EC487 Advanced Microeconomics, Part I: Lecture 1 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 1

Leonardo Felli

32L.LG.04

29 September, 2017

Course Outline

Microeconomic Theory

◮ Lecture 1: Production Theory and Profit Maximization. ◮ Lecture 2: Cost Minimization and Aggregation. ◮ Lecture 3: Pure Exchange Economy and Walrasian

Equilibrium.

◮ Lecture 4: Existence of Walrasian Equilibrium and Welfare

Theorems.

◮ Lecture 5: Walrasian Equilibrium with Production and

Externalities.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 2 / 55

Course Outline (cont’d)

Games of Complete Information

◮ Lecture 6: Static games and Nash Equilibrium. ◮ Lecture 7: Nash Theorem and Duopoly. ◮ Lecture 8: Dynamic Games, Subgame Perfection and

Bargaining Games.

◮ Lecture 9: Repeated Games. ◮ Lecture 10: The Folk Theorem.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 3 / 55

Admin

◮ My coordinates: 32L.4.02, x7525, L.Felli@lse.ac.uk ◮ PA: Emma Taverner, 32L.1.28, E.Taverner@lse.ac.uk. ◮ Office Hours: Friday 11:00-13:00 or by appointment (e-mail

L.Felli@lse.ac.uk).

◮ Course Material: available at:

http://econ.lse.ac.uk/staff/lfelli/teaching

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Textbooks

◮ Andreu Mas-Colell, Michael Whinston and Jerry Green,

Microeconomic Theory, Oxford: Oxford University Press, 1995.

◮ Martin J. Osborne and Ariel Rubinstein, A Course in Game

Theory M.I.T. Press, 1994.

◮ Drew Fudenberg and Jean Tirole, Games Theory, Cambridge:

M.I.T. Press, 1991.

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Examining

◮ This part of the course (together with the Microeconomics

part of EC451) will be officially examined in a two hours exam that will take place in the Lent term exam week (2-5 January 2018).

◮ Last year lent term exam is available with the course material

and solutions will be posted on week 10 of the current term.

◮ Exclusively for your own benefit there will be a weekly

assignment that will be discussed in the following class, starting from week 2.

◮ Again, only for your benefit, two of these assignments will be

graded: Assignment 4 due on week 5 and Assignment 10 due

• n week 11.

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Producer Theory

◮ With respect to what you have seen in EC451, we consider

now the other side of the market: the production side.

◮ Individual agent = the firm. ◮ The firm ⇐

⇒ production activity.

◮ The firm produces outputs by using inputs (both measured in

terms of flow amounts per unit time).

◮ Both outputs and inputs are services and commodities.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 7 / 55

Production Plan

◮ In particular let

◮ y o

j = quantity of commodity j produced by the firm as output,

◮ y i

j = quantity of commodity j used as input,

◮ zj = y o

j − y i j net output/input depending on whether the sign

• f zj is positive/negative.

◮ Production plan = vector of net outputs and/or inputs of all

available commodities z =      z1 z2 . . . zL     

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 8 / 55

Net Inputs and Outputs

◮ Without loss of generality we assume that:

◮ the first h commodities are net inputs ◮ while the remaining L − h commodities are net outputs.

◮ Define:

x1 = −z1, . . . , xh = −zh, y1 = zh+1, . . . , yL−h = zL

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 9 / 55

Production Plan and Possibility Set

◮ A production plan is:

z =           −x1 . . . −xh y1 . . . yL−h          

Definition (PPS)

The Production Possibility Set Z ⊂ RL (PPS) is the set of all technologically feasible production plans: the set of all vectors of inputs and outputs that are technologically feasible. The PPS Z provides a complete description of the technology identified with the firm.

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One Input One Output

Example of one input x and one output y production plan z = −x y

• .

✻ ✲

Z z2 = y z1 = x

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Short-run PPS

◮ Sometime is possible to distinguish between:

◮ immediately technologically feasible production plans

Z(xl, . . . , xh);

◮ and eventually technologically feasible production plans Z.

