Microeconomics I Chapter Three S. Farshad Fatemi Introduction Preference and Utility The Utility Maximization Problem The Expenditure Minimization Problem Relationships between Demand, Indirect Utility, and Expenditure Functions Integrability Welfare Evaluation of Economic Changes The Strong Axiom of Revealed Preferences Microeconomics I 44715 (1396-97 1st Term) - Group 1 Chapter Three Classical Demand Theory Dr. S. Farshad Fatemi Graduate School of Management and Economics Sharif University of Technology Fall 2017 1 / 43
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Definitions (Rationality, desirability and convexity): The preferencerelation % on X is
Rational: if it is complete and transitive.
Monotone: if x , y ∈ X and y � x then y � x .
Strongly Monotone: if x , y ∈ X and y ≥ x and y 6= x theny � x .
Locally non-satiated:∀x ∈ X and ∀ε > 0, ∃y ∈ X such that ||y − x || ≤ ε and y � x .
It can be shown that strong monotonicity is a strongerassumption than monotonicity; and monotonicity is also astronger assumption than local non-satiation.
Definition (homotheticity): A monotone preference relation % onX = RL
+ is homothetic if all indifference sets are related by propor-tional expansion along rays: this is if x ∼ y then αx ∼ αy for ∀α ≥ 0.
Definition (quasilinearity): A preference relation %on X = (−∞,∞)×RL−1
+ is quasilinear with respect to commodity 1 (numeraire commod-ity) if:
i) All the indifference sets are parallel displacements of each otheralong the axis of commodity 1; this is if x ∼ y thenx + αe1 ∼ y + αe1 for e1 = (1, 0, ..., 0) and ∀α ∈ R .
ii) Commodity 1 is desirable; x + αe1 � x for ∀x and ∀α > 0.
Definition (continuity): The preference relation % on X is continu-ous if it is preserved under limits; this is
for any sequence of pairs {(xn, yn)}∞n=1
with xn % yn for all n, x = limn→∞
xn
and y = limn→∞
yn,
we have x % y .
OR
A preference relation % on X is continuous if whenever a � b, thereare neighborhoods Ba and Bb around a and b, respectively, such thatfor all x ∈ Ba and y ∈ Bb, x � y .
Example for non-continuity: lexicographic preferences.
Proposition (MWG 3.C.1): Suppose that the rational preferencesrelation % on X is continuous, then there is a continuous utility func-tion u(x) that represents % .
The utility function u(.) that represents a preference relation isnot unique.
Any strictly increasing transformation of u(.) is also a utilityfunction.
Not all utility representation of a preference relation iscontinuous (one can consider a strictly increasingtransformation which is not continuous).
For analytical purposes, usually we consider the utility functionsto be differentiable (not always e.g. Leontief preferences).
We usually assume that the utility functions to be twicecontinuously differentiable, but we do not formally discuss thenecessary assumptions on preferences that implies this property.
How do restrictions on preferences imply restrictions on Utilityfunctions?
Proposition (MWG 3.D.1): If p � 0 and u(.) is continuous, thenthe utility maximization problem has a solution.
Proof: A continuous function always has a maximum value on anycompact set.
The Walrasian Demand Correspondence/FunctionThe rule that assigns the set of optimal consumption vectors in theutility maximization problem to the price-wealth pairs.
Proposition (MWG 3.D.2): Suppose that u(.) is a continuous utilityfunction representing a locally nonsatiated preference relation % de-fined on the consumption set X = RL
+. Then the Walrasian demandcorrespondence x(p,w) possesses the following properties
i) Homogeneity of degree zero in (p,w)
x(αp, αw) = x(p,w) for ∀p,w & α > 0
ii) Walras’ lawp.x = w ∀x ∈ x(p,w)
iii) Convexity/UniquenessIf % is convex (strictly convex) then x(p,w) is a convex set(singleton)
Proof:i) The feasible set remains the same.ii) Simple; using the fact that % is nonsatiated.iii) Can be proved using the fact that u(.) is quasiconcave (strictly
quasiconcave) and any linear combination of two optimalbundles is feasible. 12 / 43
In contrast for a boundary solution x∗ ≥ 0 and x∗l = 0 for some l wemight have:
MRSlk(x∗) 6= plpk
λ is the marginal (shadow) value of relaxing the budget constraint.
