Microeconomics models interactions between individual agents (consumer, government, firm, etc.) who behave according to their own private interests.  We start the course with the most classical consumer theory. In order to be able to use math apparatus for work with individual demands, we have to establish how those demands are connected with the behavior. There are two approaches to axiomatize the consumer decision-making. The preference-based approach assumes existence of a rational preference relation over the number of available alternatives. The choice-based approach supposes some consistent choice behavior. However, notice that the preference-based approach deals with an unobservable primitive (preference), while the choice-based approach works directly with the observable choices. Both approaches are closely connected and result in an establishment of a utility function – mathematical representation of consumer preference.

1   Preference order and utility

Denote by X a set of mutually exclusive alternatives available to the individual.  Consider  下 a binary relation (preference relation) on the set X that allows comparisons of pairs of elements of X . We read x 下 y as "x is at least as good as y . "

Definition:  The preference relation 下 is complete if for all x, y ∈ X :  either x 下 y or y 下 x (or both).

Completeness of preferences means that the individual is not paralyzed by indecision when asked about her preference between two alternatives and is always able to specify what she likes not less (more or her indifference).  Hence, if we ask a consumer whether she prefers pasta or sushi, she cannot reply that she does not know.

Definition:  The preference relation 下 is transitive if for all x, y, z ∈ X : if x 下 y and y 下 z, then x 下 z .

Transitivity of preference is the real guarantor of "rationality. " If a consumer says that she likes sushi not less than pasta, and pasta not less than ramen, then if we ask her about preference over

sushi and ramen, she must reply that she likes sushi not less than ramen.              Definition:  The preference relation 下 is rational if it is complete and transitive.

In addition to 下, we derive two different useful relations 〉 and ∼ .

Definition: x 〉 y ⇔ x 下 y but not y 下 x.

Definition: x ∼ y ⇔ x 下 y and y 下 x.

Working with preferences is quite abstract, that is why we would like to move to a more familiar mathematical apparatus. In order to do so, we heed to define a function that will represent consumer preference – utility function.

Definition:  A function u : X → R is a utility function that represents 下 if, for all x, y ∈ X : x 下 y ⇔ u(x) ≥ u(y).

Theorem:  Preferences 下 can be represented by a utility function if and only if 下 is rational and

X is countable.

Necessity is very straightforward and is left to you as a practice problem. First, we show sufficiency when X is finite.

Suppose for a moment that there is no indifference relation in X  (we will take them back later) or leave only one representative alternative out of each indifference and delete the rest.  Since X is finite, then we can define for each x ∈ X the number, nx , of alternatives that are worse that x. Define utility function u(x) = nx .  Now we can return all deleted indifference options and assign them with the corresponding utility number.

Now we show sufficiency when X is countable.  For each x ∈ X define the no-better-than x set as NBT (x) = {y ∈ X  : x 下 y}.  This set includes all elements worse than x and all indifferent alternatives to x.

Now let {x1 , x2 , . . . } be some (any) enumeration of set X .  The next step is to define a function d : X → R as d(xn ) = 0.5n , and specify utility function

u(x) =     工    d(z).

z∈NBT (x)

Suppose that x 下 y, then NBT (y) ⊆ NBT (x), hence, u(x) ≥ u(y). 

Example: To follow the above proof, you might want to consider the following example. Suppose that X consists of bundles with different numbers of oranges.  Hence, X = {{1}, {2}, . . . }, which is countable set as it is naturally enumerated. For simplicity of understanding, suppose that more oranges is better (this is a redundant assumption).  Consider bundle with five oranges, i.e.,  {5}, then

NBT ({5}) = {{1}, {2}, {3}, {4}, {5}}.

Now let’s use the natural enumeration for this case, i.e., xi  = {i}. Then we will assign d({i}) = 0.5i . Now compare NBT sets for {5} 〉{3}:

NBT ({3}) = {{1}, {2}, {3}}.

You may notice that NBT ({3}) ⊂ NBT ({5}). This will always be the case for any x 〉y . Hence, we can conclude the following about the constructed utility:

u({5}) = 0.51 + 0.52 + 0.53 + 0.54 + 0.55

u({3}) = 0.51 + 0.52 + 0.53

Hence, u({5}) > u({3}).

Unfortunately, when X is infinite utility does not always exist.  So when X is infinite, additional assumptions are required.

Starting from here, we will consider that X = R and we will call the elements of X bundles.       Definition: An indifference curve  (IC) is the set of all bundles that provide the same level of utility.

Example: Suppose u(x, y) = x + y . Find an IC such that u = 5.

Set x + y = 5, then y = 5 − x is the IC.

Figure 1: IC: y = 5 − x

Definition:  The marginal rate  of substitution (MRS) measures how many units of good xj

the consumer is willing to give up for one additional unit of good xi , or

MRSij  = −  |u=const .

