OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

G30D, Week 6 (Production Theory I)

Martin K. Jensen (U. B’ham)

November 2012

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Throughout these notes we’ll consider firms under thefollowing assumptions:

Each firm produces only a single output (called: non-jointproduction).Each firm uses only two types of inputs.

The first of these, non-joint production, has the importantimplication that we can look at production functions.

A production function is a function f : R2+ → R+ which to

each possible combination of inputs associates a level ofoutput.

The inputs are denoted (z1, z2). So given inputs, (z1, z2),output will be f (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Throughout these notes we’ll consider firms under thefollowing assumptions:

Each firm produces only a single output (called: non-jointproduction).Each firm uses only two types of inputs.

The first of these, non-joint production, has the importantimplication that we can look at production functions.

A production function is a function f : R2+ → R+ which to

each possible combination of inputs associates a level ofoutput.

The inputs are denoted (z1, z2). So given inputs, (z1, z2),output will be f (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Throughout these notes we’ll consider firms under thefollowing assumptions:

Each firm produces only a single output (called: non-jointproduction).Each firm uses only two types of inputs.

The first of these, non-joint production, has the importantimplication that we can look at production functions.

A production function is a function f : R2+ → R+ which to

each possible combination of inputs associates a level ofoutput.

The inputs are denoted (z1, z2). So given inputs, (z1, z2),output will be f (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Throughout these notes we’ll consider firms under thefollowing assumptions:

Each firm produces only a single output (called: non-jointproduction).Each firm uses only two types of inputs.

The first of these, non-joint production, has the importantimplication that we can look at production functions.

A production function is a function f : R2+ → R+ which to

each possible combination of inputs associates a level ofoutput.

The inputs are denoted (z1, z2). So given inputs, (z1, z2),output will be f (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Example:

f (z1, z2) = Azα1 zβ2 (1)

where A, α, β > 0. This is the Cobb-Douglas (C-D)production function; by far the most used production functionin applications such as macroeconomics.

To further specialize, assume that A = 1, α = β = 0.5, so wehave f (z1, z2) = z0.51 z0.52 . If z1 = 1 and z2 = 1 (so the firmputs one unit of each input into the production process); thelevel of output will be f (1, 1) = 10.510.5 = 1.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Example:

f (z1, z2) = Azα1 zβ2 (1)

where A, α, β > 0. This is the Cobb-Douglas (C-D)production function; by far the most used production functionin applications such as macroeconomics.

To further specialize, assume that A = 1, α = β = 0.5, so wehave f (z1, z2) = z0.51 z0.52 . If z1 = 1 and z2 = 1 (so the firmputs one unit of each input into the production process); thelevel of output will be f (1, 1) = 10.510.5 = 1.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm must produce at least y0 units of itsoutput.

The input requirement set Z (y0) is the set of all possibleinput combinations which leads to an output which is equal toor greater that y0, i.e.:

Z (y0) = {(z1, z2) ∈ R2+ : f (z1, z2) ≥ y0} (2)

An isoquant (through y0) I (y0) is the set of all possible inputcombinations which leads to exactly y0 units of the output,i.e.:

I (y0) = {(z1, z2) ∈ R2+ : f (z1, z2) = y0} (3)

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm must produce at least y0 units of itsoutput.

The input requirement set Z (y0) is the set of all possibleinput combinations which leads to an output which is equal toor greater that y0, i.e.:

Z (y0) = {(z1, z2) ∈ R2+ : f (z1, z2) ≥ y0} (2)

An isoquant (through y0) I (y0) is the set of all possible inputcombinations which leads to exactly y0 units of the output,i.e.:

I (y0) = {(z1, z2) ∈ R2+ : f (z1, z2) = y0} (3)

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm must produce at least y0 units of itsoutput.

The input requirement set Z (y0) is the set of all possibleinput combinations which leads to an output which is equal toor greater that y0, i.e.:

Z (y0) = {(z1, z2) ∈ R2+ : f (z1, z2) ≥ y0} (2)

An isoquant (through y0) I (y0) is the set of all possible inputcombinations which leads to exactly y0 units of the output,i.e.:

I (y0) = {(z1, z2) ∈ R2+ : f (z1, z2) = y0} (3)

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

That the input requirement sets are convex is a very basicassumption in production theory. It is entirely equivalent toassuming that the production function f is quasi-concave.