◮ Consider

Z(x1) =   z =   −x1 −x2 y  

• x1 = x1

  

◮ For example if the input x1 is fixed at the level x1 then we can

define a short-run or restricted production possibility set.

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Input Requirement Set and Isoquant

◮ A special feature of a technology is the input requirement set:

V (y) =

• x ∈ Rh

+

• −x

y

• ∈ Z
• the set of all input bundles that produce at least y units of
• utput.

◮ We define also the isoquant to be the set:

Q(y) =

• x ∈ Rh

+

• x ∈ V (y) and x ∈ V (y′), ∀y′ > y
• all input bundles that allow the firm to produce exactly y.

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Two Inputs One Output

✲ ✻

x1 x2 V (y) Q(y)

❖

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Production Function

Consider a technology with only one output

Definition

Production function in the case of only one output: f (x) = sup

y′

−x y′

• ∈ Z
• the maximal output associated with the input bundle x.

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Technologically Efficient

In general, we can define the technologically efficient production plan z as:

Definition

The general production plan z = −x y

• ∈ Z is technologically

efficient, if and only if there does not exist a production plan z′ = −x′ y′

• ∈ Z such that z′ ≥ z (z′

i ≥ zi ∀i) and z′ = z.

If z is efficient it is not possible to produce more output with a given input or the same output with less input (sign convention).

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Example: Cobb-Douglas Technology

◮ We definite the Cobb-Douglas technology as

f (x1, x2) = xα

1 xβ 2 ,

α > 0, β > 0

• r

Z =      −x1 −x2 y  

• y ≤ xα

1 xβ 2

  

◮ with isoquant:

Q(y) =

• (x1, x2) ∈ R2

+

• y = xα

1 xβ 2

• ◮ and input requirement set:

V (y) =

• (x1, x2) ∈ R2

+

• y ≤ xα

1 xβ 2

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 29 September, 2017 17 / 55

Cobb-Douglas Isoquant

✲ ✻

x1 x2 V (y) Q(y)

❖

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Example: Leontief Technology

◮ We definite the Leontief technology as

f (x1, x2) = min{ax1, bx2}, a > 0, b > 0

• r

Z =      −x1 −x2 y  

• y ≤ min{ax1, bx2}

  

◮ with isoquant:

Q(y) =

• (x1, x2) ∈ R2

+

• y = min{ax1, bx2}
• ◮ and input requirement set:

V (y) =

• (x1, x2) ∈ R2

+

• y ≤ min{ax1, bx2}
• ◮ where efficiency imposes

x1 = y a , x2 = y b

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 19 / 55

Leontief Isoquants

✻ ✲

x2 x1 x2 = a

b x1

✻

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Example: Perfect Substitutes

◮ We definite the technology where inputs are perfect

substitutes as f (x1, x2) = ax1 + bx2, a > 0, b > 0

• r

Z =      −x1 −x2 y  

• y ≤ ax1 + bx2

  

◮ with isoquant:

Q(y) =

• (x1, x2) ∈ R2

+

• y = ax1 + bx2
• ◮ and input requirement set:

V (y) =

• (x1, x2) ∈ R2

+

• y ≤ ax1 + bx2
• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 29 September, 2017 21 / 55

Perfect Substitutes Isoquants

✻ ✲ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡

x2 x1 x2 = y

b − a b x1

✻

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 22 / 55

Assumptions on PPS

Common assumptions on the PPS are:

• 1. Z is closed (it contains its boundaries). It is an important

property for the definition of a production function: sup is a max.

• 2. 0 ∈ Z. Shut-down property. It is an uncontroversial property

in the long run, not necessarily in the short run (inputs used with no outputs).

• 3. Free disposal, monotonicity: if z ∈ Z and z′ ≤ z then z′ ∈ Z.

Alternatively: if x ∈ V (y) and x′ ≥ x then x′ ∈ V (y). Given a feasible production plan if either one increases the quantity of inputs or reduces the quantity of output the new production plan is still feasible.

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Assumptions on PPS (cont’d)

• 4. Additivity: if z, z′ ∈ Z then z + z′ ∈ Z (stronger condition).

For f (x) this property implies f (x1 + x2) ≥ f (x1) + f (x2).

• 5. Convexity of V (y): if x, x′ ∈ V (y) then tx + (1 − t)x′ ∈ V (y)

for every 0 ≤ t ≤ 1 which means that V (y) is convex set. (rescaling of production processes). 5′. A similar condition may (or may not) be imposed on the Z: if z, z′ ∈ Z then tz + (1 − t)z′ ∈ Z for every 0 ≤ t ≤ 1, or Z is a convex set. Notice that the latter condition is stronger than the former.

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Some Results

Result

The convexity of Z implies the convexity of V (y). The opposite implication does not hold. Proof: It follows from the convexity of Z, the definition of V (y) and the following counter-example of a one-input x and one output y technology.

✲ ✻

y x Z V (x)

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Some Results (cont’d)

Result

The convexity of V (y) implies that the f (x) is quasi-concave. Proof: It follows from the convexity of V (y) and the definition of a quasi-concave f (x).

Definition (quasi-concavity)

The function f (·) is quasi-concave if and only if the set {x | f (x) ≥ k} is convex for every k ∈ R. Notice that if you choose y = k the set above is V (y).

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Some Results (cont’d)

Result

The convexity of Z implies that f (x) is (weakly) concave. Proof: Consider z = −x f (x)

• ∈ Z,

z′ = −x′ f (x′)

• ∈ Z

Convexity of Z implies that for every 0 ≤ t ≤ 1 t z + (1 − t) z′ = −(t x + (1 − t) x′) tf (x) + (1 − t)f (x′)

• ∈ Z

By definition of f (x) this means: tf (x) + (1 − t)f (x′) ≤ f (tx + (1 − t)x′) for every 0 ≤ t ≤ 1. This is the definition of a concave f (x).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 27 / 55

Returns to Scale

◮ Decreasing Returns to Scale: (DRS) if z ∈ Z then t z ∈ Z for

every 0 ≤ t ≤ 1 (graph.).

◮ Increasing Returns to Scale: (IRS) if z ∈ Z then t z ∈ Z for

every t ≥ 1 (graph.).

◮ Constant Returns to Scale: (CRS) if z ∈ Z then t z ∈ Z for

every t ≥ 0 (graph.).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 28 / 55

More Results

Result

Assumptions 0 ∈ Z and Z convex imply DRS. Proof: It follows from the definition of convexity applied at z′ = 0.

Result

A technology exhibits CRS if and only if the production function f (x) (if available) is homogeneous of degree 1.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 29 / 55

More Results (cont’d)

Proof: Assume CRS. This implies that if z ∈ Z then t z ∈ Z, for all t ≥ 0. By definition of Z, z ∈ Z means y ≤ f (x) further t z ∈ Z means t y ≤ f (t x). By definition of production function choose z, and hence x and y, so that y = f (x).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 30 / 55

More Results (cont’d)

We can re-write the latter condition as: t f (x) ≤ f (t x) We need to prove that the equality holds. Suppose it does not. Then there exists y′ such that t f (x) < y′ < f (t x) Now y′ < f (t x) implies by definition of Z that −t x y′

• ∈ Z

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 31 / 55

More Results (cont’d)

By CRS we get 1 t −t x y′

• ∈ Z
• r

−x

1 t y′

• ∈ Z

which means (1/t) y′ ≤ f (x)

• r

y′ ≤ t f (x) the latter inequality contradicts t f (x) < y′. The opposite implication is an immediate consequence of the definition of homogeneity of degree 1.

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Conditions for DRS and IRS

Weaker conditions apply for DRS and IRS technology.

Result

Consider a technology characterized by a homogenous of degree α < 1 (α > 1) production function. This technology exhibits DRS (IRS). The opposite implication does not hold.

Result

Assume that f (0) = 0 then we can prove:

◮ f (x) concave implies DRS; ◮ f (x) convex implies IRS; ◮ f (x) concave and convex implies CRS.

