microeconomics i - staff.alzahra.ac.ir

Microeconomics I

MWG, Chapter 5

Production

Alzahra University

Department of Economics

Hamid Kordbacheh

[email protected]

Edited sept 2018

Microeconomics I- Alzahra University

Hamid Kordbacheh 1

mailto:[email protected]

Overview

• Section 5.A

o Introduction

• Section 5.B

o Production set

oProperties of production set

• Section 5.C

o Profit maximization problem

o Cost maximization problem

• Section 5.D

o The geometry of cost and production relationship


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• Section 5.E

o Aggregation

• Section 5.B

o A brief sketch of welfare economics

oEfficient production

• Section 5.G

Remarks on the objectives of the Firm


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Section 5.A: Introduction

• Production: the supply side of the economy

• Firms

o Definition

Firms may be corporations or other legally recognized business.

The productive possibilities of individuals or households.

Potential productive units that are never actually organized

Thus, the theory will be able to accommodate both active

production processes and potential but inactive ones

• Boundaries of firms

o Horizontal

o Verticals

o corporate


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• Many aspect enter a full description of a firm

o Who owns it

o Who manage it

o How is it managed?

o How it is organized

o What can it do

• Of all these questions, we focus on the last one (why)

• Standard model: firms choose production plans to maximize profits


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• Assumptions

o Firms are price takes (both in input and output markets)

o Technology is endogenously given

o Firms maximize profits, as long as

The firms competitive

There is no uncertainty about profits

Managers are perfectly controlled by owners


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5.B Production Sets

• The importance of plural

• As in the previous chapters, we consider an economy with L commodities.

• A production vector, also known as an

o input- output

o or netput , vector

o or as a production plan

:is a vector 𝑦 = 𝑦1, … , 𝑦𝐿 ∈ 𝑅𝐿 that describes the (net) outputs of the L

commodities from a production process.


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5.B Production Sets

• The convention

o positive numbers denote outputs and negative numbers

denote inputs.

o Some elements may be zero, means that the process has no

net output of the commodity.

• Example5.B.1: Suppose that L=5. Then 𝑦 = −5,2,−6,3,0

means that

• To analyze the behavior of the firm, we need to start by

identifying those production vectors that are technologically

possible.


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5.B Production Sets

• The set of all production vectors that constitute plans for the firm is known

as the production set and is denote by 𝑌 ⊂ ℝ 𝐿

• Any 𝑦 ∈ 𝑌 is possible

• Any 𝑦 ∉ 𝑌 is not

• The set of the feasible production plan is limited first and foremost by

technological constraints.

• However, is any particular model, the production set may also contributed

by

o Legal restrictions

o or prior contractual commitments

• It is sometimes convenient to describe the production set Y using a function

F(.), called the transformation function.


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5.B Production Sets

• The transformation function F(.) has the property

o that 𝑌 = 𝑦 ∈ ℝ𝐿: 𝐹 𝑦 ≤ 0 and

o that 𝐹 𝑦 = 0 if and only if y is an element of the boundary of Y.

• The set of boundary points of Y, 𝑦 ∈ ℝ𝐿: 𝐹 𝑦 = 0 is known at the

transformation frontier.

• Figure 5.B.1 presents a two- good example.

•


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• If F(.) is differentiable, and if the production vector 𝑦 satisfies 𝐹(𝑦 ) = 0,

then for any commodities l and k, the ratio

𝑀𝑅𝑇𝑙𝑘 𝑦 =𝜕𝐹 𝑦 /𝜕𝑦𝑙

𝜕𝐹 𝑦 /𝜕𝑦𝑘

• Indeed, from 𝐹(𝑦 ) = 0 , we get ( by taking the total differential and setting

𝑑𝑦𝑗= 0 for 𝑗 ≠ 𝑘, 𝑙)

𝜕𝐹(𝑦 )

𝜕𝑦𝑘

𝑑𝑦𝑘 +𝜕𝐹(𝑦 )

𝜕𝑦𝑙

𝑑𝑦𝑙 = 0,

• Therefore

𝜕𝑦2

𝜕𝑦1 𝐹 𝑦 =0,𝑦=𝑦

= −𝑀𝑅𝑇12(𝑦 )


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Technologies with Distinct Inputs and Outputs

• In many actual production processes, the set of goods that can be outputs is

distinct from the set that can be inputs.

• In this case, it is sometimes convenient notationally distinguish the firm’s

inputs and outputs.

• We could for example, let

o 𝑞 = 𝑞1, … , 𝑞𝑀 : the production levels of the firm’s M outputs

o 𝑧 = 𝑧1, … , 𝑧𝐿−𝑀 ≥ 0 : the amounts of the firm’s L - M inputs.

• One of the most frequently encountered production models is that in which

there is a single output.

• An example

𝑌 = −𝑧1, … , −𝑧𝐿−1, 𝑞 : 𝑞 − 𝑓 −𝑧1, … , −𝑧𝐿−1 ≤ 0 𝑎𝑛𝑑(−𝑧1, … , −𝑧𝐿−1) ≥ 0


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• Holding the level of output fixed, we can define the marginal rate of the

technical substitution ( MRTS ) of input l for input k at 𝑧 as

𝑀𝑅𝑇𝑆𝑙𝑘 𝑧 =𝜕𝑓(𝑧 )/𝜕𝑧𝑙

𝜕𝑓(𝑧 )/𝜕𝑧𝑘

• The number 𝑀𝑅𝑇𝑆𝑙𝑘(𝑧 ) measures the additional amount of input k that

must be used to keep output at level 𝑞 = 𝑓(𝑧 ) when the amount of input l is

decreased marginally.