MRS represents the slope of ICs.

Definition:  Marginal  utility  of good xi   is how much additional satisfaction is received from consuming one extra unit of good xi , or MUi  =  .

Theorem: MRSij  =

Proof: The total differential of the utility is

du(x) =  dxi  =  MUi dxi .

If we fix an IC then du = 0, in addition, we fix the levels of all goods except xi  and xj . Hence,

0 = MUi dxi + MUj dxj  ⇒ MRSij  = −  |u=const  =  . 

Often the larger amounts of goods are preferred to smaller ones.  This feature of preferences is described by monotonicity. This property also guarantees that utility is non-decreasing.

Definition: 下 is monotone if for any x, y ∈ X : y >> x implies y > x.            Another desired property is continuity as it delivers a continuous utility function.

Definition:  下 is continuous if for any sequence of pairs {xn , yn } with xn  下 yn , x = limn→∞ xn and y = limn→∞ yn  for all n, we have x 下 y .

Theorem:  Suppose that 下 is rational, monotone and continuous, then there exists a continuous utility function u that represents 下.

Proof: Denote the diagonal in X as D and as e a vector with all 1 as its components. Then for any α ≥ 0, αe ∈ D . For any x ∈ X there exists a unique α(x)e such that α(x)e ∼ x. Why is it unique and why does it exist?

1.  Suppose that there are α 1  > α2  such that α 1 e ∼ x and α2 e ∼ x.  By monotonicity, α 1 e > α2 e, hence, we get a contradiction.

2. Suppose there is no such α that αe ∼ x. Denote by NBT (x) the set of no-better-than x bundles and by NWT (x) the set of no-worse-than x bundles.  Suppose that NBT (x) is not a closed set.

Then there exists a sequence {yn }, such that yn   ∈ NBT (x) for any n ∈ N, y = limn→∞ yn  and y  NBT (x). Or in other words, there exists a sequence {yn }, x 下 yn  with the limit y 〉x. Thus, we get a contradiction of continuity.  It means that NBT (x) is a closed set.  By analogy, we can show that NWT (x) is a closed set too. All points in D must belong to one or another set, but not both because α such that αe ∼ x does not exist.  However, note that D ⊂ NWT (x) ∪ NBT (x), while both sets are closed and D ∩ NWT (x) ∩ NBT (x) = ∅, which is impossible.

Figure 2: Intersection of IC with D

Hence, there is a unique α(x) such that α(x)e ∼ x. Now define utility u(x) = α(x). It represents 下 by construction. We will skip the proof of continuity of u(x), you can find it in Chapter 3 of MWG. 

You might remember from your earlier micro classes that ICs are usually downward sloping and convex.   None of the previous axioms would deliver these properties, which are crucial for the classical consumer theory as they guarantee "good" solutions to the constraint optimization. Hence, we need an additional axiom.  Convexity guarantees that ICs have diminishing MRS (downward sloping and convex).

Definition:  下 is  convex if for any x, y, z  ∈ X  and α  ∈ [0, 1]:  if y  下 x  and z  下 x,  then αy + (1 − α)z 下 x.

Note that if your preferences are convex then any mixtures of more desirable options should be still more desirable. This is a very strong assumption. For example, you might be indifferent between fries and ice-cream, however, having ice-cream with fries might be less pleasant than plain ice-cream.

Definition: A function f : X → R, where X is a convex set, is

1.  concave if for all x, y ∈ X and α ∈ [0, 1] : f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y);

2.  quasi-concave if for all x, y ∈ X and α ∈ [0, 1] : f(αx + (1 − α)y) ≥ min(f(x), f(y)) . Theorem:  The utility function is quasi-concave if and only if 下 is convex.

Figure 3: Convex and non-convex preferences

Proof:  First, suppose that 下 is convex.  Then for any x 下 y , u(x) ≥ u(y) and for any α ∈ [0, 1]: αx + (1 − α)y 下 y . Hence, u(αx + (1 − α)y) ≥ u(y) = min(u(x), u(y)).

On the other hand, suppose u(∙) is quasi-concave.   Then for any x  下 y:  u(αx + (1 − α)y)  ≥ min(u(x), u(y)) = u(y), and hence, αx + (1 − α)y 下 y, which means that 下 is convex. 

Generally, utility representation is not unique.  There exists an infinite number of different repre- sentations. However, all utilities must deliver the same ICs, so they are connected to each other in some way.

Theorem: If u represents 下 on X and f is a strictly increasing function with domain and range of the real numbers, then v(x) = f(u(x)) for any x ∈ X also represents 下.

On the other hand, notice that MRS characterizes ICs, and, hence, is unique for specific preferences.

