inputs and production functionsecon.queensu.ca/walras/custom/200/212/notes/13w/...inputs and...

Inputs and Production FunctionsECON 212 Lecture 9

Tianyi Wang

Queen’s Univeristy

Winter 2013

Tianyi Wang (Queen’s Univeristy) Tech Winter 2013 1 / 10

I We start modelling the supply side.I These inputs are goods and services that can be combined in someway to produce other goods called outputs.

I Only certain combinations of inputs are feasible ways to produce agiven amount of output.

I The combinations of inputs and outputs that are technologicallyfeasible is called the production set.

I Boundary of this set is called Production Function.I See class notes for graph.

Production Function with one variable

I Use only labor as input to produce output.I Q = f (L) describes the boundary.I The set: {(Q, L) : Q ≤ f (L)}I Usually write Q = Q(L).I See class notes for graph.I See class notes for effi ciency and feasibility.I See class notes for Increasing Margin Return, Decreasing MarginReturn and Decreasing Total Return.

Marginal Product and Average Product

I Average Product of Labor, APL = QL

I Graphically, the slope of ray from origin to point on productionfunction.

I Marginal Product of Labor, additional labor’s contribution to output.

MPL = lim∆L→0

Q(L+ ∆L)−Q(L)∆L

=∂Q(L)

I Graphically, the slope of production function at L.I Relationship b/w APL and MPL :

I APL ↑ ⇒ MPL > APLI APL ↓ ⇒ MPL < APLI APL at max ⇒ MPL = APL

I See class notes for math derivation.I See class notes for graph.

Two Inputs

I In general technology requires two inputs, labor and capital.I Q = Q(L,K )I As with Utility functions, we can use 2-D level curves depict 3-Dproduction surface.

I See Page 4-6 of Slide 1 for graphs.I Level curves are called Isoquants.I Isoquant represents all combinations of two inputs that produce agiven amount of output.

I Mathematically the same as indifferent curve.I See class notes for example.

Marginal Rate of Tech. Substitution

I Note: with two variables, partial derivative of one variable is derivedby taking the other variable as a constant.

I One way of characterizing production function is the MRTS

I the amount of extra K one needs to produce a given output level if onereduces L just a bit.

I Consider the production function Q = Q(L,K ).I Change both K and L, the overall effect on output is

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

∂K dKI hold Q constant, this means to set dQ = 0.I 0 = ∂Q (L,K )

∂L dL+ ∂Q (L,K )∂K dK , rearrange to get:

MRTS(L,K ) =dKdL|Q=Q̄ = −

MPLMPK

I Define MRTS as a negtive number so that it is the same as slope.I Usually assume diminishing MRTS, slope gets flatter and flatter.

I One way of characterizing production function is the MRTS

I the amount of extra K one needs to produce a given output level if onereduces L just a bit.

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

I One way of characterizing production function is the MRTSI the amount of extra K one needs to produce a given output level if onereduces L just a bit.

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

I Consider the production function Q = Q(L,K ).

I Change both K and L, the overall effect on output is

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

∂K dK

I hold Q constant, this means to set dQ = 0.I 0 = ∂Q (L,K )

MPLMPK

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

∂K dKI hold Q constant, this means to set dQ = 0.

I 0 = ∂Q (L,K )∂L dL+ ∂Q (L,K )

∂K dK , rearrange to get:

MPLMPK

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

I Define MRTS as a negtive number so that it is the same as slope.

I Usually assume diminishing MRTS, slope gets flatter and flatter.

I dQ = ∂Q (L,K )∂L dL+ ∂Q (L,K )

MPLMPK

Elasticity of Substitution

I An other way to characterize produciton function.

I Measures the extent to which firms can substitute capital for labor asthe relative productivity changes.

I Elasticity of Substitution σ is defined as

σ =d ln(KL )

d ln( MPLMPK)=

% change in kL

% change in MRTS

I See class notes for graph.

I An other way to characterize produciton function.I Measures the extent to which firms can substitute capital for labor asthe relative productivity changes.

σ =d ln(KL )

d ln( MPLMPK)=

% change in kL

% change in MRTS

σ =d ln(KL )

d ln( MPLMPK)=

% change in kL

% change in MRTS

σ =d ln(KL )

d ln( MPLMPK)=

% change in kL

% change in MRTS

Several Production Functions

I Fixed Proportions (Perfect Compliment)

I Q(L,K ) = min{aL, bK}I Also known as Leontieff production function

I Perfect Substitutes

I Q(L,K ) = aL+ bK

I Cobb-Douglas Production Functions

I Q(L,K ) = ALaKbI σ = 1, see class notes for proof.

I Constant Elasticity of Substitution (CES)

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I σ→ ∞ is perfect subs.I σ→ 0 is perfect comp.I σ = 1 is Cobb-Douglas

I Fixed Proportions (Perfect Compliment)I Q(L,K ) = min{aL, bK}

I Also known as Leontieff production function

I Q(L,K ) = aL+ bK

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Fixed Proportions (Perfect Compliment)I Q(L,K ) = min{aL, bK}I Also known as Leontieff production function

I Q(L,K ) = aL+ bK

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Q(L,K ) = aL+ bK

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Perfect SubstitutesI Q(L,K ) = aL+ bK

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Cobb-Douglas Production FunctionsI Q(L,K ) = ALaKb

I σ = 1, see class notes for proof.

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Cobb-Douglas Production FunctionsI Q(L,K ) = ALaKbI σ = 1, see class notes for proof.

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I σ→ ∞ is perfect subs.

I σ→ 0 is perfect comp.I σ = 1 is Cobb-Douglas

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

I σ→ ∞ is perfect subs.I σ→ 0 is perfect comp.

I σ = 1 is Cobb-Douglas

I Q(L,K ) = [aLσ−1

σ + bKσ−1

σ ]σ

σ−1

Returns to Scale

I Measures steepness of "production hill".

I Answer to the question of what happens when we increase all inputsby the same amount. There are three possibilities:

I Q(tL, tK ) = tQ(L,K ) Constant return to scaleI Q(tL, tK ) > tQ(L,K ) Increasing return to scaleI Q(tL, tK ) < tQ(L,K ) Decreasing return to scale

Returns to Scale

I Measures steepness of "production hill".I Answer to the question of what happens when we increase all inputsby the same amount. There are three possibilities:

Returns to Scale

I Q(tL, tK ) = tQ(L,K ) Constant return to scale

I Q(tL, tK ) > tQ(L,K ) Increasing return to scaleI Q(tL, tK ) < tQ(L,K ) Decreasing return to scale

Returns to Scale

I Q(tL, tK ) = tQ(L,K ) Constant return to scaleI Q(tL, tK ) > tQ(L,K ) Increasing return to scale

I Q(tL, tK ) < tQ(L,K ) Decreasing return to scale

Returns to Scale

Tech Progress

I Productivity improves over time. To produce the same Q, needs fewerand fewer resources.

I One of three things can happen. See class notes for graphs.

1. Neutral

I at given KL , MRTS =

MPLMPK

is unaffected.I slope of Isoquants are the same at given K

L .I technology change affects K and L in the same way.

2. Labor-saving

I at given KL , MRTS decreases.

I technology change favours K .

3. Capital-saving

I at given KL , MRTS increases.

I technology change favours L.

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

is unaffected.

I slope of Isoquants are the same at given KL .

I technology change affects K and L in the same way.

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

I technology change affects K and L in the same way.

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

Tech Progress

1. Neutral

MPLMPK

2. Labor-saving

3. Capital-saving

inputs and production functionsecon.queensu.ca/walras/custom/200/212/notes/13w/...inputs and...

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