• Indeed, taking the total differential of q=f(z) at z = z and setting 𝑑𝑧𝑗 = 0

for 𝑗 ≠ 𝑘, 𝑙 we obtain

𝑑𝑞 =𝜕𝑓(𝑧 )

𝜕𝑧𝑘𝑑𝑧𝑘 +

𝜕𝑓(𝑧 )

𝜕𝑧𝑙𝑑𝑧𝑙


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• Example 5.B.2: The Cobb-Douglas Production Function

• The Cobb-Douglas production function with two inputs in given by

𝑞 = 𝑓 𝑧1, 𝑧2 = 𝑧1𝛼𝑧2

𝛽 ,

Where 𝛼 ≥ 0 𝑎𝑛𝑑 𝛽 ≥ 0

• The marginal rate of technical substitution between the two inputs at

z=(𝑧1, 𝑧2) is

𝑀𝑅𝑇𝑆12 𝑧 =

𝜕𝑓(𝑧1, 𝑧2)𝜕𝑧1

𝜕𝑓(𝑧1, 𝑧2)𝜕𝑧2

=𝛼𝑧1

𝛼−1𝑧2𝛽

𝛽𝑧1𝛼𝑧2

𝛽−1=

𝛼𝑧2

𝛽𝑧1


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Properties of production sets

(i) Y is nonempty.

• This assumption simply says that the firm has something it can plan to do.

• Otherwise, there is no necessity to study the behavior of the firm in

question.

(ii) Y is closed.

• the set Y includes its boundary. Thus, the limit of a sequence of

technologically feasible input- output vectors is also feasible;

• This condition should be thought of as primarily technical.


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(iii) No free launch.

• Suppose that 𝑦 ∈ 𝑌 and 𝑦 ≥ 0 , so that the vector y does not use any inputs.

• The no-free-launch property is satisfied if this production vector cannot

produce output either.

• Geometrically, 𝑌 ∩ ℝ+𝐿 ⊂ 0 . For L=2

• Figure 5.B.2(a) shows a set that violates the no-free-launch property

• Figure 5.B.2(b) satisfies it.

• Figure 5.B.2(b) depicts two example of sunk costs


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(iv) Possibility of inaction

• This property says that 0 ∈ 𝑌: Complete shutdown is possible.

• Both sets in Figure 5.B.2, for example, satisfy this property.

• If we are considering a firm that could access a set of the technological

possibilities but that has not yet been organized, then inaction is possible.

• But in the case of sunk cost inaction is not possible

(v) Free disposal

• That is, if 𝑦 ∈ 𝑌and 𝑦′ ≤ 𝑦 so that 𝑦′

o Produces at most the same amount of outputs

o Using at least the same amount of outputs

then 𝑦′ ∈ 𝑌.

More succinctly, 𝑌 − ℝ+𝐿 ⊂ 𝑌 (see Figure 5.B.4).


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(vi) Irreversibility

• Suppose that 𝑦 ∈ 𝑌and 𝑦 ≠ 0. Then irreversibility says that −𝑦 ∉ 𝑌 .

• There is no way back for the firm

• Exercise 5.B.1: violation of s irreversibility


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Return to Scale

Global and local scope

• Global Returns to Scale:, NIRS, NDRS & CRS

o Super-additive, concavity

o Scale Elasticity of Output

• Local Returns to Scale

o AC vs. M

• Return to scale refers to Production set not to scarcity


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(vii) Nonincreasing returns to scale

• The production technology Y exhibits nonincreasing returns to scale if for any 𝑦 ∈ 𝑌 , we have 𝛼𝑦 ∈ 𝑌 for all scalars 𝛼 ∈ 0,1 .

• NIRS rule out increasing returns to scale

• In words, any feasible input- output vector can be scaled down

• What does this mean?

• Figure 5.B.5

• Note that nonincreasing returns to scale imply that inaction is possible [ property (iv)].

• The interesting relationship between NIRS and the presence of fixed and sunk costs (Figure 5.B.6)


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(viii) Nondecreasing returns to scale

• A production process exhibits nondecreasing returns to scale if for any

𝑦 ∈ 𝑌 , we have 𝛼𝑦 ∈ 𝑌 for any scale 𝛼 ≥ 1.

• NDRS rule out decreasing returns to scale

• In words, any feasible input-output vector can be scaled up.

• Figure 5.B.6(a)

• It does not matter for the existence of nondecreasing returns if this fixed

cost is sunk [ as in Figure 5.B.6(b)]

• Or not [as in Figure 5.B.6(a), where inaction is possible].

• Does NDRS means an increasing MP


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(ix) Constant returns to scale

• This property is the conjunction of properties (vii) and (viii).

• The production set Y exhibits constant returns to scale if 𝑦 ∈ 𝑌 implies

𝛼𝑦 ∈ 𝑌 for any scalar 𝛼 ≥ 0.

• Geometrically, Y is a cone (see Figure 5.B.7).


• Representation of the production set in the case of two input and one output

under CRS

• CRS is the most fundamental among models with convex technology

• Regulatory implication of the presence of the type of returns to scale


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Evaluating of returns to scale

• The degree of homogeneity

𝑓 t𝑧1, t𝑧2 = tn𝑓 𝑧1, 𝑧2 .

• Scale Elasticity

𝐸(𝑥) =𝜕𝑓(𝑡𝑘, 𝑡𝑙)

𝜕𝑡.

𝑡

𝑓(𝑘, 𝑙)

CRS : 𝐸(. ) = 1

CRS : 𝐸(. ) < 1

CRS : 𝐸(. ) > 1

• How can we convert a general form of technology to a standard form?


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• Exercise 5.B.2:

• Example 5.B.3: A cobb-Douglas production function

𝑓 2𝑧1, 2𝑧2 = 2𝛼+𝛽𝑧1𝛼𝑧2

𝛽= 2𝛼+𝛽𝑓 𝑧1, 𝑧2 .

