Lecture 3 - Production Functions (4 Per Page)

Econ. 300 George DeltasLecture 3 Production Functions-1-

LECTURE 3

PRODUCTION FUNCTIONS

3.1. INTRODUCTION.

We have seen a number of illustrations of optimal firm behavior models.

These models typically have three big components.

(a) Market Environment:

A description of what is “outside” the firm that is relevant to themanager’s decisions.

Examples: Consumers and their behavior, other firms and their interactionwith this particular firm, governments (taxes/regulation), etc.

(b) Internal Structure of the Firm:

A description of the firm’s capabilities that is relevant to themanager’s decisions.

Examples: Ability to transform inputs into outputs, cost of producingparticular product attributes (such as quality), behavior ofinternal labor force, etc.


(c) A Behavioral Assumption About the Management:

That is, whether the firm’s management maximizes profits orsomething else.

Elements (b) and (c) form the supply side of these models.

For the next 3 lectures we will be focusing on aspects of the internalstructure of the firm.

We will concentrate on:

P How to characterize a firm’s capabilities of transforming inputs intooutputs (and in particular, into a single output).

P How to combine the description of these capabilities with input coststo summarize the firm’s capabilities by a cost function, such as thosethat we have seen in the preceding lecture.

P How to combine this information with a summary description of themarket environment to obtain the optimal production choices of thefirm.

You can think of this part of the course as showing what lies underneaththe cost functions we have seen in Lecture 2, and how they are derivedfrom the ground-up.


3.2. SINGLE-INPUT PRODUCTION FUNCTIONS.

A single input production function is of the form

where q is the output of the firm.

x is the level of input used by the firm (e.g., labor, materials,etc).

is a transformation function that relates the input andoutput levels.

The transformation function is referred to as the productionfunction of the firm.

SOME DEFINITIONS:

The slope of the production function with respect to x is known as themarginal product of x.

The marginal product of shows the rate at which output increases as thefirm increases input use.


The ratio of total output divided by total input is known as the averageproduct of x.

Returns to Scale.

Production functions can be classified with respect to how output levelsrespond to input use.

If increasing the input x by a factor of m increases output by more than afactor of m, the production function is said to be characterized byincreasing returns to scale.

That is, for increasing returns to scale production functions, we have

where m > 1.


If increasing the input x by a factor of m increases output by less than afactor of m, the production function is said to be characterized bydecreasing returns to scale.

That is, for decreasing returns to scale production functions, we have

where m > 1.

If increasing the input x by a factor of m increases output by exactly afactor of m, the production function is said to be characterized byconstant returns to scale.

That is, for constant returns to scale production functions, we have

where m > 1.

A useful concept for measuring the extent of returns to scale is theelasticity of scale of a production function. It is defined as

and measures the percentage increase in output if inputs are increased by1 percent.


Relationship Between Returns to Scale and Average Product.

decreasing returns to scale

decreasing average product.

The opposite is true for increasing returns to scale.

[Note: There is no general relationship between returns to scale andmarginal product, we will illustrate shortly.]


Graphical Examples.

This production has decreasing returns to scale ( ).

This can also be seen directly from the definition of returns to scale:


This production has increasing returns to scale:

The nature of returns to scale can vary for different output levels:

Returns are increasing for low x, decreasing for high x.

Note: This implies that in the above definition m is supposed to be close to 1!


Note that Marginal Product may be increasing even though returns toscale (and Average Product) are decreasing.

An example of this is shown below:

Observation 1: For Average Product (and returns to scale) are

decreasing, while for they are both increasing.

Observation 2: At Marginal Product is increasing, despite the

fact that (locally) returns to scale are decreasing.

Remark: Such a production function may arise if you switch between differentproduction technologies depending on the scale of production.


Numerical and Parametric Examples.

Example 1.

The electricity output of a power plant, E, is increasing in the input ofcoal in the boiler, C.

However, there are diminishing returns to coal use.

For this plant, the relationship is given by the electricity productionfunction

This is plotted below:

Notice that both MP and AP are decreasing in C and that .


Example 2.

The handcrafting of traditional artefacts requires labor as the only(significant) input.

Experience is a key element in how quickly such artefacts are made.

Thus, the more time an artisan (or group of artisans) spends on thesehandicrafts, the more efficient he is making them.

The production function is given by

where Q is the artisan’s output and H is the number of hours he spendsin handcrafting.