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Definitions

We can now introduce few definitions:

Definition (MP)

The marginal product of input xi is MP = ∂f (x) ∂xi

Definition (AP)

The average product of input xi is AP = f (x) xi

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Definitions (cont’d)

Definition (MRTS)

The marginal rate of technical substitution between input xi and xj is

• dxi

dxj

• = ∂f (x)/∂xj

∂f (x)/∂xi this is the absolute value of the slope of the isoquant. The set of output bundles that are efficient for a given technology:

Definition (PPF)

The Production Possibility Frontier: PPF(x) =

• y
• ∃z′ ∈ Z s.t. z′ ≥ z =

−x y

• Leonardo Felli (LSE)

EC487 Advanced Microeconomics, Part I 29 September, 2017 35 / 55

Definitions (cont’d)

We can then define

Definition (MRT)

The Marginal Rate of Transformation between output ym and yn is MRT = dym dyn as the slope of the PPF.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 36 / 55

The Competitive Firm

◮ Assume that input and output prices are taken parametrically

(no influence on such prices).

◮ As you have seen in consumer theory what this means is that

whatever each firm decides in term of production does not affect the market.

◮ In other words, either firms are very small with respect to the

market.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 37 / 55

The Competitive Firm (cont’d)

◮ Alternatively we are assuming that firms are not strategic:

they do not realize that their choices trigger reactions in other firms in the market,

◮ or any of their potential choices would be taken into account

by competitors when making their own choices.

◮ We will come back to this in the second part of the course

where we will assume and analyze economic agents that are strategic.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 38 / 55

Profit Maximization

The basic producer problem is than profit maximization: max

{x,y}

p y −

h

• i=1

wi xi s.t. −x y

• ∈ Z

where p and wi are taken as parameters.

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Profit Maximization (cont’d)

Let:

◮ the h-dimensional vector of input prices be w = (w1, . . . , wh); ◮ the L-h-dimensional vector of output prices be

p = (p1, . . . , pL−h). We can re-write the producer’s problem as: max

{z}

ˆ p z s.t. z ∈ Z where ˆ p = (w, p) and z = −x y

• .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 40 / 55

Profit Maximization (cont’d)

In the case of a technology that produces only one output the profit maximization problem may be written as: max

{x,y}

p f (x) − w x The necessary first order conditions of this problem are: p ∇f (x∗) ≤ w

• r

∂f (x∗) ∂xi ≤ wi p , ∀i = 1, . . . , h and ∂f (x∗) ∂xi − wi p

• x∗

i = 0,

∀i = 1, . . . , h.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 41 / 55

Profit Maximization (cont’d)

◮ In the event that the production possibility set is convex (the

production function is concave) the first order conditions are

• f course both necessary and sufficient.

◮ In other case, the following set of sufficient conditions for a

local maximum have to be verified.

◮ The Hessian matrix of the production function has to be

negative definite at the point x∗.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 42 / 55

Profit Maximization (cont’d)

This condition can be checked by the sufficient determinant condition according to which the leading principal minors have to alternate sign starting from the negative one. For the case of two variables the first order conditions are for i = 1, 2: ∂f (x∗) ∂xi ≤ wi p and ∂f (x∗) ∂xi − wi p

• x∗

i = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 43 / 55

Profit Maximization (cont’d)

while the second order conditions are: H =  

∂2f (x∗) ∂x2

1

∂2f (x∗) ∂x1 ∂x2 ∂2f (x∗) ∂x1 ∂x2 ∂2f (x∗) ∂x2

2

  negative definite Which is implied by: ∂2f (x∗) ∂x2

i

< 0 and ∂2f (x∗) ∂x2

1

∂2f (x∗) ∂x2

2

− ∂2f (x∗) ∂x1 ∂x2 2 > 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 44 / 55