Additionally, we assume throughout that the productionfunction f is continuous.In fact, you will more often than not see production functionsthat are differentiable (for example, the Cobb-Douglasfunction). Yet, there are interesting example where f iscontinuous but not differentiable, such as fixed proportionsproduction functions:

f (z1, z2) = min{ z1β11

,z2β12} (4)

So in order to produce y0 units of output, the firm must useexactly β11y

0 units of the first input and β12y0 units of the

second input. Isoquants are “L-shaped here”.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

That the input requirement sets are convex is a very basicassumption in production theory. It is entirely equivalent toassuming that the production function f is quasi-concave.Additionally, we assume throughout that the productionfunction f is continuous.

In fact, you will more often than not see production functionsthat are differentiable (for example, the Cobb-Douglasfunction). Yet, there are interesting example where f iscontinuous but not differentiable, such as fixed proportionsproduction functions:

f (z1, z2) = min{ z1β11

,z2β12} (4)

So in order to produce y0 units of output, the firm must useexactly β11y

0 units of the first input and β12y0 units of the

second input. Isoquants are “L-shaped here”.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

That the input requirement sets are convex is a very basicassumption in production theory. It is entirely equivalent toassuming that the production function f is quasi-concave.Additionally, we assume throughout that the productionfunction f is continuous.In fact, you will more often than not see production functionsthat are differentiable (for example, the Cobb-Douglasfunction). Yet, there are interesting example where f iscontinuous but not differentiable, such as fixed proportionsproduction functions:

f (z1, z2) = min{ z1β11

,z2β12} (4)

So in order to produce y0 units of output, the firm must useexactly β11y

0 units of the first input and β12y0 units of the

second input. Isoquants are “L-shaped here”.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

That the input requirement sets are convex is a very basicassumption in production theory. It is entirely equivalent toassuming that the production function f is quasi-concave.Additionally, we assume throughout that the productionfunction f is continuous.In fact, you will more often than not see production functionsthat are differentiable (for example, the Cobb-Douglasfunction). Yet, there are interesting example where f iscontinuous but not differentiable, such as fixed proportionsproduction functions:

f (z1, z2) = min{ z1β11

,z2β12} (4)

So in order to produce y0 units of output, the firm must useexactly β11y

0 units of the first input and β12y0 units of the

second input. Isoquants are “L-shaped here”.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

That the input requirement sets are convex is a very basicassumption in production theory. It is entirely equivalent toassuming that the production function f is quasi-concave.Additionally, we assume throughout that the productionfunction f is continuous.In fact, you will more often than not see production functionsthat are differentiable (for example, the Cobb-Douglasfunction). Yet, there are interesting example where f iscontinuous but not differentiable, such as fixed proportionsproduction functions:

f (z1, z2) = min{ z1β11

,z2β12} (4)

So in order to produce y0 units of output, the firm must useexactly β11y

0 units of the first input and β12y0 units of the

second input. Isoquants are “L-shaped here”.Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The final assumption we will impose here is possibility ofinaction: f (0, 0) = 0. In words, this assumption means thatthe firm “can’t produce something from nothing” - but alsothat it is allowed to close down.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

When f is differentiable, we can define the marginalproducts of the inputs. Specifically, the marginal product ofthe first good z1 is:

MP1 =∂f (z1, z2)

∂z1(5)

The marginal product of the second good is:

MP2 =∂f (z1, z2)

∂z2(6)

Dividing these two with each other we get the Marginal Rateof Technical Substitution (MRTS):

MRTS12 =MP1

MP2(7)

The MRTS is exactly equal to minus the slope of the isoquantat (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

When f is differentiable, we can define the marginalproducts of the inputs. Specifically, the marginal product ofthe first good z1 is:

MP1 =∂f (z1, z2)

∂z1(5)

The marginal product of the second good is:

MP2 =∂f (z1, z2)

∂z2(6)

Dividing these two with each other we get the Marginal Rateof Technical Substitution (MRTS):

MRTS12 =MP1

MP2(7)

The MRTS is exactly equal to minus the slope of the isoquantat (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

When f is differentiable, we can define the marginalproducts of the inputs. Specifically, the marginal product ofthe first good z1 is:

MP1 =∂f (z1, z2)

∂z1(5)

The marginal product of the second good is:

MP2 =∂f (z1, z2)

∂z2(6)

Dividing these two with each other we get the Marginal Rateof Technical Substitution (MRTS):

MRTS12 =MP1

MP2(7)

The MRTS is exactly equal to minus the slope of the isoquantat (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

When f is differentiable, we can define the marginalproducts of the inputs. Specifically, the marginal product ofthe first good z1 is:

MP1 =∂f (z1, z2)

∂z1(5)

The marginal product of the second good is:

MP2 =∂f (z1, z2)

∂z2(6)

Dividing these two with each other we get the Marginal Rateof Technical Substitution (MRTS):

MRTS12 =MP1

MP2(7)

The MRTS is exactly equal to minus the slope of the isoquantat (z1, z2).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm initially produces 1 unit by use of 1 unit ofeach of the inputs. What will happen if the firm doubles bothinputs to 2 ?