This production function is plotted below:

Notice that both MP and AP are increasing in H and that .


Example 3.

The above two examples are numerical examples:

When the input level is specified, the production function gives anumerical answer for the level of output.

Sometimes, a parametric form of the production function may be known,but not the numerical values of the parameters.

For instance, it may be known that the handicraft production function isof the form:

but the values of a and b are unknown (though presumed positive!).

Even though the production function is in parametric form, one can stillanalyze it.

Such analysis allows us to obtain some (admittedly limited in this case)understanding of this type of production functions.


The marginal and average products are

and

Observations:

! Both MP and AP (and also returns to scale) are

– increasing in H if ,

– decreasing in H if , and

– constant if .

! The parameter a

– does not affect returns to scale and the sign of MP and AP, but

– an increase in a proportionately increases output, MP, and AP.

Insight 1: The parametric model is allows us to determine what element of theproduction function is important in determining returns to scale andwhat element is not important.


Interpretation of parameters:

The parameter a captures everything that affects the efficiency ofproduction but not how this efficiency varies with time spent at work.

Examples include:

" A new technique of assembling the artefact.

" The introduction of new easier to use materials.

" The use of better training.

The parameter b captures everything that affects the decline or theincrease in the artisan’s efficiency as he spends more time at work.

Examples include:

" Better lighting conditions that reduce eye-strain, thus reducingfatigue.

" Use of a bonus system if output reaches particular levels, providingincentives to maintain concentration after long hours of work.

Note: The above innovations also increase output, but they have a proportionatelybigger effect for workers who work longer hours.

Insight 2: The parametric model allows us to interpret the meaning of thevarious elements of the production function.


Graphical Representation:

Observation: A parametric representation of a production functiondoes not allow us to get a numerical estimate of theoutput level for any given level of input.

We must somehow obtain an estimate of the numericalvalue of a and b.

As we show next, writing a parametric model is a necessary first step toobtain such a numerical relationship between input and output.

Insight 3: The parametric model provides us with a framework to obtainestimates of the unknown elements of the production function.


Inferring Production Function Parameters.

The above 3 examples have one thing in common:

The describe a function that takes and input and yields as output aproduction level.

Issue: How does this production function comes to be known in thefirst place?

Possibilities: " One can obtain “engineering” estimates bystudying the production process.

" One can use data from observed output levels ofthe firm for given input levels.

We will here describe the second approach.

Notes: • We will return to this type of exercises repeatedly duringthe semester. This is just the “simplest case” of this type ofproblem.

• The method we describe is a demonstration of the idea ofusing data to infer parameter values. In real applications,far more sophisticated methods are used, though they allcome down to the same principles.


Step 1. Start with a parametric production function, say

Step 2. Observe that there are two unknown parameters in the aboveequation.

Step 3. Obtain information on the input, H, and associated output, q,for two different instances.

Denote this input-output pairs by and .

These pairs are known as observations.

Step 4. Use the observations to form the system of equations

Step 5. Solve this 2 equation system for the 2 unknowns, a and b.

Once the values of a and b are calculated, one can predict how changesin hours worked would affect output obtained.


Illustration:

Suppose a local crafts outfit produces

– 50 crafts in a day when using 120 man-hours

– 80 crafts in a day when using 200 man-hours

Also suppose that its production function is of the form

What is the elasticity of scale of its production function?

The elasticity of scale of the production function is given by

Therefore, we need to find out the value of the parameter b.


Given the production figures, the following system of equations musthold (see Step 4 above):

Dividing the two equations term by term, we obtain

Taking logs, we get

Therefore, and 1% increase in the number of man-hours would yield an0.92% increase in output.

Note: Had it been necessary, we could also easily have calculated the value of ausing either of the two equations in the top of the page.


Discussion 1.

This exercise is called “calibration” and is used when the amount of dataavailable is limited.

When large amounts of data are available, the parameters are obtainedvia estimation.

Estimation allows the incorporation of more than two observations toobtain two parameters.

For example, the estimates of the parameters a and b can be obtained viathe linear regression model

where the above equation is obtained by taking the log of the productionfunction, and is random variable of mean zero.

In this course, we will only solve calibration exercises (though, forcompleteness, we will sometimes briefly discuss their estimationcounterparts).


Discussion 2.

Note that the calibration approach described above relies on specifyingthe production function up to some unknown parameters.

