Production theory 1
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Production Theory 1. Short-Run v. Long-Run. Fixed input/factor of production: quantity of input is fixed regardless of required output level, e.g. capital or specialized labour Variable input/factor of production: quantity of input used depends on the level of output

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Production Theory 1

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Production theory 1

Production Theory 1


Short run v long run

Short-Run v. Long-Run

  • Fixed input/factor of production: quantity of input is fixed regardless of required output level, e.g. capital or specialized labour

  • Variable input/factor of production: quantity of input used depends on the level of output

  • Short run: at least one input/factor is fixed

  • Long run: all inputs/factors are variable


Production function

Production Function

  • A technology is a process by which inputs (e.g. labour and capital) are converted into output.

  • The output level is denoted by y.

  • The technology’s production function states the maximum amount of output possible from an input bundle.


Production function1

Production Function

One input

Output Level

y = f(x) is the

production

function

y’

y’ = f(x’) is the maximum output level obtainable from x’ input units.

x’

x

Input Level


Technology set

Technology Set

  • The collection of all feasible production plans is the technology set.


Technology set1

Technology Set

One input

Output Level

y = f(x) is the

production

function.

y’

y”

y” = f(x’) is an output level that is feasible from x’ input units.

x’

x

Input Level


Technology set2

Technology Set

One input

Output Level

y’

The technologyset

y”

x’

x

Input Level


Technology set3

Technology Set

One input

Output Level

Technicallyefficient plans

y’

The technologyset

Technicallyinefficientplans

y”

x’

x

Input Level


Technology multiple inputs

Technology: Multiple Inputs

  • What does a technology look like when there is more than one input?

  • The two input case: Input levels are x1 and x2. Output level is y.

  • Example of production function is


Preview isoquant

PREVIEW: ISOQUANT

  • An isoquant is the set of all combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output.

  • The slope of the isoquant = the marginal rate of technical substitution (MRTS) = the technical rate of substitution (TRS)

  • MRTS (TRS): The number of units of K that we can dispose of if one more unit of L becomes available while remaining on the original isoquant.


Technologies with multiple inputs

Technologies with Multiple Inputs

  • The complete collection of isoquants is the isoquant map.

  • The isoquant map is equivalent to the production function.

  • Example


Isoquants with two inputs

Isoquants with Two Inputs

K

Y=40

Y=20

L


Isoquants with two inputs1

Isoquants with Two Inputs

  • Properties

    Y/K>0, Y/L>0

    2Y/K2<0,2Y/L2<0

    Diminishing marginal product

    (Diminishing marginal utility)


Cobb douglas technology

Cobb-Douglas Technology

x2

All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.

x1


Marginal physical product

Marginal (Physical) Product

  • The marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed.


Marginal physical product1

Marginal (Physical) Product

then the marginal product of input 1 is


Marginal physical product2

Marginal (Physical) Product

then the marginal product of input 1 is


Marginal physical product3

Marginal (Physical) Product

then the marginal product of input 1 is

and the marginal product of input 2 is


Marginal physical product4

Marginal (Physical) Product

then the marginal product of input 1 is

and the marginal product of input 2 is


Marginal physical product5

Marginal (Physical) Product

  • The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if


Technical rate of substitution

Technical Rate-of-Substitution

The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution.

x2

yº100

x1


Technical rate of substitution1

Technical Rate-of-Substitution

  • How is a technical rate-of-substitution computed?


Technical rate of substitution2

Technical Rate-of-Substitution

  • How is a technical rate-of-substitution computed?

  • The production function is

  • A small change (dx1, dx2) in the input bundle causes a change to the output level of


Technical rate of substitution3

Technical Rate-of-Substitution

Along an individual isoquant, dy = 0, therefore the changes dx1 and dx2 must satisfy the following,


Technical rate of substitution4

Technical Rate-of-Substitution

which rearranges to

or


Technical rate of substitution5

Technical Rate-of-Substitution

is the rate at which input 2 must be givenup as input 1 increases so as to keepthe output level constant. It is the slopeof the isoquant = MRTS = TRS.


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