Microeconomic Theory I, part 1

Marcin Pęski


Instructor: Marcin Pęski, mpeski@gmail.com. Office hours: Thursday, 11.30-1.30pm, GE207,
TA: Rami Abou-Seido. Office hours: TBA
Lecture: Tue, 9-11pm, BA3012, Th:, 9-11pm, AB114,
Tutorial Th, 2-4pm, WW121.
Please fill the AVAILABILITY FORM. I need this information to schedule office hours and one additional discussion section outside of the regular time.


Lectures and assigned readings

The required textbook is Microeconomic Theory by MasCollel, Whinston and Green (MWG for short). Below, you can find a more detailed description of the topics together with required readings. The assignment of topics to particular days is tentative and it may move as we go along.
Date Topics Readings Extras Problem sets due Solutions
7.09 Choices and preferences. Choice correspondence. Weak Axiom of Revealed Preference. Preference representation. MWG 1, WARP and preference relations↓, Lexicographic preferences↓
12.09 Utility theory and classical demand theory. Consumption space. Utility representation. Budget sets. Utility maximization. Walrasian demand correspondence and its properties. Price and wealth effects. MWG 3.A-D, 2, continuous utility
14.09 Classical demand theory II. Kuhn-Tucker conditions. Properties of indirect utility. Expenditure minimization. Properties of Hicksian demand and expenditure function. Envelope Theorems. Shepherd’s Lemma and Roy’s identity. Slutsky equation. The Law of Compensated Demand. MWG 3D-3G, separating hyperplane theorem ,Upper contour sets can be deduced from the expenditure function↓, Equivalence between dual and primal problems↓, 5↓ Problem set 1 PS1 solutions
19.09 Classical demand theory III. Aggregate demand. Integrability. Welfare comparisons. MWG 3H, 4A-B, 3I Aggregate demand↓
21.09 Firm Theory. Production sets. Profit maximization and cost minimization. Properties of Aggregate supply. Le Chatelier Principle. MWG 5 Cost function and returns to scale↓ Problem set 2 PS2 solutions
26.09 Comparative statics I. Implicit Function Theorem. Robust comparative statics. Single-crossing condition. Increasing differences condition. Comparative statics 1-3
28.10 Comparative statics II. Multivariate comparative statics. Supermodularity. Comparative statics 4 Problem set 3 PS3 solutions
3.10 Introduction to choice under uncertainty. Examples. State-dependent and state-independent expected utility. Subjective and objective uncertainty. Acts. Axioms of the Anscombe-Aumann Theory. Uncertainty 1-2, MWG 6.A
5.10 Expected utility theory I: The (Anscombe-Aumann State-Dependent) Expected Utility Representation Theorem.
MWG 6.B, 6.F
Problem set 4 PS4 solutions
10.10 Expected utility theory II. State-independent utility. Non-expected utility theories. Allais, Machina, and Ellsberg paradoxes. Uncertainty 4-5, MWG 6.B
12.10 Expected utility theory III: Risk-aversion. Certainty equivalent. First and second-order stochastic dominance. MWG 6.C-D Asset management↓ Problem set 5 PS5 solutions
17.10 Monotone statics under uncertainty. Stochastic dominance ordering. Comparison of lotteries. Marginal likelihood ration property. Uncertainty 7 Problem set 6 PS6 solutions
19.10 Midterm
The lecture notes will be posted on the course website.

Problem sets, past exams and solutions


Lecture 1. Choices and preferences.