If sufficient data is available one can compare two different parametricforms and decide which is the best approximation to reality.

For example, one could test whether

or

is a better model of the production of handicrafts.

Indeed, if one has a large enough sample, no assumptions about theparametric form of the production function need be made


3.3. MULTIPLE INPUT PRODUCTION FUNCTIONS.

Most production processes require more than one input.

In general, a production function can be written as

where the x’s are a list of inputs.

[The entire vector will be denoted by .]

The slope of the production function with respect to input i, , is knownas the marginal product of .

The marginal product of shows the rate at which output increases as thefirm increases input i.

[Average product is harder to define in multi-input production functionsbecause one needs to fix the input proportions.]


Returns to Scale.

A production function has increasing returns to scale if

where m > 1 (but close to 1).

A production function has decreasing returns to scale if


A production function has constant returns to scale if


In multiple input production functions, returns to scale (and the elasticityof scale) depend not only on the level of output, but also on the directionin which we increase it.


Example.

Consider the production function

If the firm only uses input 1, it is characterized by constant returns toscale.

If the firm uses only input 2, it is characterized by increasing returns toscale.

If the firm uses an equal amount of inputs 1 and 2, then for low levels ofthe input it is characterized by increasing returns, for high levels of theinput it is characterized by decreasing returns.

[For low x the quadratic term dominates the cubic, for high x the opposite is true.]


Marginal Rate of Technical Substitution.

A firm with a multiple input production function can trade-off betweenthe two inputs while keeping output constant.

The rate at which the firm can replace one input for another whilekeeping total output constant is called the marginal rate of technicalsubstitution.

Formally, the marginal rate of technical substitution of input i forinput j is equal to the number of units of input j than are required tomake up for the loss of one unit of input i, so that total output remainsconstant.

In other words, if is the use of input i and is the use of input j, then

[the negative sign is because we are considering a loss of a unit of i.]

The MRTS can be expressed in terms of input marginal products.

This we show next using the concept of total differential.


Consider the total differential of a firm’s output:

where is the marginal product of input i.

Since the concept of MRTS is operative for a fixed level of output, weconsider changes in input levels such that , or equivalently

Focusing on the MRTS of two inputs, i and j, we will hold all otherinputs fixed. This yields:

This can be re-written as

Observe that since we have explicitly held output constant, the left handside of the equation is the .

Thus,


Numerical Illustration.

It follows directly from the above that the MRTS depends on the levelsof all inputs used by the firm.

Consider the production function

Notice that input 3 enhances the productivity of input 1.

The is given by

Notice that it takes more and more units of to make up for the loss ofone unit of as the utilization of goes up.

This is not surprising given that input 3 increases the productivity ofinput 1 but leaves the productivity of input 2 unaffected.

Therefore: Even though the MRTS between two inputs represents thetradeoff between these inputs holding all other inputsconstant, it matters at what level the other inputs are heldconstant!


Graphical Depiction of Multi-input Production Functions.

Multi-input production functions are hard to graph on a flat surface.

One technique that helps visualize them is to graph on a two dimensionalsurface the set of input combinations that are sufficient to produce agiven output level (holding all other inputs constant).

These set of input combinations is known as isoquants.

Illustration:

Consider the production function for an aluminum producer, which isgiven by

where K denotes the level of capital the firm uses, L denotes the level oflabor, and E denotes the energy inputs (mineral inputs will be ignored ).

Aside/Definition: Production functions of the form

are known as Constant Elasticity of Substitutionproduction functions.


An isoquant that plots capital versus labor can be obtained by:

• solving the above equation for K, and

• fixing the values of output and energy to some levels and .

Solving for K yields

Fixing the values of q and E to some levels yields

[Technical point: The values of and must be such that otherwisethere would be no positive values of K and L that would satisfy theproduction function equation. ]

If no labor is used, the required level of capital is

If no capital is used, the required level of labor is:


The isoquant is plotted below:

The absolute value of the slope of the isoquant is the , since

Notice that as we move along the isoquant from left to right decreases.

That is, as the input mix changes from being relative capital-rich torelative labor-rich, the gain of one unit of labor frees up a smaller andsmaller amount of capital.

Equivalently, as the input mix changes from being relative capital-richto relative labor-rich, the marginal value of labor relative to capitaldeclines.


Similarly, as the input mix changes from being relatively labor-rich torelatively capital rich, decreases.

That is, capital becomes less valuable, relative to labor, the more capitalyou have, relative to labor.