Example: The demand function for a Cobb-Douglas utility function.
It can be shown that if preferences are continuous, strictly convex,and locally nonsatiated then the Walrasian demand function is alwayscontinuous at all (p,w) � 0. Furthermore, if the determinant ofthe bordered Hessian of u(.) is nonzero at x∗, the Walrasian demandfunction is differentiable.
The Utility Maximization ProblemThe Indirect Utility Function
The Indirect Utility FunctionThe utility value of the utility maximization problem.
Proposition (MWG 3.D.3): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated preference relation %defined on the consumption set X = RL
+. Then the indirect utilityfunction v(p,w) possesses the following properties:
i. Homogeneity of degree zero in (p,w)
v(αp, αw) = v(p,w) ∀p,w & α > 0
ii. Strictly increasing in w and non-increasing in pl
Proposition (MWG 3.E.1): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated preference relation %defined on X = RL
+ and that the price vector is p � 0. We have:
i) If x∗ is optimal in the UMP when wealth is w > 0, then x∗ isoptimal in the EMP when the required utility level is u(x∗).Moreover, the minimized expenditure level in this EMP isexactly w .
ii) If x∗ is optimal in the EMP when the required utility level isu > u(0), then x∗ is optimal in the UMP when wealth is p.x∗.Moreover, the maximized utility level in this UMP is exactly u.Students need to go through the proof in MWG
The Expenditure FunctionGiven the prices and the required utility, the value of the EMP isdenoted e(p, u) which is called the expenditure function. Its value isp.x∗ where x∗ is any solution to the EMP.
Proposition (MWG 3.E.2): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated preference relation %defined on X = RL
+. Then the expenditure function e(p, u) is
i. Homogeneous of degree one in p
e(αp, u) = αe(p, u) ∀p, u & α > 0
ii. Strictly increasing in u and nondecreasing in pl ; ∀liii. Concave in p
iv. Continuous in p and u
Proof:
i. The constraint set remains unchanged and the minimizationproblem is equivalent to the original one.
ii. Use the continuity of u(.) for the first part and the optimizationcondition for the second part.
iii. Can be proved using the definition of expenditure function.
Proposition (MWG 3.E.3): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated preference relation %defined on X = RL
+. Then for any p � 0, the Hicksian demandh(p, u) is
i. Homogeneous of degree zero in p
h(αp, u) = h(p, u) ∀p, u and α > 0
ii. No excess Utility
∀x ∈ h(p, u) : u(x) = u
iii. Convexity/UniquenessIf % is convex (strictly convex) then h(p, u) is a convex set(singleton)
Note: the Walrasian and Hicksian demand functions are equivalentas:
The Hicksian demand function is also called compensated demandfunction since it can show the level of wealth compensation requiredto keep the consumer at the same level of utility after a change inprices.
The Hicksian Demand and the Compensated Law of Demand
Proposition (MWG 3.E.4): Suppose that u(.) is a continuous utilityfunction representing a locally nonsatiated preference relation % andthat h(p, u) consists of a single element for all p � 0. Then h(p, u)satisfies the compensated law of demand:
Relationships between Demand, Indirect Utility,and Expenditure Functions
Proposition (MWG 3.G.1): (Shephard’s lemma) Suppose that u(.)is a continuous utility function representing a locally non-satiated andstrictly convex preference relation % defined on X = RL
+. Then
∀p, u ∴ h(p, u) = 5pe(p, u)
Or
∀p, u ∴ hl(p, u) =∂e(p, u)
∂pl∀l = 1, .., L
The Economic interpretation of this proposition can be seen if we lookmore carefully at the following:
Relationships between Demand, Indirect Utility,and Expenditure Functions
Walrasian Demand Function Proposition (MWG 3.G.2): Supposethat u(.) is a continuous utility function representing a locally non-satiated and strictly convex preference relation % defined on X = RL
+.Suppose also that h(., u) is continuously differentiable at (p, u). Then
i. Dph(p, u) = D2pe(p, u)
ii. Dph(p, u) is a negative semidefinite matrix.