There are three possibilities: The output will exactly double(y = 2), it will less than double (y < 2), or it will more thandouble (y > 2).

In the first case there is Constant returns to scale (CRS), inthe second Decreasing returns to scale (DRS), and in thethird Increasing returns to scale (IRS).

More formally we have the following definition:

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm initially produces 1 unit by use of 1 unit ofeach of the inputs. What will happen if the firm doubles bothinputs to 2 ?

There are three possibilities: The output will exactly double(y = 2), it will less than double (y < 2), or it will more thandouble (y > 2).

In the first case there is Constant returns to scale (CRS), inthe second Decreasing returns to scale (DRS), and in thethird Increasing returns to scale (IRS).

More formally we have the following definition:

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm initially produces 1 unit by use of 1 unit ofeach of the inputs. What will happen if the firm doubles bothinputs to 2 ?

There are three possibilities: The output will exactly double(y = 2), it will less than double (y < 2), or it will more thandouble (y > 2).

In the first case there is Constant returns to scale (CRS), inthe second Decreasing returns to scale (DRS), and in thethird Increasing returns to scale (IRS).

More formally we have the following definition:

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Imagine that a firm initially produces 1 unit by use of 1 unit ofeach of the inputs. What will happen if the firm doubles bothinputs to 2 ?

There are three possibilities: The output will exactly double(y = 2), it will less than double (y < 2), or it will more thandouble (y > 2).

In the first case there is Constant returns to scale (CRS), inthe second Decreasing returns to scale (DRS), and in thethird Increasing returns to scale (IRS).

More formally we have the following definition:

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

Definition

Consider a production function f : R2+ → R. If for every s > 1 and

every (z1, z2) ∈ R2+:

f (sz1, sz2) = sf (z1, z2)Then f is said to exhibit constant returns to scale.

f (sz1, sz2) < sf (z1, z2)Then f is said to exhibit decreasing returns to scale.

f (sz1, sz2) > sf (z1, z2)Then f is said to exhibit increasing returns to scale.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

If given a specific combination of inputs (z1, z2) and a specificscale s > 1, we have one of the three possibilities above, thenwe say that f exhibits respectively, constant, decreasing, andincreasing returns, at (z1, z2) for the scale s. Thisterminology allows for the possibility the f has, say, increasingreturns in some “zones” of outputs and decreasing returns inother “zones”.

Next, a mathematical definition. If a function f : R2+ → R

satisfies:f (sz1, sz2) = st f (z1, z2) (8)

for all s ≥ 0 and (z1, z2) ∈ R2+, then f is said to be

homogenous of degree t.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

If given a specific combination of inputs (z1, z2) and a specificscale s > 1, we have one of the three possibilities above, thenwe say that f exhibits respectively, constant, decreasing, andincreasing returns, at (z1, z2) for the scale s. Thisterminology allows for the possibility the f has, say, increasingreturns in some “zones” of outputs and decreasing returns inother “zones”.

Next, a mathematical definition. If a function f : R2+ → R

satisfies:f (sz1, sz2) = st f (z1, z2) (8)

for all s ≥ 0 and (z1, z2) ∈ R2+, then f is said to be

homogenous of degree t.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

As you should check, a production function exhibits constantreturns to scale if and only if it is homogenous of degreet = 1. If (but not only if!) a production function ishomogenous of degree t < 1, then it exhibits decreasingreturns. Finally, if (but not only if!) f is homogenous ofdegree t > 1, it exhibits increasing returns to scale.

Consider the C-D production function f (z1, z2) = Azα1 zβ2 .

Notice thatf (sz1, sz2) = A(sz1)α(sz2)β = sα+βAzα1 z

β2 = sα+βf (z1, z2).

It follows that a C-D production function is homogenous ofdegree t = α + β.In particular, if α + β = 1, then this exhibits CRS. Ifα + β < 1, there is DRS. If α + β > 1, there is IRS.Note also that this function is concave if and only ifα + β ≤ 1, and strictly concave if and only if α + β < 1.By the way: You need to know what a concave function is!