Thus,

when 𝛼 + 𝛽 = 1, we have constant returns to scale;

when 𝛼 + 𝛽 < 1, we have decreasing returns to scale; and

When 𝛼 + 𝛽 > 1, we have increasing returns to scale

• A transcendental production function

𝑓 . =∝0 𝑥𝑖∝𝑖 𝑒

12 𝐵𝑖𝑗

𝑛𝑗=1 𝑥𝑖𝑥𝑗

𝑛

𝑖=1

𝑛

𝑖=1


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(x) Additivity

• If 𝑦 ∈ 𝑌 is being produced by a firm and another firm enters and produces

𝑦′ ∈ 𝑌, then the net result is the vector 𝑦 + 𝑦′.

• Hence, the aggregate production set (the production set describing feasible

production plans for the economy as a whole) must satisfy additivity

whenever unrestricted entry, or (as it is called in the literature) free entry, is

possible.

• Additivity is also related to the idea of entry.


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(xi) Convexity

• That is, if y, 𝑦′ ∈ 𝑌 and 𝛼 ∈ 0,1 , then 𝛼𝑦 + (1 − 𝛼)𝑦′ ∈ 𝑌.

• Figures 5.B.5(a) & 5.B.5(b)

• The convexity assumption can be interpreted as incorporating two ideas

about production possibilities.

• The first in nonincreasing returns. In particular, if inaction is possible (i

.e.,if 0 ∈ 𝑌 ), then convexity implies that Y has nonincreasing returns to

scale.

• To see this, note that for any 𝛼 ∈ 0,1 , we can write

• 𝛼𝑦 = 𝛼𝑦 + 1 − 𝛼 0.

• Hence, if 𝑦 ∈ 𝑌 and 0 ∈ 𝑌, convexity implies that 𝛼𝑦 ∈ 𝑌 .


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• Second, convexity captures the idea that “unbalanced” input combinations

are not more productive than balanced ones.

• In particular, if production plans y and 𝑦′produce exactly the same amount

of output but use different input combinations, then a production vector that

uses a level of each input that is the average of the level used in these two

plans can do at least as well as either y or 𝑦′.

• Exercise 5.B.3 illustrates these two ideas for the case of a single-output

technology.

• Exercise 5.B.3: Show that for a single-output technology, Y is convex if

and only if the production function f (z) is concave.


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• (xii) Y is a convex cone.

• This is the conjunction of the convexity (xi) and constant returns to scale (ix) properties.

• Formally, Y is a convex cone if for any production vector y,𝑦′ ∈ 𝑌 and constants 𝛼 ≥ 0 and 𝛽 ≥ 0 , we have 𝛼𝑦 + 𝛽𝑦′ ∈ 𝑌.

• The production set depicted in Figure 5.B.7 is a convex cone.

• An important fact is given in proposition 5.B.1.

• Proposition 5.B.1: The production set Y is additive and satisfies the nonincreasing returns condition if and only if it is a convex cone.

• Proof – Part A: First, we want to show that if Y is a convex cone, then it is additive and satisfies the nonincreasing returns condition.


Hamid Kordbacheh 29

• The definition of a convex cone directly implies the nonincreasing returns

and additivity properties.

• Proof – Part B: second, we want to show that if nonincreasing returns and

additivity hold, then for any y, 𝑦′ ∈ 𝑌 and any 𝛼 ≥ 0 , and 𝛽 ≥ 0 , we

have 𝛼𝑦 + 𝛽𝑦′ ∈ 𝑌 .

To this effect, let k be any integer such that k > max 𝛼, 𝛽 .

By additivity,𝑘𝑦 ∈ 𝑌and 𝑘𝑦′ ∈ 𝑌 .

Since 𝛼/𝑘 ≤ 1 and 𝛼𝑦 = 𝛼, 𝑘 𝑘𝑦 , the nonincreasing returns

condition implies that 𝛼𝑦 ∈ 𝑌.

Similarly, 𝛽𝑦′ ∈ 𝑌 .

Finally, again by additivity, 𝛼𝑦 + 𝛽𝑦′ ∈ 𝑌 .


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• Proposition 5.B.1 provides a justification for the convexity assumption in

production. Informally , we could say that

• If feasible input-output combinations can always be scaled down, and

• If the simultaneous operation of several technologies without mutual

interface is always possible, then, in particular, convexity obtains.

• Proposition 5.B.2: For any convex production set 𝑌 ⊂ ℝ𝐿with 0 ∈ 𝑌,there

is a constant returns, convex production set 𝑦′ ⊂ ℝ𝐿+1 such that 𝑌= 𝑦 ∈ ℝ𝐿: (𝑦, −1) ∈ 𝑌′ .

• Proof: simply let

𝑌′ = 𝑦′ ∈ ℝ𝐿+1: 𝑦′ = 𝛼 𝑦, −1 for some 𝑦 ∈ 𝑌 and 𝛼 ≥ 0 .


Hamid Kordbacheh 31

5.C: Profit maximization and cost minimization

• The study of the market behavior of the firm.

• Assumptions:

o there is a vector of prices quoted for the L goods, denoted by p

o The prices are independent of the production plans of the firm

o The firm’s objective is to maximize its profit.

o The firm’s production set Y satisfies the properties of

Nonemptiness

Closedness

Free disposal


Hamid Kordbacheh 32

• Given a price vector 𝑝 ≫ 0 and a production vector 𝑦 ∈ ℝ𝐿, the profit

generated by implementing y is

𝑝. 𝑦 = 𝑝1𝑦1

𝐿

𝑙=1

• Given the technological constraints represented by its production set Y, the

firm’s profit maximization problem (PMP) is then

max𝑝. 𝑦

𝑠. 𝑡. 𝑦 ∈ 𝑌


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• Using a transformation function to describe Y, F(.), we can equivalently

state the PMP as

max𝑦

𝑝𝑦

𝑠. 𝑡 𝐹(𝑦) ≤ 0

Given a production set Y, the firm’s profit function 𝜋(𝑝) associates to every p

is:

𝜋 𝑝 = 𝑚𝑎𝑥 𝑝. 𝑦: 𝑦 ∈ 𝑌

• Similarly, we define the firm’s supply correspondence at p, denoted y(p),

as the set of profit-maximizing vectors

𝑦 𝑝 = 𝑦 ∈ 𝑌: 𝑝. 𝑦 = 𝜋(𝑝) .

