高级微观经济学 (黄有光) Topic5-Production

Advanced Microeconomics

Topic 5: Production & Cost

Primary Readings: DL – Chapter 2 & JR – Chapter 5

In this lecture, we will present a general framework of production technology. We will focus on what choices could be made; and the issue of what choices would be made will be deferred to the next lecture when we look into the firm’s behaviour.

The first part will describe production possibilities in physical terms; while the second part will recast this description into a cost function framework.

The treatment in this lecture is a bit abstract and quite general. Y ou are required to understand the relevance of this abstract framework in terms of particular technological processes.

5.1 Production Possibility Sets

There are many ways to describe the technology of a firm, such as, production functions, graphs, or systems of inequalities. But in mathematical term, these representations can all be expressed as a set.

∙The firm uses and produces a total of m commodities.

∙A particular production plan is y in R m:

∙y i > 0 implies that a net amount y i of i-th commodity is produced;

∙y j < 0 implies that a net amount –y j of j-th commodity is used;

∙y is called a netput vector.

∙Production possibility set of a firm is a subset Y⊂R m. A firm may select any vector y ∈Y as its production plan.

Properties of Production Possibility Set

∙Closed: If the limit of any converging sequence of vectors in Y is in Y.

∙Convex: Convex combinations of its elements remain to be inside.

∙Free disposal: If y∈Y implies that y’∈Y for all y’≤y.

∙Meaning that: commodities (inputs or outputs) can be thrown away.

Input Requirement Set: V(q) = {z: (-z, q) ∈Y }

Isoquant: Q (q ) = {z : (-z , q ) ∈ Y , (-z , q’) ∉ Y ∀ q’ ≥ q , q’ ≠ q }

∙ The isoquant Q (q ) is usually the boundary closest to the origin of V (q ).

Proposition : If Y is convex, so is V (q ).

∙ We normally do not require that the production possibility set is convex. If so, it will rule

out "start-up costs" and other sorts of returns to scale. (Do you see why?)

5.2 Production Functions

Transformation Function of the Production Possibility Set

For most production possibility sets, it is possible to describe them in item of single inequality of the form T (y ) ≤ 0. That is,

Y = {y : T (y ) ≤ 0}

A function T that describes Y this way is called a transformation function .

Efficient Production

∙

A production point y ∈Y is efficient is there is no y’ ∈Y , y’ ≠ y , with y’ ≥ y .

∙ An efficient production implies that it is not possible to either unilaterally increase the output(s)

or unilaterally decrease the input(s) while still remaining in Y .

Production Functions (Joan Robinson)

∙ For those technologies that have a single output can be described by a production function , which has both the theoretical and empirical appeal.

∙ The netput vector has the form: (-z , q ), where q is the output.

∙

If the technology has a transformation function T , i.e., Y = {(-z , q ): T (-z , q ) ≤ 0}, then under certain regularity conditions, we can solve T (-z , q ) = 0 for all q , which leads to another function: q = f (z ). This function f is the production function .

∙ The specification of q = f (z ) involves the notion of efficiency since it represents the maximum

output level that can be achieved with the input, i.e.,

f (z ) = max{q’: T (-z , q’) ≤ 0}.

∙ With a single output, the input requirement set V (q ) is convex if and only if the corresponding production function f (z ) is a quasiconcave function.

MRTS and Separable Production Functions

∙ With a given production function q = f (z ), the marginal rate of technical substitution

(MRTS) between two inputs i and j is defined as follows:

./)(/)()(j

i ij

z f z f MRTS

∂∂∂∂=z z z

∙

Normally, MRTS ij depends on the specification of all inputs. We can use MRTS to define separable

production functions , which involves regrouping the inputs into several mutually exclusive and exhaustive subsets. For details, refer to p.221 of Jehle & Reny.