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

As you should check, a production function exhibits constantreturns to scale if and only if it is homogenous of degreet = 1. If (but not only if!) a production function ishomogenous of degree t < 1, then it exhibits decreasingreturns. Finally, if (but not only if!) f is homogenous ofdegree t > 1, it exhibits increasing returns to scale.Consider the C-D production function f (z1, z2) = Azα1 z

β2 .

Notice thatf (sz1, sz2) = A(sz1)α(sz2)β = sα+βAzα1 z

β2 = sα+βf (z1, z2).

It follows that a C-D production function is homogenous ofdegree t = α + β.In particular, if α + β = 1, then this exhibits CRS. Ifα + β < 1, there is DRS. If α + β > 1, there is IRS.Note also that this function is concave if and only ifα + β ≤ 1, and strictly concave if and only if α + β < 1.By the way: You need to know what a concave function is!

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

As you should check, a production function exhibits constantreturns to scale if and only if it is homogenous of degreet = 1. If (but not only if!) a production function ishomogenous of degree t < 1, then it exhibits decreasingreturns. Finally, if (but not only if!) f is homogenous ofdegree t > 1, it exhibits increasing returns to scale.Consider the C-D production function f (z1, z2) = Azα1 z

β2 .

Notice thatf (sz1, sz2) = A(sz1)α(sz2)β = sα+βAzα1 z

β2 = sα+βf (z1, z2).

It follows that a C-D production function is homogenous ofdegree t = α + β.

In particular, if α + β = 1, then this exhibits CRS. Ifα + β < 1, there is DRS. If α + β > 1, there is IRS.Note also that this function is concave if and only ifα + β ≤ 1, and strictly concave if and only if α + β < 1.By the way: You need to know what a concave function is!

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

As you should check, a production function exhibits constantreturns to scale if and only if it is homogenous of degreet = 1. If (but not only if!) a production function ishomogenous of degree t < 1, then it exhibits decreasingreturns. Finally, if (but not only if!) f is homogenous ofdegree t > 1, it exhibits increasing returns to scale.Consider the C-D production function f (z1, z2) = Azα1 z

β2 .

Notice thatf (sz1, sz2) = A(sz1)α(sz2)β = sα+βAzα1 z

β2 = sα+βf (z1, z2).

It follows that a C-D production function is homogenous ofdegree t = α + β.In particular, if α + β = 1, then this exhibits CRS. Ifα + β < 1, there is DRS. If α + β > 1, there is IRS.

Note also that this function is concave if and only ifα + β ≤ 1, and strictly concave if and only if α + β < 1.By the way: You need to know what a concave function is!

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

As you should check, a production function exhibits constantreturns to scale if and only if it is homogenous of degreet = 1. If (but not only if!) a production function ishomogenous of degree t < 1, then it exhibits decreasingreturns. Finally, if (but not only if!) f is homogenous ofdegree t > 1, it exhibits increasing returns to scale.Consider the C-D production function f (z1, z2) = Azα1 z

β2 .

Notice thatf (sz1, sz2) = A(sz1)α(sz2)β = sα+βAzα1 z

β2 = sα+βf (z1, z2).

It follows that a C-D production function is homogenous ofdegree t = α + β.In particular, if α + β = 1, then this exhibits CRS. Ifα + β < 1, there is DRS. If α + β > 1, there is IRS.Note also that this function is concave if and only ifα + β ≤ 1, and strictly concave if and only if α + β < 1.

By the way: You need to know what a concave function is!

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

As you should check, a production function exhibits constantreturns to scale if and only if it is homogenous of degreet = 1. If (but not only if!) a production function ishomogenous of degree t < 1, then it exhibits decreasingreturns. Finally, if (but not only if!) f is homogenous ofdegree t > 1, it exhibits increasing returns to scale.Consider the C-D production function f (z1, z2) = Azα1 z

β2 .

Notice thatf (sz1, sz2) = A(sz1)α(sz2)β = sα+βAzα1 z

β2 = sα+βf (z1, z2).

It follows that a C-D production function is homogenous ofdegree t = α + β.In particular, if α + β = 1, then this exhibits CRS. Ifα + β < 1, there is DRS. If α + β > 1, there is IRS.Note also that this function is concave if and only ifα + β ≤ 1, and strictly concave if and only if α + β < 1.By the way: You need to know what a concave function is!

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

We assume from now on that f is continuous, thatf (0, 0) = 0 (possibility of inaction), and quasi-concave (theinput requirement sets are convex). In addition we will assumethat f is strictly increasing in both coordinates, so addingmore of one of the inputs leads to greater output.