• We use the term supply correspondence to keep the parallel with the

demand terminology of the consumption side.

• Recall however that y(p) is more properly thought of as the firm’s net

supply to the market. Microeconomics I- Alzahra University

Hamid Kordbacheh 34

• In particular, the negative entries of a supply vector should be interpreted as

demand for inputs.

• Figure 5.C.1

• The optimizing vector y(p) lies at the point in Y associated with the highest

level of profit.

• In the Figure, y(p) therefore lies on the iso-profit line that intersects the

production set farthest to the northeast and is, therefore, tangent to the

boundary of Y at y(p)

• An iso-profit line is a line in ℝ2 along which all points generate equal

profits, i.e.

𝑝. 𝑦 = 𝜋

• where 𝜋 denotes a given level of profits.


Hamid Kordbacheh 35

• In general, y(p) may be a set rather than a single vector.

• Also, it is possible than no profit-maximizing production plan exists.

o For example, the price system may be such that there is no bound on

how high profits may be. In this case, we say that 𝜋 𝑝 = +∞.

o Rigorously, to allow for the possibility that 𝜋 𝑝 = +∞ (as well as

for other cases where no profit-maximizing production plan exists),

the profit function should be defined by

𝜋 𝑝 = 𝑠𝑢𝑝 𝑝. 𝑦: 𝑦 ∈ 𝑌

• Exercise 5.C.1: Prove that, in general, if the production set Y exhibits

nondecreasing returns to scale, then either 𝜋 𝑝 ≤ 0or 𝜋 𝑝 = +∞.


Hamid Kordbacheh 36

max𝑦

𝑝𝑦

𝑠. 𝑡 𝐹(𝑦) ≤ 0

• If the transformation function F(.) is differentiable, then first-order conditions can be used to characterize the solution to the PMP.

• If 𝑦⋆ ∈ 𝑦(𝑝) , then, for some 𝜆 ≥ 0, 𝑦⋆ must satisfy the first-order conditions

𝑝𝑙 = 𝜆𝜕𝐹(𝑦⋆)

𝜕𝑦𝑙 for 𝑙 = 1, … , 𝐿

• or, equivalently, in matrix nonation,

𝑝 = 𝜆𝛻𝐹 𝑦⋆ (5.C.1)

• In words, the price vector p and the gradient 𝛻𝐹 𝑦⋆ are proportional (Figure 5.C.1 depicts this fact).


Hamid Kordbacheh 37

• Condition (5.C.1) also yields the following ratio equality:

𝑝𝑙

𝑝𝑘 = 𝑀𝑅𝑇𝑙𝑘 𝑦⋆ 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑙, 𝑘

• For L = 2, this says that the slope of the transformation frontier at the

profit-maximizing production plan must be equal to the negative of the

price ratio, as shown in Figure 5.C.1.

• Were this not so, a small change in the firm’s production plan could be

found that increase the firm’s profits.


Hamid Kordbacheh 38

• When Y relates to a single-output technology with differentiable production function f (z), we can view the firm’s decision as simply a choice over its input levels z.

• In this special case, we shall let the scalar 𝑝 > 0 denote the price of the firm’s output and the vector 𝑤 ≫ 0 denote is input prices.

• The input vector 𝑧∗maximizes profit given(p,w) if it solves

max 𝑝𝑓 𝑧 − 𝑤. 𝑧. 𝑧 ≥ 0

• If 𝑧∗ is optimal, then the following first-order conditions must be satisfied for 𝑙 = 1, … , 𝐿 − 1:

p𝜕𝑓(𝑧∗)

𝜕𝑧𝑙≤ 𝑤𝑙 , 𝑤𝑖𝑡ℎ 𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑖𝑓𝑧𝑙

∗ > 0,

• or, in matrix notation,

𝑝𝛻𝑓 𝑧∗ ≤ 𝑤 𝑎𝑛𝑑 𝑝𝛻𝑓 𝑧∗ − 𝑤 . 𝑧∗ = 0.


Hamid Kordbacheh 39

• Thus, the marginal product of every input l actually used (i-e., with 𝑧𝑙∗ > 0)

must equal its price in terms of output, 𝑤𝑙/𝑝.

𝜕𝑓(𝑧∗)

𝜕𝑧𝑙=

𝑤𝑙

𝑝, 𝑖𝑓 𝑧𝑙

∗ > 0.

• Marginal product of input l

• Note also that for any two inputs l and k with (𝑧𝑙∗, 𝑧𝑘

∗) ≫ 0, condition

(5.C.2) implies that

𝑀𝑅𝑇𝑆𝑙𝑘 𝑧∗ =𝜕𝑓(𝑧∗)/𝜕𝑧𝑙

𝜕𝑓(𝑧∗)/𝜕𝑧𝑘=

𝑤𝑙

𝑤𝑘,

• That is, the marginal rate of technical substitution between the two inputs is

equal to their price ratio, the economic rate of substitution between them.

• The importance of convexity condition!