WARP and preference relations

I wanted to clarify few points in the proof that WARP is necessary and almost sufficient for the existence of rationalizing preference relation. Recall that the choice structure satisfies WARP if and only if If for some B, B’ ∈ ℬ, if{x, y} ⊆ B and {x, y} ⊆ B and x ∈ C(B) and y ∈ C(B), then x ∈ C(B).
  • We want to show if (i) Warp is satisfied and (ii) the choice structure contains all 2- and 3-element subsets, then the revealed preference relation * is transitive.\begin_inset Separator latexpar\end_inset
    • Suppose that x*y, and y*z.
    • The definition of the revealed preference relation implies that x ∈ C({x, y}) and y ∈ C({y, z}).
    • Applications of WARP:\begin_inset Separator latexpar\end_inset
      • Because x ∈ C({x, y}) , WARP implies that if y ∈ C({x, y, z}), then x ∈ C({x, y, z}),
      • Because y ∈ C({y, z}) , WARP implies that if z ∈ C({x, y, z}), then y ∈ C({x, y, z}).
    • Because C({x, y, z}) ≠ Ø, the above implies that x ∈ C({x, y, z}).
    • By one more application of WARP, it must be that x ∈ C({x, z}).
    • By the definition of the revealed preference relation, x*z.
  • Given the same assumptions, we want to show that the revealed preference relation rationalizes the choices. We need to show two implications:\begin_inset Separator latexpar\end_inset
    • If x ∈ C(B), then x*y for each y ∈ B,\begin_inset Separator latexpar\end_inset
      • For each y ∈ B, we can apply WARP to sets B and B’ = {x, y} to conclude that x ∈ C({x, y}).
      • By the definition of the revealed preference relation, we have x*y (for each y ∈ B).
    • If x*y for each y ∈ B, then x ∈ C(B).\begin_inset Separator latexpar\end_inset
      • The definition of the revealed preference relation implies that x ∈ C({x, y}) for each y ∈ B.
      • WARP applied to set {x, y} and B implies that if y ∈ C(B), then x ∈ C(B).
      • Because C(B) ≠ Ø, the above implies that x ∈ C(B).\
  • Additionally, in class, I forgot to say that the revealed preference relation is the unique preference relation that rationalizes the choices. (If this wasn’t true, our Identification question would not be satisfactorily answered. ) This is because we observe choices from all 2-element sets.

Lexicographic preferences

Suppose that X = [0, 1] × [0, 1] is a consumption space over money and leisure. For each (c, l), (c, l) ∈ X, say that
(c, l)(c’, l) if and only if either c < c’,  or c = c and l ≤ l’.
These are preferences of a workoholic. We are going to show that these preferences do NOT have utility representation.
Indeed, suppose that the lexicographic preferences are represented by an utility function u:[0, 1]2 → ℝ.
For each c ∈ [0, 1], define f(c) = u(c, 0). Function f is (strictly) increasing.
For each c ∈ [0, 1], define Δ(c) = u(c, 16) − u(c, 0). Then, Δ(c) > 0 for each c ∈ [0, 1]. Moreover, for each c’ > c, u(c’, 0) > u(c, 16), which implies that
f(c) − f(c) = u(c’, 0) − u(c, 0) > u(c, 16) − u(c, 0) = Δ(c). for each c’ > c.
It follows that function f is discontinuous at each point c ∈ [0, 1], with a jump at least Δ(c) > 0:
liminfccf(c) ≥ f(c) + Δ(c).
In particular, function f has uncountably many points of the right-hand side discountinuity (there are uncountably many elements of the interval [0, 1]).
But then, we have a contradiction with the following Lemma:
Lemma. An increasing function f:[0, 1] → ℝ has at most countably many points of right-hand side discontinuity.
Proof: Let
Aε = {c ∈ (0, 1):f(c) ≥ f(c) + ε for each c’ > c}
be the set of c such that the right-hand side discontinuity is of the size at least ε.
Notice that for each ε > 0, we have
f(1) ≥ f(0) + ε|Aε|, 
where |Aε| is the number of elements in the set Aε. (This is because at each point in Aε, the value of function f jumps by at least ε, and the function is cinreasing, hence it never losses value.) Let Δ* = f(1) − f(0). Hence, it must be that
|Aε| ≤ (Δ*)/(ε) < ∞, 
and the set Aε is finite.
A = ∪ε > 0Aε, 
be the set of all points of right hand side discontinuity. Becuase for each x ∈ Aε ⊆ A1 ⁄ n, for each integer n > 1 ⁄ ε, we have
A = ∪ε > 0Aε = ∪nA1 ⁄ n, 
is a countable collection of finite sets, hence countable. QED.

Lecture 3. Classical demand theory II.