iii. Dph(p, u) is a symmetric matrix.
iv. Dph(p, u)p = 0
Note:(ii) implies that compensated own price effects are nonpositive.(iii) implies that for compensated price cross derivatives we must have:
∂hk(p, u)
∂pl=∂hl(p, u)
∂pk
We also define two goods as substitutes and complements based on
the sign of ∂hk (p,u)∂pl
. And (iv) implies that every good has at least onesubstitute.
Relationships between Demand, Indirect Utility,and Expenditure Functions
Proposition (MWG 3.G.3): Suppose that u(.) is a continuous util-ity function representing a locally non-satiated and strictly convexpreference relation % defined on X = RL
+. Then
∀p,w , u = v(p,w) ∴ Dph(p, u) = Dpx(p,w)+Dwx(p,w)x(p,w)T
Or
∀p,w , u = v(p,w) ∴∂hl(p, u)
∂pk=∂xl(p,w)
∂pk+∂xl(p,w)
∂wxk(p,w) ∀l , k
Note:The Slutsky equation describes the relationship between the slope ofthe ordinary and compensated demand functions.
Relationships between Demand, Indirect Utility,and Expenditure Functions
Proposition (MWG 3.G.4): (Roy’s Identity) Suppose that u(.) isa continuous utility function representing a locally non-satiated andstrictly convex preference relation % defined on X = RL
+. Supposealso that v(.) is differentiable at (p, w)� 0. Then
x(p, w) = − 1
5wv(p, w)5p v(p, w)
Or
xl(p, w) = −∂v(p,w)
∂pl∂v(p,w)
∂w
∀l = 1, ..., L
Note:Roy’s Identity and Shepard’s lemma are parallel results for UMP andEMP.
Welfare Analysis with Partial Information So far, we learnt howit is possible to calculate the welfare effect of a price change when weknow the consumer’s expenditure function. The latter is not alwaysthe case.Firstly, a test is introduced to evaluate whether a price change im-proves the welfare or not.
Proposition (MWG 3.I.1): Suppose that the consumer has a locallynonsatiated rational preference relation %. If (p1 − p0).x0 < 0 , thenthe consumer is strictly better off under price-wealth situation (p1,w)than under (p0,w).
Proof: (p1 − p0).x0 < 0 ⇒ p1.x0 < p0.x0 = w which means x0 isaffordable under (p1,w) and is an interior point in the budget con-straint. Then since preferences are nonsatiated then there should bea better option available under (p1,w).
Proposition (MWG 3.I.2): Suppose that the consumer has a dif-ferentiable expenditure function. Then if (p1 − p0).x0 > 0, thereis a sufficiently small α ∈ (0, 1) such that for all α < α we havee((1 − α)p0 + αp1, u0) > w , and so the consumer is strictly betterunder price-wealth situation (p0,w) than under ((1−α)p0 +αp1,w).
Approximation of the Welfare Effect using the Walrasian De-mand CurveSince Hicksian demand is not directly observable, the Area Variation(AV) has been used extensively in the literature:
Proposition (MWG 3.J.1): If the market demand function x(p,w)Satisfies SARP then there is a rational preference relation % thatrationalizes x(p,w), that is, such that for all (p,w), x(p,w) � y forevery y 6= x(p,w) with y ∈ Bp,w .
Note: For L = 2 the SARP and the WARP are equivalent. But forL > 2, the SARP is stronger than the WARP.