Hamid Kordbacheh 40

Proposition 5.C.1

• Proposition 5.C.1: suppose that 𝜋 . is a profit function of the production

set Y and that y(.) is the associated supply correspondence. Assume also

that Y is closed and satisfies the free disposal property. Then

i. y(.) Is homogenous of degree zeroone.

ii. 𝜋 . is homogeneous of degree

iii. 𝜋 . is convex.

iv. If Y is convex, then 𝑌 = 𝑦 ∈ ℝ𝐿: 𝑝. 𝑦 ≤ 𝜋 𝑝 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝 ≫ 0 .

v. If Y is convex, then y(p) is a convex set for all p. Moreover, if Y is

strictly convex, then y(p) is single-valued (if nonempty).

vi. ( Hotelling’s lemma) If 𝑦(𝑝 )consists of a single point, then 𝜋 . is

differentiable at 𝑝 and 𝛻𝜋 𝑝 = 𝑦(𝑝 ) .

vii. If y(.) is a function differentiable at 𝑝 , then 𝐷𝑦 𝑝 = 𝐷2 𝜋 𝑝 is a

symmetric and positive semidefinite matrix with 𝐷𝑦 𝑝 𝑝 = 0.


Hamid Kordbacheh 41

Comments on proposition 5.C.1

property (ii):

• Why should profit function be convex?

• properties (ii),(iii),(vi), and (vii) are the important ones.

• Exercise 5.C.2: Prove that 𝜋(p) is a convex function.

• Property (iii)

• if Y is closed, convex, and satisfies free disposal, then 𝜋(𝑝) provides an

alternative (“dual”) description of the technology.

• This property is called recoverability results: if you know 𝛑(𝐩) , you an

recover the production set Y

• It seems that π(p) covers less information about the firm than its technology

set Y


Hamid Kordbacheh 42

Recoverability

• If Y is convex, Y and π (p) contain the exact same information.

• i.e. π (p) contains a complete description of the productive possibilities open to

the firm. To illustrate this:

o First, we know π(p) is generated from Y

o Choose a positive price vector p >>0, and find the set 𝑦|𝑝. 𝑦 ≤ 𝜋(𝑝) .

o Since y(p) is the optimal point in Y, then 𝑌 ⊂ {𝑦|𝑝. 𝑦 ≤ 𝜋 𝑝 }, and that any

point not in {𝑦|𝑝. 𝑦 ≤ 𝜋 𝑝 }, cannot be in Y.

o Thus by reiterating this process we can recover all points of the entire

transformation frontier, effectively recovering the set Y as 𝑌 = {𝑦|𝑝. 𝑦≤ 𝜋 𝑝 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑝 ≫ 0}, whenever Y is convex.

o The importance of this result is that π (p) is analytically much easier to work

with than


Hamid Kordbacheh 43

Property (V)

• If the production set Y strictly convex, the tangency condition or the supply

correspondence y(p) is single-valued (if nonempty)

• If the production set Y is weakly convex, then the supply correspondence

y(p) is a convex set for all p.

• What is the intuition behind this?


Hamid Kordbacheh 44

Property (vi)

• Relates supply behavior to the derivatives of the profit function.

• As in proposition 3.G.1, the fact that 𝛻𝜋 𝑝 = 𝑦(𝑝 )can also be established

by the related arguments of the of the envelope theorem and of first-order

conditions.

• Note that the law of supply holds for any price change.

• By the sign convention, this implies that

o If the price of an output increases (all other prices remaining the

same),then the supply of the output increases; and

o If the price of an input increases, then the demand for the input decreases.


Hamid Kordbacheh 45

Property (vii)

If y(p) is single-valued and differentiable at 𝑝

• 𝐷2𝑦 𝑝 is s a symmetric, positive semi−definite matrix

• Own-substitution effects are nonnegative [𝜕𝑦𝑖(𝑝)/𝜕𝑝𝑖 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 i],

• The substitution effects are symmetric [𝜕𝑦𝑖(𝑝)/𝜕𝑝𝑗 = 𝜕𝑦𝑗(𝑝)/𝜕𝑝i for all i

, j] (Young’s theorem )

• These seams from the convexity of 𝜋 𝑝


Hamid Kordbacheh 46

• In nondifferentiable terms, the law of supply can be expressed as

𝑝 − 𝑝 . 𝑦 − 𝑦 ≥ 0 (5. C. 3)

• For all p, 𝑝 amd 𝑝 , 𝑦 ∈ 𝑦(𝑝)and 𝑦 ∈ 𝑦(𝑝 ).

𝑝 − 𝑝 . 𝑦 − 𝑦 = 𝑝. 𝑦 − 𝑝. 𝑦 + (𝑝 . 𝑦 − 𝑝 . 𝑦) ≥ 0

• Where the inequality follows from the fact that

o 𝑦 ∈ 𝑦(𝑝), i.e., y is profit maximizing given prices p , which, in turn,

implies that 𝑝. 𝑦 ≥ 𝑝. 𝑦 and

o 𝑦 ∈ 𝑦(𝑝 ), i.e., 𝑦 is profit maximizing given prices 𝑝 , which, in turn,

implies that 𝑝 . 𝑦 ≥ 𝑝 . 𝑦.


Hamid Kordbacheh 47

Cost Minimization

• An important implication of the firm choosing a profit-maximizing

production plan is that there is no way to produce the same amount of outputs

at a lower total input cost.

o Thus, cost minimization is a necessary condition for profit maximization.

o This observation motivates us to an independent study of the firm’s cost

minimization problem.

• The cost minimization problem is also of interest because:

o It leads us to a number of result and constructions that are empirically and

technically very useful

o As well shall see in ch.12, when a firm is a not a price taker in its output

market,

We can no longer use the profit function for analysis,

But, as long as the firm is a price taker in its input market, the result

following from the cost minimization problem continue to be valid


Hamid Kordbacheh 48

• To be concrete, we focus our analysis on the signal-output case.

• As usual, we let

• z be a nonnegative vector of inputs,

• f (z) the production function,

• q the amounts of output, and

• 𝑤 ≫ 0 the vector of input prices.