Equivalence between dual and primal problems

Proposition. Suppose that preferences are continuous and locally non-satiated. For each p ∈ RL +  +  and each u ≥ u(0),
v(p, e(p, u)) = u.
Proof. We need to show that (a) v(p, e(p, u)) ≥ u and (b) v(p, e(p, u)) ≤ u.
For (a), take h ∈ h(p, u). By the definition of the dual problem, u(h) ≥ u and ph = e(p, u). This implies that h satisfies the constraints of the primal problem (p, e(p, u)) (i.e., h ∈ B(p, e(p, u)), and thus
v(p, e(p, u)) ≥ u(h) ≥ u.
For (b), take x ∈ x(p, e(p, u)). Because the assumptions imply that Walras Law holds, we have5
px = e(p, u).
Suppose that u(x) > u ≥ u(0) and consider replacing x by
xε = (1 − ε)x + ε0
for some ε > 0. Then, for each ε > 0,
pxε = (1 − ε)px < e(p, u).
Moreover, by continuity of the utility function, for sufficiently small ε,
u(xε) > u.
But then, xε satisfies the constraints of the dual problem and leads to lower expenditure. Contradiction.

Upper contour sets can be deduced from the expenditure function

Assume that preferences are continuous, monotonic and convex. For each u,
{x:u(x) ≥ u} = {x:px ≥ e(p, u) for each p ∈ RL + }.
Inclusion “ ⊆ ” follows form the definition of the expenditure function. For the other inclusion, we need to show that for each x = (x1, ..., xL), if u(x) < u, then there exists p ∈ RL +  such that px < e(p, u).
Because the preferences are convex, the set {x:u(x) ≥ u} is convex. The serparating hyperplane theorem shows that there exists p ∈ RL, p ≠ 0 (but, possibly, pl < 0 for some l) such that
(1) px < infy:u(x) ≥ upy.
Below, we show that it must be that pl ≥ 0 for each l. Given that, notice that the right-hand side of the above inequality is equal to e(p, u). Thus, px < e(p, u).
To finish our proof, we need to show that if (1↑) holds, then it must be that pl ≥ 0 for each l. On the contrary, suppose not and there is l such that pl < 0. For each a > 0 and ε > 0, define a bundle
xa, ε =  x + a(ε, ..., ε, 1lth position, ε, ..., ε)  =  (x1 + aε, ..., xl − 1 + aε, xl + a, xl + 1 + aε, ..., xL + aε).
pxa, ε = px + a(pl + εl’ ≠ lpl).
Because pl < 0, we can find ε* > 0 small enough so that pl + εl’ ≠ lpl < 0 and for each a > 0,
(2) pxa, ε* < px.
Also, we can find a* > 0 large enough so that
(3) u(xa*, ε*) ≥ u.
(Indeed, choose any y such that u(y) ≥ u and notice that we can find a* large enough so that for each l, a*ε* ≥ yl − xl. Then, for each l
xa*, ε*l ≥ xl + a*ε* ≥ xl + yl − xl ≥ yl, 
in which case the monotonicity of preferences implies (3↑). But then, (2↑) and (3↑) contradicts (1↑).

Generalized Envelope Theorems

We follow Paul Milgrom, “The envelope Theorems”, 1999. Consider a problem
V(t) = maxx ∈ Kf(x, t),  where
  • K ⊆ RL is a compact set,
  • f:RL × [ − 1, 1] → R is continuous in (x, t) and it has partial derivative ft(x, t) that is continuous in (x, t)
Let x*(t) be the set of maximizers of the above problem.
Theorem. (Generalized Envelope Theorem) For each t,
limsupt’ → t(V(t) − V(t))/(t’ − t)  = maxx ∈ x*(t)ft(x, t),  liminft’ → t(V(t) − V(t))/(t’ − t)  = minx ∈ x*(t)ft(x, t).
In particular, if x*(t) is unique, then V is differentiable at t and (d)/(dt)V(t) = ft(x*(t), t).
Proof. First, we show that
limsupt’ → t(V(t) − V(t))/(t’ − t) ≥ maxx ∈ x*(t)ft(x, t).
Indeed, because the derivative ft(x, t) is continuous, there exists x0 ∈ argmaxx ∈ x*(t)ft(x, t). Moreover, by the definition of the partial derivative,
ft(x0, t) = limt’ → t(f(x0, t) − f(x0, t))/(t’ − t).
The claim follows from the fact that
V(t) ≥ f(x0, t) and V(t) = f(x0, t).
Next, we show that
limsupt’ → t(V(t) − V(t))/(t’ − t) ≥ maxx ∈ x*(t)ft(x, t).
If not, then, there exists a sequence tn → t and xn such that
V(tn) = f(xn, tn)
limsupn → ∞(f(xn, tn) − V(t))/(tn − t) > maxx ∈ x*(t)ft(x, t).
By ocmpactness of K, we can choose a subseqnece xn → x. By the continuuity of function f, it must be that x is an optimal choice for t, i.e., x ∈ x*(t). Because V(t) ≥ f(xn, t), we have
limsupn → ∞(f(xn, tn) − V(t))/(tn − t) ≤ limsupn → ∞(f(xn, tn) − f(xn, t))/(tn − t) = limsupn → ∞ft(xn, tn)
for some tn’ ∈ (t, tn) (the last inequality follows from the Median Value Theorem). The continuity of the partial derivative implies that
limsupn → ∞ft(xn, tn) = ft(x, t).