Unconditional Factor Demand and Supply Function

The solution to the profit maximization problem if it exists provides the unconditional factor demands: x(p, w) = x∗ By substitution it is possible to obtain the supply function of the producer: y(p, w) = f (x(p, w)). Comparative statics results obtained by differentiating the FOC (they are identities in (p, w)).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 45 / 55

Properties of Factor Demands

• 1. Non-positive own factor demands price effects (SOC)

(generalizes to h inputs): ∂x1 ∂w1 ≤ 0 ∂x2 ∂w2 ≤ 0

• 2. Symmetry (generalizes to h inputs):

∂x1 ∂w2 = ∂x2 ∂w1

• 3. Complementary inputs (generalizes to h inputs):

∂x1 ∂w2 = ∂x2 ∂w1 < 0

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Properties of Factor Demands (cont’d)

• 4. Substitutability of inputs (it does not generalize):

∂x1 ∂w2 = ∂x2 ∂w1 > 0

• 5. Finally positive output price effects (generalizes to h inputs):

∂x1 ∂p > 0 ∂x2 ∂p > 0 (If x1 and x2 are complementary inputs.)

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Properties of Supply Function

Comparative static results obtained differentiating the supply function of the firm: y(p, w) = f (x(p, w))

• 6. Own price effect non-negative:

∂y ∂p ≥ 0

• 7. Symmetry:

−∂xi ∂p = ∂y ∂wi for i = 1, 2.

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Summary of the Properties

Such comparative statics properties can be summarized as:     

∂y ∂p ∂y ∂w1 ∂y ∂w2

− ∂x1

∂p

− ∂x1

∂w1

− ∂x1

∂w2

− ∂x2

∂p

− ∂x2

∂w1

− ∂x2

∂w2

     s.t.   + a b a + c b c +  

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 49 / 55

Other Properties

• 8. Both x(p, w) and y(p, w) are homogeneous of degree 0.

Proof: If you increase both input and output prices by a factor t > 0 you obtain: max

x

(tp) f (x) − (tw) x = max

x

t [p f (x) − w x] which clearly is solved by the same vector x(p, w) that solves: max

x

p f (x) − w x Further, by definition of supply function: y(tp, tw) = f (x(tp, tw)) = f (x(p, w)) = y(p, w)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 50 / 55

Profit Function

Definition (Profit Function)

The following is defined as the profit function π(p, w) = max

x

p f (x) − w x = p f (x(p, w)) − w x(p, w) Properties:

• 1. Price effects:

∂π ∂wi ≤ 0 ∂π ∂p ≥ 0.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 51 / 55

Properties of the Profit Function

• 2. The profit function π(p, w) is homogeneous of degree 1 in

(p, w). Proof: It follows from the homogeneity of degree 0 of x(p, w) and, for any scalar α, π(αp, αw) = α p f (x(αp, αw)) − α w x(αp, αw) = α p f (x(p, w)) − α w x(p, w) = α [p f (x(p, w)) − w x(p, w)] = α π(p, w)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 52 / 55

Properties of the Profit Function (cont’d)

• 3. Hotelling Lemma (which proves property 1):

∂π ∂p = y(p, w) ≥ 0 and ∂π ∂wi = −xi(p, w) ≤ 0 Proof: It follows by Envelope Theorem applied to π(p, w) = max

x

p f (x) − w x = p f (x(p, w)) − w x(p, w)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 53 / 55

Properties of the Profit Function

• 4. The profit function π(p, w) is convex in (p, w).

Proof: Consider the two price vectors (p, w) and (p′, w′) and for every scalar λ ∈ (0, 1) let p′′ = λ p + (1 − λ) p′ and w′′ = λ w + (1 − λ) w′

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 54 / 55

Properties of the Profit Function (cont’d)

◮ Then :

π [p′′, w′′] = p′′ f (x(p′′, w′′)) − w′′ x(p′′, w′′) = λ [p f (x(p′′, w′′)) − w x(p′′, w′′)] + (1 − λ) [p′ f (x(p′′, w′′)) − w′ x(p′′, w′′)] ≤ λ π(p, w) + (1 − λ) π(p′, w′) which proves convexity of π(p, w).

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