost obvious, requires only very weak logical behavior

8. The 1st 3 axioms The first 3 axioms:
Completeness
Transitivity
Reflexivity
These formalize the notion that the consumer’s choices are rational or logically consistent.
These require that individuals can rank choices  ordinal measure of satisfaction/utility
Does not require individuals to assign some level or degree of satisfaction with these choices  not cardinal measure
 Utility function is only a ranking.
If A is preferred to B, then U(A) > U(B)

9. Utility is an ordinal measure
Ordinality  individuals can rank bundles
If U(A)=50 and U(B)=25, then A is preferred to B, but we cannot say that A is preferred twice as much as B.
Utility function is not unique – can take a monotonic transformation
Any transformation that preserves rankings
U0(x) = x2 or U1(x) = 2ln(x) will work just as well

10. 4. Continuity & differentiability
Mathematical assumptions about utility function
Continuity
If A is strictly preferred to B, then there is a bundle “close” to A that is also preferred to B
Differentiability
Utility function is twice differentiable

11. 5. Non-satiation
The utility function is monotonically increasing in the consumption of each good
“More is preferred to less”

12. 6. Substitution Consumers can make trade-offs among goods
Assume bundles are perfectly divisible
The maximum x2 a consumer will give up to get 1 unit of x1 is the amount that will leave her indifferent between old and new situation
Indifference curves (analogous to isoquants)
Slope of indifference curve represents trade-offs person is willing to make
Indifference curve – locus of consumption bundles that yield same level of utility

13. Indifference curves slope down
An explicit function for an indifference curve:
Substitute into utility function to get identity:
Differentiate identity wrt x1:
Downward sloping indifference curves is implied by assumption of nonsatiation
Indifference curves are mathematically the same idea as isoquants for production functions
Rearrange terms:

14. Marginal rate of substitution
A negatively sloped indifference curve means consumers are willing to make trade-offs  less of one good for more of the other
Marginal rate of substitution (MRS)

15. Indifference curves are convex
The marginal value of any good decreases as more of that good is consumed.
Diminishing MRS  ¶2x1 / ¶x22 > 0
Differentiate MRS wrt 𝑥 1
See notes #8 slides for derivation

16. Graphical illustration
At point a, consumer is willing to give up more x2 for a unit of x1 than at point b.

17. Indifference curves cannot cross
A and B are on same indifference curve
U(A) = U(B) = U1
A and C are on same indifference curve
U(A) = U(C) = U2
By transitivity, it must be true that
U(B) = U(C)
But…C includes more of both goods than B. By nonsatiation:
U(B) < U(C)
Contradiction  curves cannot cross

18. Indifference maps
We can fully describe a utility function, and hence an individuals preferences in (x2, x1) space by an indifference map.
U2 > U1

19. Utility maximization
Suppose a consumer gets utility from 2 goods and has income M. The indirect utility function is:
Assume person spends all her money.
V (p1, p2, M) is quasi-convex in (p1, p2)
we are not going to prove this

20. FONC
– Slope of indifference curve
– Slope of budget line

21. Tangency of indifference curve and budget line

22. SOSC
Strict quasi-concavity of the utility function assures us that the indifference curves will be strictly convex
We do NOT know sign of U11 and U22 !!  Diminishing marginal utility not implied

23. Marshallian demand functions
By the IFT, if the SOSC are satisfied then the FONC can, in principle, be solved simultaneously for the explicit choice functions (Silb., p. 262, lays the IFT conditions out more formally):
xim(p1, p2, M) i=1,2
lm(p1, p2, M)
xim(p1, p2, M) are utility maximizing demands
also referred to as Marshallian demand functions
or money-income-held-constant demands

24. Elasticities

25. Marshallian demands are HOD(0) in prices and income
For prices & income (p1, p2, M), the FONC:
For prices & income (tp1, tp2, tM), the FONC:
Since the FONC are the same for both sets of prices, the solutions must also be the same. Therefore, xim(p1, p2, M) = xim(tp1, tp2, tM)

26. Comment on homogeneity
HOD(0) implies that “relative” price matter
Consumption choices will not change with inflation if all prices and wages are increased at the same rate
Consumption opportunities do not change if prices and income change by same proportion.

27. Marshallian demands are invariant to positive monotonic transformation
The demand functions that solve:
Are identical to the demand functions that solve:
see Silb p , 264-5
Read his discussion on why “diminishing marginal utility” has no meaning with ordinal utility.

28. Proof of invariance to monotonic transformation
Since FONCs are identical, xim that solve FONCs must be identical.
Note that U1/U2 = p1/p2 holds if F ' >0 or F ' <0. But if F ' <0, then an increase in both x1 and x2 would decrease utility. Hence, F ' <0 would correspond with minimizing utility. If F ' >0, then U and Û will move in same direction, and U will achieve a maximum IFF Û does. (See Silb p. 264 Proposition 1)

29. Comment on monotonic transformation
This proposition highlights the ordinal nature of preferences.
A positive monotonic transformation preserves the ranking of all bundles
A positive monotonic transformation says nothing about the behavior of the individual

30. Envelope theorem The indirect objective function is:
Apply envelope theorem:

31. Characteristics of indirect utility function
Indirect utility function is:
Non-increasing in prices
Proof: ¶V / ¶pi < 0 from previous slide
Non-decreasing in income
Proof: ¶V / ¶M > 0 from previous slide
Indirect utility function is HOD(0) in prices and income
Proof: since Marshallian demands are HOD(0):
Indirect utility fctn is quasi-convex in prices (p1, p2)
We will not prove this…
Why is indirect utility HOD(0), but indirect profit and indirect cost functions are HOD(1)?  Utility function does not have any parameters explicitly. All parameters are in the constraint.

32. Roy’s identity Solving for xi, and using lm = ¶V / ¶M :
This relationship is known as Roy’s identity

33. Interpreting l
See Silb., p
At any given consumption point, additional utility, Ui, can be gained by consuming “a little more” xi.
The marginal cost of this extra xi is pi.
 The marginal utility per dollar spent on xi is Ui / pi

34. Use Envelope Theorem to Interpret l
The Lagrange multiplier is always interpreted as the marginal effect on the optimal value of the objective function as the constraint changes.
lm(p1, p2, M) is the marginal utility of income

35. Comparative statics Earlier, we used the IFT to solve the FONC:
xim(p1, p2, M) i=1,2
lm(p1, p2, M)
If we substitute these explicit choice functions back into the FONC, we get the following identities:

36. Comparative statics for p1
Differentiate the identities wrt p1, then express in matrix form:

37. Apply Cramer’s rule to solve
Signs are indeterminate
If M increases, cannot use less of both goods – would violate budget constraint
Find ¶x2m / ¶p1 on your own
Why is sign indeterminate?
Parameters show up in the constraint.

38. Show: Cannot use less of both goods (when income increases)
Substitute solution back into budget constraint:
Differentiate wrt M:

39. Engel Aggregation
Convert to elasticity form:

40. Cournot aggregation
Note that if p1 increases, you could consume less of both goods.

41. Cournot and Engel Aggregation
Cournot and Engel aggregation can be useful when estimating demand functions
Can be used as a restriction in a regression
Or, can test whether estimated coefficients are consistent with this.

42. Slutsky (later we will prove)
Find the sign, using comp. static results: