EC487 Advanced Microeconomics, Part I: Lecture 2 Leonardo Felli - - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 2

Leonardo Felli

32L.LG.04

6 October, 2017

Properties of the Profit Function

Recall the following property of the profit function π(p, w) = max

x

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

◮ Hotelling’s Lemma:

∂π ∂p = y(p, w) ≥ 0 and ∂π ∂wi = −xi(p, w) ≤ 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 2 / 49

Hessian of the Profit Function

Consider now the Hessian matrix of the profit function π(p, w): H =        

∂2π ∂p2 ∂2π ∂p∂w1

· · ·

∂2π ∂p∂wh ∂2π ∂w1∂p ∂2π ∂w2

1

· · ·

∂2π ∂w1∂wh

. . . . . . ... . . .

∂2π ∂wh∂p ∂2π ∂wh∂w1

· · ·

∂2π ∂w2

h

        By Hotelling’s Lemma the matrix H is: H =       

∂y ∂p ∂y ∂w1

· · ·

∂y ∂wh

− ∂x1

∂p

− ∂x1

∂w1

· · · − ∂x1

∂wh

. . . . . . ... . . . − ∂xh

∂p

− ∂xh

∂w1

· · · − ∂xh

∂wh

      

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 3 / 49

Properties of the Hessian of the Profit Function

Result

Let F : A → R where A ⊂ Rn is a convex open set. Let F(·) be twice differentiable. Then the function F(·) is convex if and only if the Hessian matrix of F(·) is positive semi-definite on A.

◮ Recall that the profit function π(p, w) is convex in (p, w). ◮ This together with Young theorem implies that H is

symmetric and has a non negative main diagonal.

◮ Assume now that π(p, w) is defined on an open and convex

set S of prices (p, w) then convexity of π(p, w) is equivalent to the Hessian Matrix H being positive semi-definite.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 4 / 49

Profit Maximization

Profit maximization can be achieved in two sequential steps:

• 1. Given y, find the choice of inputs that allows the producer to
• btain y at the minimum cost;

this generates conditional factor demands and the cost function;

• 2. Given the cost function, find the profit maximizing output

level.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 5 / 49

Comments

◮ Notice: step 1 is common to firms that behave competitively

in the input market but not necessarily in the output market.

◮ Only in step 2 we impose the competitive assumption on the

• utput market.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 6 / 49

Cost Minimization

We shall start from step 1: min

x

w x s.t. f (x) ≥ y The necessary first order conditions are: y = f (x∗), w ≥ λ∇f (x∗) and [w − λ∇f (x∗)] x∗ = 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 7 / 49

Cost Minimization (cont’d)

• r for every input ℓ = 1, . . . , h:

wℓ ≥ λ∂f (x∗) ∂xℓ with equality if x∗

ℓ > 0.

The first order conditions are also sufficient if f (x) is quasi-concave (the input requirement set is convex). Alternatively, a set of sufficient conditions for a local minimum are that f (x) is quasi-concave in a neighborhood of x∗.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 8 / 49

Cost Minimization (cont’d)

This can be checked (sufficient condition) by means of the bordered hessian matrix and its minors. In the case of only two inputs f (x1, x2) we have: wℓ ≥ λ∂f (x∗) ∂xℓ , ∀ℓ = 1, 2 with equality if x∗

ℓ > 0

and SOC:

• f1(x∗)

f2(x∗) f1(x∗) f11(x∗) f12(x∗) f2(x∗) f21(x∗) f22(x∗)

• > 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 9 / 49

Cost Minimization (cont’d)

In the case the two first order conditions are satisfied with equality (no corner solutions) we can rewrite the necessary conditions as: MRTS =

• dx2

dx1

• = ∂f (x∗)/∂x1

∂f (x∗)/∂x2 = w1 w2 and y = f (x∗) Notice a close formal analogy with consumption theory (expenditure minimization).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 10 / 49

Conditional Factor Demands and Cost Function

This leads to define:

◮ the solution to the cost minimization problem:

x∗ = z(w, y) =    z1(w, y) . . . zh(w, y)    the conditional (to y) factor demands (correspondences).