This property appears to hold relatively generally and is known asDiminishing Marginal Rate of Technical Substitution.

This is not a “law” that must hold for production function.

It just turns out to be often true.

The most commonly offered rationale for why diminishing marginal rateof technical substitution holds is motivated by factors such as thesedescribed below:


Suppose that you have a certain supply of labor, but no capital, and youproduce of output.

As you add capital to the production process, labor gets freed-up (alwaysholding output constant).

The first unit of capital you would add would be the one that saves themost labor: a piece of heavy equipment saves a lot of muscle power!

The second unit of capital you would add would also save a lot of labor,though not as much: perhaps some automated machinery that reducessome repetitive tasks.

The third unit of capital would save even less labor: perhaps a fastcomputer with a drawing

There are some jobs that people are much better suited to doing thanmachines.

Thus, the fifth and sixth unit of capital will be used for assignments thatpeople are much better at doing (relative to machines), e.g., using smart“robots” in assembly lines that can do a variety of tasks, or that need tobe mobile.

At the end of the process, it may not even be technically feasible toreplace labor with capital (e.g., product design will need at least somehuman intervention).


Increasing output to shifts the isoquant out:

This reflects that fact that holding the energy input constant, more laborand capital are needed to produce a higher output level.

This can also be seen mathematically from the expression for theisoquant

and the expressions for and .


Increasing the energy input to shifts the isoquant in.

This reflects the fact that less labor and capital are needed to produce agiven output if more electricity is available.

This can also be seen mathematically from the expression for theisoquant

and the expressions for and .


More Generally . . .

The concept of isoquants can be used generally, not necessarily withinthe context of a particular parametric numerical example.

In general, isoquants do not have to

R Touch the axes (some input may be essential and cannot besubstituted away).

R Have the same shape as one moves to higher level of output.

R Have decreasing MRTS.


Some Standard Special Cases.

Even though the concepts we have covered are applicable generally,there are some particular parametric examples that are very commonlyused.

These parametric examples have special properties that make them:

! Important benchmarks (or points of departure) for more generalproduction functions.

! Tractable and thus easily amenable to algebraic manipulation.

These examples are:

i The Cobb-Douglas production function.

ii The Leontieff (or Fixed Proportions) production function.

iii The Linear production function.

We turn to a detailed examination of these productions, while notforgetting that these are just special cases (and that we will also applythe ideas we have learned more broadly).


A. Cobb-Douglas.

The two input Cobb-Douglas production function is of the form.

where the two inputs are capital (K) and labor (L) and A, , and areconstants.

The marginal products of capital and labor are

and

The is given by

Notice the more capital rich is the input mix, the lower the relative valueof K regardless of output level (and similarly for labor).


The same conclusion is obtained when looking at the isoquants.

Solving the production function for capital we obtain

Notice that as L goes to zero, the required level of capital to produceoutput q goes to infinity.

Similarly, one can show that as K goes to zero, the required level oflabor to produce output q goes to infinity.

Therefore, the isoquants of the Cobb-Douglas look like this:


The Cobb-Douglas production function isoquants have an importantproperty:

The MRTS along a ray from the origin is independent of the scale ofproduction.

This can be seen geometrically in the figure below :

This is a property that is shared by all three of our special caseproduction functions, and contributes to their tractability.

It may be a reasonably assumptions in many real world cases, especiallyfor limited changed in output.

However, for large enough changes in output, it is unlikely to hold!

Typically, for large output levels, capital becomes more productiverelative to labor.


Returns to scale:

Increasing all inputs by a factor of m yields an output of

Therefore, if , then

and the production function exhibits increasing returns to scale.

If the production function exhibits constant returns to scale.

If the production function exhibits decreasing returns to scale.

The elasticity of scale for a Cobb-Douglas production function is:


Generalization:

The Cobb-Douglas can easily be generalized for more than two inputs.

An example of a four input Cobb-Douglas would be

where E is the level of energy inputs and M is the level of other materialsused in production.


B. Leontieff (or Fixed Proportions) Production Function.

A production process is characterized by a Leontieff production functionif inputs must be used in fixed proportions.

Example 1: 2 units of capital and 3 units of labor are required toproduce 1 unit of output.

Example 2: units of capital and units of labor are required toproduce 1 unit of output.

Example 3: units of capital and units of labor arerequired to produce 1 units of output.