• The cost minimization problem (CMP) (we assume free disposal of output):

min 𝑤. 𝑧

𝑧 ≥ 0 (CMP)

𝑠. 𝑡. 𝑓 𝑧 ≥ 𝑞. • The optimized value of the CMP is given by the cost function c( w ,

q ).


Hamid Kordbacheh 49

• The corresponding optimizing set of input (or factor ) choices, denoted by z( w , q ), is known as the conditional factor demand correspondence

• Discuss Figure 5.C.2(a) for a case with two inputs.

• Condition (5.C.4), like condition (5.C.2) of the PMP, implies that for any two inputs l and k with (𝑧𝑙, 𝑧𝑘) ≫ 0, we have

𝑀𝑅𝑇𝑆𝑙𝑘 =𝑤𝑙

𝑤𝑘

• This correspondence is to be expected because, as we have noted, profit maximization implies that input choices are cost minimizing for the chosen output level q.

• For L = 2, Condition (5.C.4) entails that the slope at 𝑧∗ of the isoquant associated with production level q is exactly equal to the negative of the ratio of the input prices −𝑤1/𝑤2.

• Figure 5.C.2(a) depicts this fact as well.


Hamid Kordbacheh 50

• Proposition 5.C.2: suppose that the c( w , q ) is the cost function of a single-output technology Y with production function f (.) and that z ( w , q ) is the associated conditional factor demand correspondence. Assume also that Y is closed and satisfies the free disposal property. Then:

i. z(.) is homogeneous of degree zero in w.

ii.c(.) is homogenous of degree one in w and nondecreasing in q.

iii.c(.) is a concave function of w.

iv.If the sets 𝑧 ≥ 0: 𝑓(𝑧) ≥ 𝑞 are convex for every q, then Y = −𝑧, 𝑞 : 𝑤. 𝑧 ≥ 𝑐 𝑤, 𝑞 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑤 ≫ 0


Hamid Kordbacheh 51

v. If the set 𝑧 ≥ 0: 𝑓(𝑧) ≥ 𝑞 is weakly convex, then z( w , q ) is a

convex set. Moreover, if 𝑧 ≥ 0: 𝑓(𝑧) ≥ 𝑞 is a strictly convex set, then

z( w , q ) is single-valued.

vi. (Shepard’s lemma) if 𝑧(𝑤 , 𝑞)consists of a single point, then c(.) is

differentiable with respect to w at 𝑤 and .

vii. If z(.) is differentiable at 𝑤 , then 𝐷𝑤𝑧 𝑤 , 𝑞 = 𝐷𝑤2 𝑐 𝑤 , 𝑞 ,is a

symmetric and negative semidefinite matrix with 𝐷𝑤𝑧 𝑤 , 𝑞 𝑤 = 0.

viii. If f (.) is homogenous of degree one (i.e., exhibits constant returns to

scale ), then c(.) and z(.) are homogenous of degree one in q.

ix. If f (.) is concave, then c(.) is a convex function of q ( in particular,

marginal costs are nondecreasing in q).


Hamid Kordbacheh 52

• Example 5.C.1: Profit and Cost Functions for the Cobb-Douglas Production

Function.

• Here we drive the profit and cost functions for the Cobb-Douglas

production function Example 5.B.2,

𝑓 𝑧1, 𝑧2 = 𝑧1𝛼𝑧2

𝛽.

• Recall from Example 5.B.3 that

𝛼 + 𝛽 = 1corresponds to the case of constant returns to scale,

𝛼 + 𝛽 < 1corresponds to decreasing returns, and

𝛼 + 𝛽 > 1corresponds to increasing returns.

• The conditional factor demand equations and cost function have exactly

the same form, and are derived in exactly the same way, as the expenditure

function in Section 3.E


Hamid Kordbacheh 53

• see Example 3.E.1; the only difference in the computation is that we now

do not impose 𝛼 + 𝛽 = 1

• Conditional factor demand equations:

𝑧1 𝑤1, 𝑤2, 𝑞 = 𝑞1/(𝛼+𝛽)(𝛼𝑤2/𝛽𝑤1)𝛽/(𝛼+𝛽),

𝑧2 𝑤1, 𝑤2, 𝑞 = 𝑞1/(𝛼+𝛽)(𝛽𝑤1/𝛼𝑤2)𝛼/(𝛼+𝛽)

Cost function

𝑐 𝑤1, 𝑤2, 𝑞

= 𝑞1/(𝛼+𝛽)[ 𝛼/𝛽 𝛽/ 𝛼+𝛽 + 𝛼/𝛽 −𝛼/ 𝛼+𝛽 ]𝑤1𝛼/ 𝛼+𝛽 𝑤2

𝛽/ 𝛼+𝛽 .


Hamid Kordbacheh 54

5. E Aggregation In Production

• The importance of aggregation in microeconomics

• Could we construct aggregate supply based upon individual PMP

• MWG: There is no problem aggregating in production, “If firms maximize

profits taking prices as given, then the production side of the economy

aggregates beautifully.


Hamid Kordbacheh 55

• Consider J units (firms, plants) with production sets Y1,...,YJ, with profit

functions 𝜋𝑗 𝑝 and supply correspondences 𝑦𝑗 𝑝 , j = 1,...,J.