◮ the minimand function of the cost minimization problem:

c(w, y) = w z(w, y) the cost function.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 11 / 49

Properties of the Cost Function

• 1. c(w, y) is non-decreasing in y.

Proof: Suppose not. Then there exist y′ < y′′ such that (denote x′ and x′′ the corresponding solution to the cost minimization problem) w x′ ≥ w x′′ > 0 If the latter inequality is strict we have an immediate contradiction of x′ solving the cost minimization problem.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 12 / 49

Properties of the Cost Function (cont’d)

If on the other hand w x′ = w x′′ > 0 then by continuity and monotonicity of f (·) there exists α ∈ (0, 1) close enough to 1 such that f (α x′′) > y′ and w x′ > w αx′′ which contradicts x′ solving the cost minimization problem.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 13 / 49

Properties of the Cost Function (cont’d)

• 2. c(w, y) is non-decreasing with respect to wℓ for every

ℓ = 1, . . . , h. Proof: Consider w′ and w′′ such that w′′

ℓ ≥ w′ ℓ but w′′ k = w′ k

for every k = ℓ. Let x′′ and x′ be the solutions to the cost minimization problem with w′′ and w′ respectively. Then by definition of c(w, y) c(w′′, y) = w′′ x′′ ≥ w′ x′′ ≥ w′ x′ = c(w′, y).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 14 / 49

Properties of the Cost Function (cont’d)

• 3. c(w, y) is homogeneous of degree 1 in w.

Proof: The feasible set of the cost minimization problem f (x) ≥ y does not change when w is multiplied by the factor t > 0. Hence ∀t > 0, minimizing (t w) x on this set leads to the same answer as minimizing w x. Let x∗ be the solution, then: c(t w, y) = (t w) x∗ = t c(w, y).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 15 / 49

Properties of the Cost Function (cont’d)

• 4. c(w, y) is a concave function in w.

Proof: Let ˆ w = t w + (1 − t) w′ for t ∈ [0, 1]. Let ˆ x be the solution to the cost minimization problem for ˆ w. Then c( ˆ w, y) = ˆ w ˆ x = t w ˆ x + (1 − t) w′ ˆ x ≥ t c(w, y) + (1 − t) c(w′, y) by definition of c(w, y) and c(w′, y) and f (ˆ x) ≥ y.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 16 / 49

Properties of the Conditional Factor Demands

• 5. Shephard’s Lemma:

If z(w, y) is single valued with respect to w then c(w, y) is differentiable with respect to w and ∂c(w, y) ∂wℓ = zℓ(w, y) Proof: By constrained Envelope Theorem and the definition

• f cost function

c(w, y) = w z(w, y) = w z(w, y) − λ(w, y) [f (z(w, y) − y]

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 17 / 49

Properties of the Conditional Factor Demands (cont’d)

• 6. z(w, y) is homogeneous of degree 0 in w.

Proof: By Shepard’s Lemma and the following result.

Result

If a function G(x) is homogeneous of degree r in x then (∂G/∂xℓ) is homogeneous of degree (r − 1) in x for every ℓ = 1, . . . , L. Proof: Differentiate with respect to xℓ the identity that defines homogeneity of degree r: G(k x) ≡ kr G(x) ∀k > 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 18 / 49

Properties of the Conditional Factor Demands (cont’d)

We obtain: k ∂G(k x) ∂xℓ = kr ∂G(x) ∂xℓ ∀k > 0

• r

∂G(k x) ∂xℓ = k(r−1) ∂G(x) ∂xℓ ∀k > 0 This is the definition of homogeneity of degree (r − 1) of the function ∂G(k x) ∂xℓ .