Generally, in a Leontieff production function units of capital and

units of labor are required to produce q units of output.

Additional units of either labor or capital only are not useful inincreasing output.

Mathematically, the corresponding production function is given by

You can double check this by substituting in the above of K and of Land confirming that the output will indeed be q.


A more typical representation of the Leontieff production function is

where, in the context of the previous example, a equals , b equals, and equals .

Discussion on Notation:

Do not get hang up on what symbols are used to represent particular parameters.

The symbols are just notation.

For example the production function given as

would denote a fixed-proportions production function in which units of

labor and units of capital are required to produce q units of output.

Also, the dependence of the production function on K and L will often besuppressed when there is no risk of ambiguity. That is, we will often write

instead of


Isoquants:

Example 1 (revisited):

The isoquants for and are plotted below:

The isoquant for reflects the fact that this output level can beproduced with 2 units of K and 3 units of L, and additional units ofeither K or L do not increase output.

The isoquant for any other level of q is similarly constructed.

Notice that the “corners” of all isoquants line on a line, the slope ofwhich gives us the ratio of capital to labor in the production function.





General case:

The isoquants for and are plotted below.

It can be easily seen that if the Leontieff production function hasincreasing returns to scale.

If the Leontieff production function has increasing returns toscale.

If the Leontieff production function has constant returns to scale.

In fact, the elasticity of scale of the Leontieff production function isequal to .


Generalization.

The Leontieff production function can be generalized to more than twoinputs.

For example, a four input Leontieff production function would be givenby:

where K, L, E, and M are the capital, labor, energy, and other materialsinputs, and a, b, c, d, and are parameters.


C. Linear Production Function.

A production process is characterized by a linear production function ifinputs are perfect substitutes in producing a given level of output.

Example 1: 2 units of capital or 3 units of labor are required toproduce 1 unit of output.

Example 2: units of capital or units of labor are required toproduce 1 unit of output.

Example 3: units of capital or units of labor are requiredto produce 1 units of output.

Generally, in a linear production function units of capital orunits of labor are required to produce q units of output.

Mathematically, the corresponding production function is given by

You can double-check this by either units of labor or units oflabor in the above function and observing that the output will be q.


A more typical representation of the linear production function is

where, in the context of the previous example, a equals , b equals, and equals .

The marginal rate of technical substitution of capital for labor is

The MRTS is equal to a constant and does not depend on input mix orscale of output.

A direct implication of this is that isoquants are straight lines.


Isoquants.


The isoquants that corresponds to and are given below:

The intercepts correspond to the level of K or L needed to produce agiven level of output.

The slope is equal to the ratio of marginal products of the two inputs.



The isoquants that corresponds to and are given below:


General Case:


It can be easily seen that if the linear production function hasincreasing returns to scale.

If the linear production function has increasing returns to scale.

If the linear production function has constant returns to scale.

In fact, the elasticity of scale of the linear production function is equal to.


Generalization.

The linear production function can be generalized to more than twoinputs.

For example, a four input linear production function would be given by:

where K, L, E, and M are the capital, labor, energy, and other materialsinputs, and a, b, c, d, and are parameters.


Production Functions with a Fixed Input.

Sometimes, the choice of one input is fixed at some particular value (atleast temporarily).

Examples: Labor may be fixed because a labor contract makes firingdifficult.

Capital may be fixed because it takes time to bring newcapacity on line.

A two input production function with one fixed input can be thought ofas a single input production function.

Example: The Cobb-Douglas production function

for which capital is fixed at a level of 9, is equivalent to theproduction function

This is similar to our examples of single-input productionfunctions, with labor being the only input.


More generally, a k input production function with of these inputs

being fixed at some levels can be thought of as a input production

function.

Short-run vs. long-run.

Inputs differ in the ease with each they can be adjusted.

For many production processes, energy inputs and some materials thatcan be obtained from “liquid” markets are the easiest to adjust.

Next in ease of adjustment are materials that are obtained via long termcontracts, and some types of non-specialized labor.

Next in ease of adjustment are specialized labor and “off-the-shelf”capital.

The most difficult to adjust is specialized capital and other physicalinfrastructure.

Therefore, a production function of the form

where K represents capital inputs, L labor inputs, and M energy andmaterials, represents a firm’s capabilities only if enough time haselapsed so that the firm can adjust all inputs.