• Definition - Aggregate Supply Correspondence: The sum of the 𝑦𝑗 𝑝 is

called aggregate supply correspondence:

• 𝑦 𝑝 = 𝑦𝑗 𝑝 = {𝑦 ∈ 𝑅𝐿|𝑦 = 𝑦𝑗 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑦𝑗 ∈ 𝑦𝑗(𝑝)}, 𝑗 = 1, … , 𝐽}𝐽𝑗

𝐽𝑗

• The law of supply also holds for the aggregate supply function.

o From proposition 5.C.1 we can conclude that 𝐷𝑦 𝑝 is symmetric and

positive semidefinite.

o The positive and semidefiniteness of 𝐷𝑦 𝑝 implies the law of supply in

the aggregate.

o We can Also prove this aggregate supply directly:

We know from 5.C.3

𝑝 − 𝑝 [𝑦𝑗 𝑝 − 𝑦𝑗 𝑝 ] ≥ 0 for all j=1,…,J

adding over j

𝑝 − 𝑝 [𝑦 𝑝 − 𝑦 𝑝 ]


Hamid Kordbacheh 56

Representative producer

• Definition of Aggregate Production Set: The sum of the individual 𝑌𝑗 is

called aggregate production set:

• 𝑌 = 𝑌𝑗 = {𝑦 ∈ 𝑅𝐿|𝑦 = 𝑦𝑗 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑦𝑗 ∈ 𝑌𝑗}, 𝑗 = 1, … , 𝐽}𝐽𝑗

𝐽𝑗

• Let π*(p) and y*(p) be the profit function and the supply correspondence

of the aggregate production set Y

• In a purely competitive environment the maximum profit obtained by every

firm maximizing profits individually is the same as the profit obtained if all

J firms where they coordinate their choices in a joint profit maximization.

𝜋∗(𝑝) = π𝑗(𝑝)

𝐽

𝑗

• In other words, there exists a representative producer


Hamid Kordbacheh 57

Proposition [5.E.1]: For all p>> 0 we have

(i) 𝜋∗ 𝑝 = 𝜋𝑗(𝑝)𝐽𝑗

(ii) 𝑦∗ 𝑝 = 𝑦𝑗 𝑝 = { 𝑦𝑗|𝑦𝑗 ∈ 𝑦𝑗(𝑝)𝐽𝑗 }

𝐽𝑗

• What do these mean?

Proof (i)

• Since π* is the maximum value function obtained from the aggregate

maximization problem 𝜋∗ 𝑝 ≥ 𝑝𝑦 𝑝 = 𝑝 𝑦𝑗 p = 𝑝. 𝑦𝑗 𝑝 𝐽𝑗

𝐽𝑗 such

that 𝜋∗ ≥ 𝜋𝑗(𝑝)𝐽𝑗

• To show equality, note that there are 𝑦𝑗 in Y𝑗 such that 𝑦 = 𝑦𝑗𝐽𝑗 then

𝑝. 𝑦 ≤ 𝜋𝑗(𝑝)𝐽𝑗 for all 𝑦 ∈ 𝑌 thus 𝜋∗(𝑝) ≤ 𝜋𝑗(𝑝)

𝐽𝑗

• Together these imply 𝜋∗ 𝑝 = 𝜋𝑗(𝑝)𝐽𝑗



Hamid Kordbacheh 58

Proof (ii)

• Here we have to show that Σ𝑦𝑗 𝑝 ⊂𝑦∗(𝑝) and 𝑦∗(𝑝)⊂ Σ𝑦𝑗 𝑝

• Consider 𝑦𝑗∈Σ𝑦𝑗 then 𝑝 𝑦𝑗 = p𝑦𝑗 = 𝜋𝑗 𝑝 = 𝜋∗(𝑝)

• This argument results in 𝑦𝑗 ⊂ 𝑦∗ 𝑝

• To get the second direction we start with y ∈ 𝑦∗ 𝑝 then that 𝑦 = 𝑦𝑗𝐽𝑗 for

with 𝑦𝑗 ∈ 𝑌𝑗

• Since 𝑝𝑦 = 𝑝 𝑦𝑗 = p𝑦𝑗 = 𝜋𝑗 𝑝 = 𝜋∗(𝑝)) we get 𝑦∗ 𝑝 ⊂ Σ𝑦𝑗 𝑝

• The same aggregation procedure can also be applied to derive aggregate

cost.

• The intuition behind these: decentralization results or laissez faire

• Discuss Figure 5.E.1


Hamid Kordbacheh 59

5.F Efficient production

• Providing the basic backgrounds about efficient production plans to study

welfare economics discussions (Chapters 10 and 16)

• The aim is to show that the production plans which are not efficient are

wasteful.

• Definition:[5.F.1] A production vector is efficient if there is no y' ∈ Y such

that 𝑦′≥ 𝑦 and 𝑦′≠𝑦

• There is no way to increase output with given inputs or to decrease input

with given output (sometimes called technical efficiency)

• Discuss Figure 5.F.1


Hamid Kordbacheh 60

• Proposition[5.F.1]: If y ∈ Y is profit maximizing for some 𝑝≫0, then y is

efficient.

• This is a version of the fundamental theorem of welfare economics. (Chapter

16)

• It also tells us that a profit maximizing firm does not choose interior points in

the production set.

Proof:

• It can be showed by means of a contradiction:

• Suppose that there is a 𝑦′∈𝑌 such that 𝑦′≠𝑦 such that a y' ∈ Y and that 𝑦′ ≥ 𝑦.

• Because 𝑝≫0 we get 𝑝𝑦′ ≥ 𝑝𝑦, contradicting the assumption that y solves the

PMP.

• For interior points (y" ), by the same argument we see that this is neither

efficient nor optimal

• The result also holds for nonconvex production sets - see Figure 5.F.2

• The assumption 𝑝≫0 cannot be relaxed to 𝑝 ≥ 0, this only works with

convex Y (Exercise 5.F.1)


Hamid Kordbacheh 61

• Note that the converse of the 1st FTWE is not necessarily true. It cab be

showed by Figure 5.E.2

• Proposition[5.F.2] Suppose that Y is convex. Then every efficient

production y ∈ Y is profit maximizing for some 𝑝 ≥ 0 and 𝑝≠0

• Note that this is restricted to convex production sets

• Proof

• Suppose that y is efficient. Construct the set 𝑃𝑦 = 𝑦′ ∈ ℝ𝐿: 𝑦′ ≫ 𝑦 This

set is convex.