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 19 / 49

Properties of the Conditional Factor Demands (cont’d)

Notice that

• 6. The Lagrange multiplier of the cost minimization problem is

the marginal cost of output: ∂c(w, y) ∂y = λ(w, y) Proof: By constrained Envelope Theorem applied to the cost function: c(w, y) = w z(w, y) − λ(w, y) [f (z(w, y) − y]

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 20 / 49

Law of Supply

• 7. Let the set W of input prices w be open and convex, if

z(w, y) is differentiable in w then: H =    

∂2c ∂w2

1

· · ·

∂2c ∂w1 ∂wh

. . . ... . . .

∂2c ∂wh ∂w1

· · ·

∂2c ∂w2

h

    =   

∂z1 ∂w1

· · ·

∂z1 ∂wh

. . . ... . . .

∂zh ∂w1

· · ·

∂zh ∂wh

   is a symmetric and negative semi-definite matrix.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 21 / 49

Law of Supply

Proof: Symmetry follows from Shephard’s lemma and Young Theorem: ∂zℓ ∂wi = ∂ ∂wi ∂c(w, y) ∂wℓ

• =

∂ ∂wℓ ∂c(w, y) ∂wi

• = ∂zi

∂wℓ While negative semi-definiteness follows from the concavity of c(w, y), the fact that W is open and convex and the following result.

Result

Let F : A → R where A ⊂ Rn is a convex open set. Let F(·) be twice differentiable. Then the function F(·) is concave if and only if the Hessian matrix of F(·) is negative semi-definite on A.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 22 / 49

Euler Theorem

Result (Euler Theorem)

If a function G(x) is homogeneous of degree r in x then: r G(x) = ∇G(x) x Proof: Differentiating with respect to k the identity: G(k x) ≡ kr G(x) ∀k > 0 we obtain: ∇G(kx) x = rk(r−1) G(x) ∀k > 0 for k = 1 we obtain: ∇G(x) x = r G(x).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 23 / 49

Returns to Scale

We now introduce a set of new properties closely related to the

• nes of the expenditure function in consumer theory.
• 8. If f (x) is homogeneous of degree one (i.e. exhibits constant

returns to scale), then c(w, y) and z(w, y) are homogeneous

• f degree one in y.

Proof: Let k > 0 and consider: c(w, k y) = min

x

w x s.t. f (x) ≥ k y (1) By definition of c(w, y) if x∗ is the solution to the cost minimization problem we have y = f (x∗).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 24 / 49

Returns to Scale (cont’d)

Hence by homogeneity of degree 1 of f (x) we obtain: k y = k f (x∗) = f (k x∗) which implies that k x∗ is feasible in Problem (1). Therefore: k c(w, y) = k [w x∗] = w (k x∗) ≥ c(w, k y).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 25 / 49

Returns to Scale (cont’d)

Let now ˆ x be the solution to Problem (1). Necessarily: f (ˆ x) = k y

• r, by homogeneity of degree 1:

f [(1/k) ˆ x] = (1/k) f (ˆ x) = y which implies that [(1/k) ˆ x] is feasible in the problem that defines c(w, y). Therefore we get: c(w, k y) = w ˆ x = k w [(1/k) ˆ x] ≥ k c(w, y) which concludes the proof.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 26 / 49

Returns to Scale (cont’d)

8′. In other words, a technology that exhibits CRS has a cost function that is linear in y: c(w, y) = c(w)y.

• 9. A technology that exhibits CRS has equal and constant

marginal (∂c(w, y)/∂y) and average cost functions (∂c(w, y)/∂y) = (c(w, y)/y).

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 27 / 49

Returns to Scale (cont’d)

Proof: Homogeneity of degree 1 and Euler theorem imply cy(w)y = c(w, y) or cy(w) = c(w, y) y .

✲ ✻ ✻ ✲ ✱✱✱✱✱✱✱✱✱✱✱✱✱

y cy(w) = c(w,y)

y

y c(w, y) = cy(w) y

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 28 / 49

Returns to Scale (cont’d)

• 10. If f (x) is convex (IRS technology), then the cost function

c(w, y) is concave in y.