If there is only sufficient time to adjust energy and materials, theproduction function is given by

where and are the current (fixed for a period of time) level ofcapital and labor.

This could be thought of as the very short run production function of thefirm.

If there is sufficient time to also adjust labor inputs, the productionfunction is given by

This could be thought of as the short-run production function of the firm.

Thus, the production function

in which all inputs are thought of as being flexible is the long-runproduction function of the firm.

There is no set amount of time the corresponds to the short-run and thelong-run.

The long-run is simply the amount of time that is long enough that thefirm can choose the level of all inputs.


This amount of time varies from industry to industry.

Relationship with single-input production functions.

Reconsider the illustration using the Cobb-Douglas production function

in which capital is fixed at a level .

This short-run production function is a single input production, but onein which capital is a parameter.

Different levels of capital result in different values for that parameter,and thus to different short-run production functions.

For example, if a sample of firms share the same Cobb-Douglasproduction function, but have different levels of capital (or if a firm’slevel of capital varies over time), the short-run production function willbe differ across firms (or over time).

Even though may differ from firm to firm and time to time, for anygiven firm and any given time period it is not a decision variable for thefirm, but rather a parameter that is held constant.


Returns to Scale.

When there are only 2 inputs, the extent to which isoquants shift out asoutput increases reflects the returns to scale of the production function.

For example, the isoquants of the linear production function

for , and are more distantly spaced as we move tohigher output levels.

The bigger spacing between isoquants reflects the fact that an increase ofoutput by a unit requires progressively more a more inputs.

The converse is true for a production function with increasing returns toscale.


Technical Change.

Technical innovations increase the output a firm can produce with thesame level of inputs.

We distinguish two types of innovations:

– Innovations that are neutral with respect to the inputs.

These affect the productivity (or marginal product) of both inputsequally, thus leaving MRTS unchanged.

– Innovations that are biased towards one (or more) of the inputs.

These affect the productivity (or marginal product) of the two inputsunequally, thus changing the MRTS between the inputs.

We will consider these in the context of our benchmark examples.


Cobb-Douglas Production Function

Recall that the marginal rate of technical substitution for the Cobb-Douglas is given by:

Input neutral technical change increases A. This does not affect theMRTS because it increases and proportionately.

An increase in " increases output (assuming K > 1) by increasing themarginal products of both capital and labor, since

and

However, the increase in the marginal product of capital isproportionately larger, leading to an increase in : It now takesmore units of labor to make up for the loss of one unit of capital for anygiven labor-capital ratio.

Therefore, an increase in " is due to technical change that is biasedtowards capital.

The converse is true for an increase in $.


Leontieff Production Function

Can’t use MRTS because production function not differentiable.

Technical change is neutral if a and b increase proportionately.

Isoquants shift in towards the origin.

It takes proportionately less of both inputs to produce the same amountof output.


Technical change is biased towards capital if the ratio of increases,and biased towards labor if decreases.

Isoquants shift more towards the L-axis than the K-axis if technicalchange favors capital.

It takes less capital (relative to labor) to produce the same level ofoutput.

The converse is true if technical change favors labor.


Linear Production Function

Recall that the marginal rate of technical substitution for the linearproduction function is given by:

Therefore, a proportional increase in " and $ is neutral towards the twoinputs.

If " increases proportionately more than $, then technical change favorscapital.

Otherwise, technical change favors labor.


Calibration/Estimation of Production Function Parameters.

If the general form of the production function is known, but the value ofassociated parameters are not known, then these parameters can be inprinciple be calibrated (or estimated) by using observed data.

The procedure follows the same steps as these we have used for thesingle-input production functions.

Step 1. Start with a parametric production function, say

where a, b, and are parameters to be estimated.

Step 2. Obtain information on the inputs, K and L, and the output, q,for as many times as there are unknown parameters.

Denote this input-output combinations (observations) by, , and .

Step 3. Use the observations to form the system of equations

and then solve this system for the unknown parameters.


As we discussed in the case of single input production function, inpractice more observations than parameters are used to estimateproduction functions.

The estimated parameters are those that provide the “best fit” to theentire set of observations.

However, estimation follows the same underlying idea as that shownabove.

The identification of the parametric form of the production functionfollows similar principles.

A researcher can compare the three production functions we examinedabove and determine which best helps explain the input-outputobservations.

In fact, researchers typically examine more than the three productionfunctions that have been introduced above.

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Lecture 3 - Production Functions (4 Per Page)