Denote the prices of the inputs by p1 > 0 and p2 > 0respectively. Imagine that the firm must produce y units ofthe output. We know that in order to do so, it must choose(z1, z2) in Z (y) = {(z1, z2) ∈ R2

+ : f (z1, z2) ≥ y} (the inputrequirement set).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

We assume from now on that f is continuous, thatf (0, 0) = 0 (possibility of inaction), and quasi-concave (theinput requirement sets are convex). In addition we will assumethat f is strictly increasing in both coordinates, so addingmore of one of the inputs leads to greater output.

Denote the prices of the inputs by p1 > 0 and p2 > 0respectively. Imagine that the firm must produce y units ofthe output. We know that in order to do so, it must choose(z1, z2) in Z (y) = {(z1, z2) ∈ R2

+ : f (z1, z2) ≥ y} (the inputrequirement set).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The cost minimization problem consists in, for a given“target” output y , finding the minimum cost inputcombination. Formally, the problem reads:

min p1z1 + p2z2

s.t.

{f (z1, z2) ≥ y(z1, z2) ∈ R2

+

(9)

If f is continuous and strictly monotone, this problem has asolution. If f is strictly quasi-concave, the problem will have aunique solution (z1(p1, p2, y), z2(p1, p2, y)) which is the firm’sconditional demand function (conditional here refers to thefact that this demand function is conditional upon outputbeing y).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The cost minimization problem consists in, for a given“target” output y , finding the minimum cost inputcombination. Formally, the problem reads:

min p1z1 + p2z2

s.t.

{f (z1, z2) ≥ y(z1, z2) ∈ R2

+

(9)

If f is continuous and strictly monotone, this problem has asolution. If f is strictly quasi-concave, the problem will have aunique solution (z1(p1, p2, y), z2(p1, p2, y)) which is the firm’sconditional demand function (conditional here refers to thefact that this demand function is conditional upon outputbeing y).

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The actual cost associated with producing y as cheaply aspossible is exactly:

C (p1, p2, y) = p1z1(p1, p2, y) + p2z2(p1, p2, y) (10)

C (p1, p2, y) is called the cost function (or sometimeslong-run cost function to stress the fact that in the long runboth factors can be varied).

Let us define an isocost line as the set of input combinationswhich costs the same amount C > 0, so:

L(C ) = {(z1, z2) ∈ R2+ : p1z1 + p2z2 = C} (11)

At the lectures, I will illustrate the cost-minimization problemusing isoquants and isocost lines.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The actual cost associated with producing y as cheaply aspossible is exactly:

C (p1, p2, y) = p1z1(p1, p2, y) + p2z2(p1, p2, y) (10)

C (p1, p2, y) is called the cost function (or sometimeslong-run cost function to stress the fact that in the long runboth factors can be varied).

Let us define an isocost line as the set of input combinationswhich costs the same amount C > 0, so:

L(C ) = {(z1, z2) ∈ R2+ : p1z1 + p2z2 = C} (11)

At the lectures, I will illustrate the cost-minimization problemusing isoquants and isocost lines.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The actual cost associated with producing y as cheaply aspossible is exactly:

C (p1, p2, y) = p1z1(p1, p2, y) + p2z2(p1, p2, y) (10)

C (p1, p2, y) is called the cost function (or sometimeslong-run cost function to stress the fact that in the long runboth factors can be varied).

Let us define an isocost line as the set of input combinationswhich costs the same amount C > 0, so:

L(C ) = {(z1, z2) ∈ R2+ : p1z1 + p2z2 = C} (11)

At the lectures, I will illustrate the cost-minimization problemusing isoquants and isocost lines.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)

OutlineThe production function, isoquants

Basic AssumptionsMarginal Products and the MRTS

Decreasing, Constant, and Increasing Returns to ScaleThe Cost-Mimimization Problem

The actual cost associated with producing y as cheaply aspossible is exactly:

C (p1, p2, y) = p1z1(p1, p2, y) + p2z2(p1, p2, y) (10)

C (p1, p2, y) is called the cost function (or sometimeslong-run cost function to stress the fact that in the long runboth factors can be varied).

Let us define an isocost line as the set of input combinationswhich costs the same amount C > 0, so:

L(C ) = {(z1, z2) ∈ R2+ : p1z1 + p2z2 = C} (11)

At the lectures, I will illustrate the cost-minimization problemusing isoquants and isocost lines.

Martin K. Jensen (U. B’ham) G30D, Week 6 (Production Theory I)