• 11. A technology that exhibits IRS has a decreasing marginal cost

function (∂c(w, y)/∂y) and average cost function (∂c(w, y)/∂y) ≤ (c(w, y)/y)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 29 / 49

Returns to Scale (cont’d)

✲ ✻ ✻ ✲

y

c(w,y) y

y c(w, y) cy(w, y)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 30 / 49

Returns to Scale (cont’d)

• 12. If f (x) is concave (DRS technology), then the cost function

c(w, y) is convex in y.

• 13. A technology that exhibits DRS has an increasing marginal

cost function (∂c(w, y)/∂y) and average cost function (∂c(w, y)/∂y) ≥ (c(w, y)/y)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 31 / 49

Returns to Scale (cont’d)

✲ ✻ ✻ ✲

y y cy(w, y)

c(w,y) y

c(w, y)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 32 / 49

Profit Maximization with Single Output

◮ Assume that the output market is competitive. ◮ The profit maximization problem is then:

max

y

p y − c(w, y) The necessary FOC are: p − ∂c(w, y∗) ∂y ≤ 0 with equality if y∗ > 0. The sufficient SOC for a local maximum: ∂2c(w, y∗) ∂y2 > 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 33 / 49

Profit Maximization with Single Output (cont’d)

◮ Clearly the SOC imply at least local DRS in a neighborhood

• f y∗.

◮ Notice that if y∗ > 0 the optimal choice of the firm is:

p = ∂c(w, y∗) ∂y = MC(y∗)

◮ In words, price equal to marginal cost. ◮ This condition defines the solution to the profit maximization

problem: the supply function: y∗(w, p)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 34 / 49

Profit Maximization with Single Output (cont’d)

The two profit maximization problems we have considered so far produce the same outcome for equal (w, p). In fact: max

y

py − c(w, y) where c(w, y) = min

x

w x s.t. f (x) ≥ y yields max

x

p y − w x s.t. f (x) = y which is the very first problem we considered.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 35 / 49

Short Run

Short run cost minimization arises when one or more inputs may be fixed, xh = ¯ xh, while the remaining inputs may be varied at will. The short run variable cost function: cS(w, y, ¯ xh) = wh ¯ xh + min

x1,...,xh−1 h−1

• ℓ=1

wℓ xℓ s.t. f (x1, . . . , xh−1, ¯ xh) ≥ y Alternatively: cS(w, y, ¯ xh) = min

x

w x s.t. f (x) ≥ y xh = ¯ xh

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 36 / 49

Short Run (cont’d)

If z(w, y) denotes the long run conditional factor demands, that solve: c(w, y) = min

x w x

s.t. f (x) ≥ y Let ¯ x = (¯ x1, . . . , ¯ xh) be the input vector that achieves the minimum long run cost of producing ¯ y with prices ¯ w: ¯ x = (¯ x1, . . . , ¯ xh) = z( ¯ w, ¯ y)

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 37 / 49

Short Run vs. Long Run

We can then characterize the relationship between short and long run total costs, or alternatively, short run and long run variable costs (more familiar). Notice that c(w, y) ≡ cS(w, y, zh(w, y)) (2)

• r

c(w, y) y ≡ cS(w, y, zh(w, y)) y

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 38 / 49

Short Run vs. Long Run (cont’d)

moreover by Envelope Theorem ∂c(w, y) ∂y ≡ ∂cS(w, y, zh(w, y)) ∂y (3) We shall now focus on a neighborhood of ( ¯ w, ¯ y) and set ¯ xh = zh( ¯ w, ¯ y). From (2) above by Envelope Theorem we get: ∂c(w, y) ∂y = ∂cS(w, y, ¯ xh) ∂y Recall that Envelope Theorem implies that only the first order effect is zero.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 39 / 49

Short Run vs. Long Run (cont’d)