2   Budget sets

A consumer needs to choose a bundle of L goods to purchase in the market.  We will denote a

lx1 」                                         lp1 」

x2                                                                         p2

. . .                                               . . .

「xL                                                                     「pL

choices are limited by a number of physical constraints. The most natural is that the consumption usually cannot be negative. Another example – it is impossible to consume more than 24 hours of leisure in a day.  Or often you might not be able to buy less than one loaf of bread (or less than a half of a loaf). For simplicity, we will not consider these special situations and will assume that the consumer is bounded only by two constraints: non-negative bundles and budget. The classical consumer theory can be extended to the most other special situations when required. For simplicity, we will also suppose that all prices are positive, even though generally this is not required.

Definition: A bundle x ∈ X is called feasible if px = p1 x1 + ∙ ∙ ∙ + pL xL  ≤ I .                              Clearly, the chosen bundle must be feasible, otherwise, consumer cannot physically buy it. Hence, the set of feasible bundles is {x ∈ RL  : px ≤ I}, which we will call a budget set.  Note that the budget set is convex.

Figure 4: Budget set

3   Choice

Now we turn our attention to the second approach in axiomatization. The primitive object of the

theory in this section is a choice structure instead of preference relation.

Definition: A choice structure (B, C(∙)) is represented by two elements:

1. B is a set of non- empty budget sets;

2. C(∙) is a choice rule that assigns non- empty set of chosen elements C(B) ⊂ B for every budget set B ∈ B .

Examples: Suppose that X = {x, y, X} and B = {{x, y}, {x, y, X}}.

1. C1 ({x, y}) = {x} and C1 ({x, y, X}) = {x};

2. C2 ({x, y}) = {x} and C2 ({x, y, X}) = {y, X}.

As with preference relation we needed rationality, in the same way we need an assumption about the choice structure such that the actual choices "make sense. "

Definition:  The choice structure (B, C(∙)) satisfies the  weak  axiom  of revealed preference (WARP) if for some B ∈ B with x, y ∈ B we have x ∈ C(B), then for any D ∈ B with x, y ∈ D and y ∈ C(D): x ∈ C(D).

This axiom basically states that when you go to a restaurant that serves sushi and pasta (and potentially other dishes) and it is known that you choose pasta at least sometimes, then when you go to a different restaurant that serves the same sushi and the same pasta, and it is known that you order sushi from time to time, then it has to be the case that you also order pasta on some occasions.

Note that C1 ( ∙ ) satisfies WARP (trivially), while C2 ( ∙ ) does not.  To satisfy WARP, C2 ({x, y, z}) has to be either {x, y} or {x, y, z}. For simplicity, we will suppose that X is finite. All the results below can be generalized to infinite case with some additional assumptions.

Now we would like to connect the idea of preference relation with the choice structure. Note that 下 always generate a choice structure defined as C(B, 下) = {x ∈ B : x 下 y for every y ∈ B}.

Theorem: Suppose that 下 is a rational preference relation, then the choice structure generated by 下, (B, C(∙ , 下)), satisfies WARP.

Proof: Suppose that for some B ∈ B, we have x, y ∈ B and x ∈ C(B, 下). This implies that x 下 y . Suppose there is also D such that x, y ∈ D and y ∈ C(D, 下). By definition of the choice structure we get that y 下 x since they both are in D .  However, we already established that x 下 y , hence, x ∼ y ⇒ x ∈ C(D, 下). 

So rational preference implies "rational" choice. And what about the opposite direction?

Note that the choice structure delivers less information about the consumer rather than preference as you observe only the best elements under available budgets.  However, not all possible budgets might be available and inconsistencies might arise in unobservable sets.

Example: Suppose that X = {x, y, z}, B = {{x, y}, {y, z}, {x, z}}, C({x, y}) = {x}, C({y, z}) = {y} and C({x, z}) = {z}.  This choice structure trivially satisfies WARP. However, we get x 〉 y , y  〉 z and z  〉 x, which contradicts transitivity.  So there is no rational preference that could generate this choice structure, even though it satisfies WARP.

As you can see the problem with this example is that the choice form the budget set {x, y, z} is not observable. So we need to add a condition that could guarantee that B is rich enough.

Theorem: If (B, C(∙)) is a choice structure such that WARP is satisfied and B includes all subsets of X  of up to three elements, then there is a unique rational preference  下 that rationalizes C(∙) relative to B.

Proof:  The idea behind the proof relies on the preference generated by the choice structure.  All two elements budgets guarantee that the generated preference is complete. While all three element

budgets deliver transitivity. For more detail, see MWG Chapter 1.D.                                         

4   Revealed Preference

In this section, we observe some choices made by consumer under different prices and/or income. What can we tell about this consumer? We are interested in whether these choices can be explained by maximizing a utility function.