Lecture 4. Consumer theory IV

Aggregate demand

Suppose that there is I consumers with (Walrasian) demand functions xi(p, w). All individual demands are derived from consumer optimization proiblem (hence, they satisfy all the required properties). Consider an aggregate demand function
x(p, w1, ..., wI) = ixi(p, wi).
All consumers choose their demands given the same economy-wide price vector p, and given their private wealth levels.
The observer (empirical economist) observes only the aggregate demand, prices p, and the aggregate wealth level w = w`1 + ... + wI. Is there a way of describeing the relationship between the demand, prices and wealth using a demand function?
There exists function x*(p, w)such that
x*(p, w1 + ... + wI) = ixi(p, wi)
for all p and w1, ..., wI if and only if there exists functions ai(p) for each agent i and b(p) such that for each agent i,
vi(p, w) = ai(p) + b(p)w
(for each agent i, the preferences admit indirect utility functions of the Gorman form with the same coefficient on w).
Notice that quasi-linear and homothetic preferences have Gorman representation.
One direction is easy to do using Roy’s identity. For the other direction, assume that there is aggregate demand function and I = 2. Then, for all w1, w, it must be that
x1(w1) + x2(w − w1) = x*(w)
(we fix prices p and suppress them from notation). Take derivative with respect to w1 and get for all w, w1,
x1(w1) − x2(w − w1) = 0.
Thus, for all w1, w2, we have
(4) x1(w1) = x2(w2).
Take deroivative with respect to w1 again, and obtain
(5) x1’’(w) = 0.
It follows from (4↑) and (5↑) that it must be that
xi(p, w) = ai(p) + b(p)w.
The rest of the argument. how to get from the above representation of demands to Gorman form for indirect utility, is difficult.

Lecture 5 Theory of Firm

Cost function and returns to scale

Let f(z1, ..., zn) be a production function and let c(w;q) = minf(z) ≥ qwz is the cost function.
We are going to show that if the production function is concave (decreasing returns to scale), then the cost function is convex in q. Indeed, take arbitrary q1, q2 and let z1 ∈ z(w;q1) and z2 ∈ z(w;q2) be two optimal factor demands for prices w and quantities, respectively, q1 and q2. The concavity of the production function implies that for each α
f(αz1 + (1 − α)z2)  ≥ αf(z1) + (1 − α)f(z2) ≥ αq1 + (1 − α)q2.
It follows that
c(w;αq1 + (1 − α)q2)  ≤ w(αz1 + (1 − α)z2) = αwz1 + (1 − α)wz2  = αc(w;q1) + (1 − α)c(w, q2).
Because the above ineqyualities hold for any q1, q2, α, c(w;q)is convex in q.