• Since 𝑦′ ≫ 𝑦 then 𝑌⋂𝑃𝑦 = ∅

• There is some 𝑝 ≥ 0 such that 𝑝𝑦′ ≥ 𝑝𝑦" for every 𝑦′ ∈ 𝑃𝑦 and 𝑦" ∈ 𝑌.

• This implies 𝑝𝑦′ ≥ 𝑝𝑦" for every 𝑦′ ≫ 𝑦"

• The interesting property of the 2nd FTWE is 𝑝≥0


Hamid Kordbacheh 62

• The 2nd FTWE does not allow for input prices to be negative

o If some 𝑝 < 0 then we could have 𝑝𝑦′ < 𝑝𝑦 for some 𝑦′ ≫ 𝑦" with

y′ − y sufficiently large.

• It remains to show that y maximizes the profit

o Take an arbitrary 𝑦" ∈ 𝑌. y was fixed,

o Then 𝑝𝑦′ ≥ 𝑝𝑦" for every 𝑦′ ∈ 𝑃𝑦

o 𝑦′ ∈ 𝑃𝑦 can be chosen arbitrary close to y, such 𝑝𝑦 ≥ 𝑝𝑦" still has to

hold.

o

o I.e. y maximizes the profit given p.


Hamid Kordbacheh 63

5.G Remarks on the objectives of the Firm

What are the objectives of a firm?

• Until now we have assumed that the firm maximizes its profit under a price

vector p assumed to be fixed.

A firm can have a number of other objectives:

• Profit satisficing ( agency problem)

• Sales or Revenue Maximization (maximising the size of the business or

predatory pricing)

So only if p is fixed we can rationalize profit maximization

Do the shareholders maximize the profit of their firm in the absence of

agency problem?

• Conflict of interest

• Uncertainty problem


Hamid Kordbacheh 64

For a deeper analysis:

• Consider a production possibility set Y owned by consumers i=1,…I ,

• The consumers own the shares 𝜃𝑖 of profit with 𝜃𝑖 = 1

• 𝑦∈𝑌 is a production decision.

• 𝑤𝑖 is non-profit wealth

• Consumer i maximizes utility

max𝑥𝑖

𝑢 𝑥𝑖

𝑠. 𝑡. 𝑝. 𝑥𝑖 ≤ 𝑤𝑖 + 𝜃𝑖𝑝. 𝑦.

• With fixed prices the budget set described by 𝑝. 𝑥𝑖 ≤ 𝑤𝑖 + 𝜃𝑖𝑝. 𝑦, increases if p·y increases

• With higher 𝑝.𝑦, each consumer i is better off.

• Here maximizing profits 𝑝.𝑦 makes sense.

• Problems arise (e.g.) if

o Profits are uncertain.

o Prices depend on the action taken by the firm.

o Firms are not controlled by its owners.


Hamid Kordbacheh 65

The uncertain profits

• Suppose that the output of a firm is uncertain. It is important to know

whether output is sold before or after uncertainty is resolved.

• If the goods are sold on a spot market (i.e. after uncertainty is resolved),

then also the owner’s attitude towards risk will play a role in the output

decision.

• Maybe less risky production plans are preferred (although the expected

profit is lower).

• If there is a futures market the firm can sell the good before uncertainty is

resolved the buyer bears the risk. Profit maximization can still be optimal


Hamid Kordbacheh 66

A Price setter firm

• Consider a two good economy with goods 𝑥1 and 𝑥2; 𝐿 = 2,

• 𝑤𝑖 = 0.

• Suppose that the firm can influence the price of good 1, 𝑝1 = 𝑝(𝑥1)

• We normalize the price of good 2, such that 𝑝2 = 1

• 𝑧 units of 𝑥2 are used to produce 𝑥1 with production function 𝑥1 = 𝑓(𝑧)

• The cost is given by 𝑝2𝑧 = 𝑧.

• Two extreme cases

1. In an input oriented decision making all consumers unanimously want

max𝑥𝑖

𝑢 𝑥𝑖

𝑠. 𝑡. 𝑝. 𝑥𝑖 ≤ 𝑤𝑖 + 𝜃𝑖𝑝. 𝑦.

Which results in

max𝑧≥0

𝑝 𝑓 𝑧 𝑓(𝑧) − 𝑧


Hamid Kordbacheh 67

• Assume an extreme preferences of the owners good 1. (Output oriented)

• The aggregate amount of 𝑥1the consumers can buy is 1

𝑝1 .𝑝1 𝑓 𝑧 𝑓 𝑧 − 𝑧 = 𝑓 𝑧 − 𝑧/𝑝1(𝑓 𝑧 )

• Then max𝑥𝑖

𝑢 𝑥𝑖

𝑠. 𝑡. 𝑝. 𝑥𝑖 ≤ 𝑤𝑖 + 𝜃𝑖𝑝. 𝑦

• 𝑟𝑒𝑠𝑢𝑙𝑡𝑠 𝑖𝑛 max 𝑓 𝑧 − 𝑧/𝑝1(𝑓 𝑧 )

• There are different optimization problems in which the solutions are different

• e.g Derive the F.O.C for two O.P given that 𝑝1(𝑓 𝑧 = 𝑧

For a heterogeneous preferences nothing changes.


Hamid Kordbacheh 68

Ownership of the firm

Who make the decisions?

• Owners (typically in small businesses)

• Professional managers.

• Possible goals of managers

o Survive

o Beat the competition

o Maximize sales or revenues

o Maximize net income

o Maximize market share

o Minimize costs

o Maximize the value or the price of (stock) shares

• Agency theory


Hamid Kordbacheh 69

microeconomics i - staff.alzahra.ac.ir

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