We start with an example. Suppose we observe a consumer who makes the following choices.

1. p1  = (2, 1), I1  = 10 ⇒ x1  = (4, 2)

2. p2  = (1, 2), I2  = 10 ⇒ x2  = (2, 4)

Figure 5: x1  and x2

Are these choices consistent with utility maximization? We can try to you WARP here, however, we do not observe entire choice sets and without some additional restrictions we will not be able to contradict anything. The only result we can get is to to obtain a set of indifference as in the proposed example. Note that B 1  = {(x1(1), x2(1)) : 2x1(1) + x2(1)  ≤ 10}, x1  ∈ B 1  and x2  ∈ B 1  because x2p1  = 8 < 10, so x2 is feasible when x1 is chosen. On the other hand, B2  = {(x1(2), x2(2)) : x1(2) +2x2(2)  ≤ 10}, x2  ∈ B2  and x1  ∈ B2  because x1p2  = 8 < 10, so x1  is feasible when x2  is chosen. Hence, we get that x1  ∈ C(B1 ),

x2  ∈ C(B2 ) and x1 , x2  ∈ B 1 ∩ B2 , hence, by WARP x1 , x2  ∈ C(B1 ), C(B2 ). So the only thing we can conclude is x1  ∼ x2 . See Figure 5 for illustration.

However, notice that when x1   is chosen and x2   is feasible, x2   costs only 8, while x1   costs 10. Why would the more expensive bundle be chosen when a cheaper equivalent is feasible? This does not make much sense, but to be able to use this logic we add an assumption of monotonicity of preferences.

Lemma: Suppose a consumer with rational and monotone preferences 下 chooses the consumption bundle x* facing prices p and income I .  Then x*  下 x for all bundles x such that px = I and x*  〉 x for all bundles x such that px < I .

Proof:  All bundles x such that px = I are feasible, and since x*  is chosen, it has to be the case that x*  下 x. We can conclude the same for bundles x such that px < I . We need to prove that the indifference is not possible in this case. Suppose x ∼ x*  and px < I . However, due to monotonicity for bundles x in the interior of the set there exists a different bundle y which is also feasible and y 〉 x, hence, y 〉 x ∼ x* . Given that y is feasible and was not chosen, we get a contradiction. 

Now if we come back to our example and take into account monotonicity, then we have x1  ∈ C(B1 ) and x2    C(B1 ), x2  ∈ C(B2 ) and x1    C(B2 ).  At the same time, WARP requires that x1 , x2  ∈ C(B1 ), C(B2 ), thus, it is not satisfied.

However, what if x1 was (2,6) instead of (4,2). Would it satisfy WARP? See Figure 6 for illustration.

Figure 6: x1  and x2

Again, we get that x1  ∈ C(B1 ) and x1  〉 x2 . However, when x2  ∈ C(B2 ), x1p2  = 14 > 10, hence, x1   B2 . So these choices satisfy WARP.

Now consider a different example. Suppose that there are three consumption goods and we observe three choices.

1. p1  = (10, 10, 10), I1  = 300 ⇒ x1  = (10, 10, 10)

2. p2  = (10, 1, 2), I2  = 130 ⇒ x2  = (9, 25, 7.5)

3. p3  = (1, 1, 10), I3  = 110 ⇒ x3  = (15, 5, 9)

We can approach these choices in the same manner. First, we need to find out which bundles are feasible in which budget sets.  The easiest way to do so is to draw a table with the cost of each bundle under each set of prices.

Table 1: Cost of the bundles

Hence, we can conclude that x1 , x3   ∈ B 1 , x1 , x2   ∈ B2   and x2 , x3   ∈ B3 .   Note that WARP is satisfied here since we never have two chosen bundles feasible in two different budget sets. However, monotonicity suggests us that x1  〉 x3 , x2  下 x1  and x3  〉 x2 , which clearly contradicts transitivity. The problem of WARP is that it does not take into account cycles as above.  Even monotonicity does not help because the problem is with the set of budgets B which is not rich enough.

Definition:  The set of choices {(p , x )}ttt∈T  satisfies the Generalized Axiom of Revealed Pref- erence (GARP) if whenever there is a subset of observations satisfying

pt1 xt2   ≤ pt1 xt1

pt2 xt3   ≤ pt2 xt2

. . .

ptn − 1 xtn   ≤ ptn − 1 xtn − 1

ptn xt1   ≤ ptn xtn

then all inequalities hold as equalities.

Example: In the case with three observations, we have the following cycle.

p1 x3  ≤ p1 x1  (290 < 300) ⇒ x1  〉 x3

p3 x2  ≤ p3 x3  (109 < 110) ⇒ x3  〉 x2

p2 x1  ≤ p2 x2  (130 ≤ 130) ⇒ x2  下 x1