Lecture 11. Expected utility theory III

Asset management

Consider a decision maker with Bernoulli utility function u:R → R. We assume that u(.) is strictly increasing, strictly concave, and (sufficiently many times) differentiable. The decison maker has wealth w > 0 and she considers investing it into one of two assets:
  • safe asset, with rate of return 1 + r for each dollar invested (r < 0 implies that the “safe” asset is a bad storage technology),
  • risky asset, with retrun rate 1 + z, where z is random, and it is drawn from distribution F.
Let α ∈ [0, w] denote the amount of investment into a risky asset and w − α be the remaining investment into a safe asset. Also, let
(α) = u((w − α)(1 + r) + α(1 + z))dF(z)
be the expected utility from risky investment α. Finally, let
αu = argmaxα(α)
be the optimal investment.
Claim 1. \strikeout off\uuline off\uwave off(α) is strictly concave in α.
Proof: Because u(x) is strictly concave in x,  we have for each α, β ∈ [0, w] st. α ≠ β and each γ ∈ (0, 1),
u(γ[(w − α)(1 + r) + α(1 + z)] + (1 − γ)[(w − β)(1 + r) + β(1 + z)])  <  γu((w − α)(1 + r) + α(1 + z)) + (1 − γ)u((w − β)(1 + r) + β(1 + z)).
(γα + (1 − γ)β)  =  u((w − (γα + (1 − γ)β))(1 + r) + (γα + (1 − γ)β)(1 + z))dF(z)  =  u(γ[(w − α)(1 + r) + α(1 + z)] + (1 − γ)[(w − β(1 + r) + β(1 + z))])dF(z)  >  [γu((w − α)(1 + r) + α(1 + z)) + (1 − γ)u((w − β)(1 + r) + β(1 + z))]dF(z)  =  γ(α) + (1 − γ)(β).
The claim leads to the following corollary.
Corollary\strikeout off\uuline off\uwave off 1. The optimal investment αu is unique.
The next claim implies that if the expected rate of return on the risky asset is strictly higher than r, zdF(z) > r, then the optimal investment into the risky asset is non-zero.
Claim 2. Suppose that zdF(z) > r. Then, αu > 0.
Proof:. We will show that (d)/(dα)(0) > 0. In particular, if the DM invests all her wealth into the safe asset, she can imcrease her expected utility by re-investing some of the wealth into the risky asset. Indeed, notice that
(d)/(dα)(0)  =  (d)/(dα)[[u((1 + r)w + α(z − r))]dF(z)]|α = 0  =  u((1 + r)w)(z − r)dF(z)|α = 0 > 0.
The last claim shows that more risk-averse DM invests less into the risky asset. QED.
Claim 3. Suppose that v is more risk averse than u. Then, αv ≤ αu.
Proof. Recall that if v is more risk averse than u, then there exists striclty increasing and concave function ψ such that v(x) = ψ(u(x)) for each x.
If αv is the unique optimum of v, then the first order conditions for the optimality imply that
0  = (d)/(dα)(αv)  = (d)/(dα)[v((1 + r)w + α(z − r))]|α = αvdF(z)  = (z − r)v((1 + r)w + αv(z − r))dF(z)  = (z − r)ψ(u((1 + r)w + αv(z − r)))u(((1 + r)w + αv(z − r)))dF(z).
Next, we show that for each x,
(6) xψ(u((1 + r)w + αvx)) ≤ xψ(u((1 + r)w)).
Indeed, if x > 0, then, because αv ≥ 0,
(1 + r)w + αvx ≥ (1 + r)w, 
which, because u(.) is increasing, implies that
u((1 + r)w + αvx) ≥ u((1 + r)w), 
which, because ψ is decreasing (remember that ψ is concave) implies that
ψ(u((1 + r)w + αvx)) ≤ ψ(u((1 + r)w)).
Similarly, if x < 0, then
ψ(u((1 + r)w + αvx)) ≥ ψ(u((1 + r)w)), 
which together with x < 0 implies (6↑).
Let ψ* = ψ(u((1 + r)w)). Inequality (6↑) implies that n nb
0 =  (d)/(dα)(αv)  =  (z − r)ψ(u((1 + r)w + αv(z − r)))u(((1 + r)w + αv(z − r)))dF(z)  ≤  ψ’*(z − r)u(w(1 + r) + αv(z − r))dF(z)  =  ψ’*(d)/(dα)(αv).
Becuase ψ* ≥ 0 (this is due to the fact that ψ is an increasing function), the above inequality implies that
(d)/(dα)(αv) ≥ 0 = (d)/(dα)(αu), 
where the last eqyuality comes from the first order cponditions for the utility function u. Becuase is concave, we have
αu ≥ αv.