Since (3) is an identity in (w, y) we can differentiate both sides with respect to y: ∂2cS(w, y, ¯ xh) ∂y2 + ∂2cS(w, y, ¯ xh) ∂y ∂¯ xh ∂zh(w, y) ∂y = ∂2c(w, y) ∂y2 and with respect to wh: ∂2cS(w, y, ¯ xh) ∂y ∂wh + ∂2cS(w, y, ¯ xh) ∂y ∂¯ xh ∂zh(w, y) ∂wh = ∂2c(w, y) ∂y ∂wh

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 40 / 49

Short Run vs. Long Run (cont’d)

Now ∂2cS(w, y, ¯ xh) ∂y ∂wh = 0 since ∂cS(w, y, ¯ xh) ∂wh = ¯ xh is independent of y. Hence by Shephard’s Lemma: ∂2cS(w, y, ¯ xh) ∂y ∂¯ xh = ∂zh(w, y)/∂y ∂zh(w, y)/∂wh

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 41 / 49

Short Run vs. Long Run (cont’d)

which implies by substitution: ∂2cS(w, y, ¯ xh) ∂y2 + (∂zh(w, y)/∂y)2 ∂zh(w, y)/∂wh = ∂2c(w, y) ∂y2 which delivers: ∂2cS(w, y, ¯ xh) ∂y2 ≥ ∂2c(w, y) ∂y2 since (∂zh(w, y)/∂y)2 ∂zh(w, y)/∂wh ≤ 0

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 42 / 49

Short Run vs. Long Run (cont’d)

This allows us to conclude that the loss function: l(w, y) = c(w, y) − cS(w, y, ¯ xh) ≤ 0 reaches a local maximum at ¯ x. Proof: We have shown above that the FOC are satisfied: ∂cS(w, y, ¯ xh) ∂y = ∂c(w, y) ∂y and we just proved that the SOC hold: ∂2c(w, y) ∂y2 ≤ ∂2cS(w, y, ¯ xh) ∂y2

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 43 / 49

Le Chatelier Principle

In other words: c(w, y) ≤ cS(w, y, ¯ xh) A similar approach proves that for every ℓ ∈ {1, . . . , h}: 0 ≥ ∂zS

ℓ

∂wℓ ≥ ∂zℓ ∂wℓ Moving to profit maximization: 0 ≥ ∂xS

ℓ

∂wℓ ≥ ∂xℓ ∂wℓ and 0 ≤ ∂yS ∂p ≤ ∂y ∂p All these results are known as Le Chatelier Principle.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 44 / 49

Aggregation

◮ The question we address is when can we speak of an

aggregate supply function in similar terms as the aggregate demand you have seen before?

◮ The question above is closely related to the question: under

which conditions can we speak of a representative producer?

◮ Recall that the key problem when constructing an aggregate

demand was the presence of income effects in consumer theory.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 45 / 49

Aggregate Supply

◮ The absence of a budget constraint implies that individual

firms’ supply are not subject to income effects.

◮ Hence aggregation of production theory is simpler and

requires less restrictive conditions.

◮ Consider J production technologies: (Z 1, . . . , Z J) ◮ Let zj(p, w) =

−xj(p, w) yj(p, w)

• be firm j’s production plan.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 46 / 49

Law of Supply

◮ We have seen that the matrix of cross and own price effects of

production plan zj(p.w): Dzj(p, w) is symmetric and positive semi-definite.

◮ This is also known as law of supply.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 47 / 49

Aggregate Law of Supply

◮ Define now the following aggregate optimal production plan:

z(p, w) =

J

• j=1

zj(p, w) = −

j xj(p, w)

• j yj(p, w)
• ◮ Does an aggregate law of supply hold?

◮ Since both symmetry and positive semi-definiteness are

preserved under sum then Dz(p, w) is also symmetric and positive semi-definite.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 48 / 49

Representative Producer

Result

In a purely competitive environment the maximum profit obtained by every firm maximizing profits separately is the same as the profit obtained if all J firms where they coordinate their choices in a joint profit maximization: π(p, w) =

J

• j=1

πj(p, w) In other words, there exists a representative producer.

Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 49 / 49