Production | Firms 1
The Production Set
Production requires inputs | factors of production | for example,labour and capital equipment.
A production set is the set of outputs feasible given a particular combination of inputs,The production function
describes the maximum output given a particular combination of inputs,Both are illustrated below.
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Output = y
Input = x 0
x2
x1
Production set
y = f(x)= Production function
Isoquant
y = y
An isoquant maps all the combinations of the (two) inputs that that are just su–cient to produce a given output.
Technology is assumed to be monotonic and convex,producing isoquants like those above,Monotonicity (or free
disposal) says that more of one input will result in at least as much output,Convexity says that if two difierent
combinations of inputs produce output y,then a weighted average will produce at least y.
Production | Firms 2
Marginal Products
When one of the factors is increased a small amount,￠x1,output rises by ￠y,The marginal product is:
MP1(x1;x2)= ￠y￠x
1
= f(x1 +￠x1;x2)?f(x1;x2)￠x
1
… @f@x
1
Marginal product gives the slope of the production function y = f(x1;x2).
How much additional input 2 is required to just continue producing exactly y units of output,given a decrease in
the amount of input 1? The answer is the technical rate of substitution (TRS),If output remains constant,then:
￠y = MP1(x1;x2)￠x1 +MP2(x1;x2)￠x2 =0
This equation can be solved for the change in input 2 divided by the change input 1,It is the slope of the isoquant.
TRS= ￠x2￠x
1
=?MP1(x1;x2)MP
2(x1;x2)
The \law" of diminishing marginal product states that marginal product will decrease as the amount of the input
used is increased,keeping all other inputs constant,Why is this considered a \law"?
Diminishing TRS is also assumed | it follows from the convexity of the isoquant.
Production | Firms 3
The Long and Short Run
All factors can be varied in the long run,At least one is flxed in the short run.
Suppose input 2 is land,and can only be varied in the long run,The short run production function is written
y = f(x1;x2),It looks identical to the earlier function | getting atter as x1 increases,Why?
When all inputs are increased by a common factor k what happens to production? Returns to scale can be:
1,Constant,Output also increases by a factor of k | f(kx1;kx2)= kf(x1;x2).
2,Decreasing,Output increases by less than k | f(kx1;kx2) < kf(x1;x2).
3,Increasing,Output increases by more than k | f(kx1;kx2) > kf(x1;x2).
Production | Firms 4
Short Run Proflt Maximisation
Proflt is total revenue minus total cost,If x2 is flxed at x2 in the short run and the prices of output and the two
inputs are given by p,w1 and w2 respectively,then short run proflt is,… = py?w1x1?w2x2.
Isoproflt lines are drawn by rearranging to give y as a function of x1,Hence y =(…+w2x2)=p+(w1=p)x1.
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0
y
x1
y = f(x1;x2)
Isoproflt lines
(…+w2x2)=p
x?1
y?,,,,,,,,,,,,,
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The flrm is assumed to maximise proflts,Pushing the isoproflt lines upward increases proflt,However,they cannot
go higher than the production set as that output combination would be infeasible,Hence;
max… =maxx
1
pf(x1;x2)?w1x1?w2x2 =) w1p = MP1(x?1;x2)
The slope of the isoproflt line is equal to the slope of the production function,This means the value of the marginal
product is set equal to the price of the input,pMP1(x?1;x2)= w1 | a natural condition,Why?
Production | Firms 5
Comparative Statics
The analysis so far can be used to show what happens when various prices change,The slope of the isoproflt line is
w1=p and so is afiected whenever w1 or p change.
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0
y
x1 0
y
x1
y = f(x1;x2) y = f(x1;x2)?
x1
y?
x01
y0
x1
y
x01
y0
w1 p0
w01 p
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In the flrst graph,w1 increases to w01 and the result is a fall in the factor demand for input 1 and a fall in y.
In the second graph,p increases to p0 and the result is a rise in the demand for input 1 and a rise in the output y.
Changing w2 has no efiect upon the slope and so nothing changes in the short run | except proflt levels.
Production | Firms 6
Long Run Proflt Maximisation
In the long run all factors are variable,The flrm solves the following problem:
maxx
1;x2
pf(x1;x2)?w1x1?w2x2 =) pMP1(x?1;x?2)= w1 and pMP1(x?1;x?2)= w2
Given the optimal amount of input 2 (x?2) the function pMP1(x1;x?2)= w1 deflnes the inverse factor demand curve
which is the relationship between optimal factor demand and factor price,derived from the previous slide.
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pMP1(x1;x?2)= w1
0
w1
x1
The factor demand curve is always downward sloping,There are no \Gifien" production inputs,Why not?
A proflt maximising flrm with constant returns to scale makes zero proflt,Why?
Production | Firms 7
Cost Minimisation
An equivalent way to think about the flrm’s production decision is cost minimisation.
minx
1;x2
w1x1 +w2x2 subject to f(x1;x2)? y
That is,the flrm minimises their cost of producing at least an output of y,Isocost curves connect all the input
combinations of equal cost and are linear.
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0
x2
x1 0
x2
x1
f(x1;x2)? y
y = y
Isocost lines w1x1 +w2x2 = c
The second graph is an isoquant showing the constraint in the above equation,f(x1;x2)? y at all points above it.
The similarity with consumer theory is striking,The slope of the isocost lines is?w1=w2.
Production | Firms 8
The TRS Condition
The isocost line is moved downward as far as it will go giving another tangent condition.
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x2
x1
x?2
x?1
y = y
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The isoquant slope (the TRS) is set equal to the isocost slope giving the TRS condition:
TRS=?MP1(x
1;x?2)
MP2(x?1;x?2) =?
w1
w2
This is exactly what proflt maximisation had revealed,The two methods are equivalent.
The conditional factor demands are the optimal factor demands in terms of the amount of output the flrm wishes to
produce and the factor prices,x1(w1;w2;y) and x2(w1;w2;y).
Production | Firms 9
The Cost Function
The (long run) cost function is the minimum cost required to produce a certain output,It solves the problem:
c(y)= minx
1;x2
(w1x1 +w2x2) subject to f(x1;x2)? y
It can be written in terms of the conditional factor demands,c(y)= w1x1(w1;w2;y)+w2x2(w1;w2;y).
The short run cost function is the solution to the same problem when one of the factors is flxed,e.g,x2 = x2.
Constant costs that have to be paid no matter which level of output is produced are called flxed costs,Constant
costs that have to paid no matter which level of output is produced as long as it is not zero are called quasi-flxed
costs,All other costs are called variable costs,For example:
c(y)=
8<
:
1+k+y2 if y > 0
1+y2 if y =0
The above example has flxed costs of 1,quasi-flxed costs of k and variable costs of y2.
There are no flxed costs in the long run,although there could be quasi-flxed costs.
Production | Firms 10
Average Cost
Costs can always be written as c(y)= cv(y)+F where cv(y) is variable and includes any quasi-flxed costs.
Average cost is the minimum cost per unit of producing a total output y,Hence:
AC(y)= c(y)y = cv(y)y + Fy = AVC(y)+AFC(y)=Average variable cost+Average flxed cost
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AC
y
AC
y
AC
y
AFC AVC AC
AFC slopes downward,AVC will eventually slope upward,So AC is usually drawn with a quadratic shape.
Production | Firms 11
Marginal Cost
Marginal cost measures the change in cost given a small change in output,So:
MC(y)= ￠c(y)￠y = c(y+￠y)?c(y)￠y = ￠cv(y)￠y
The last equality follows because flxed costs do not change as output changes,by deflnition.
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0
Costs
y 0
MC
y
ACMC
AVC MC
Variable cost
The marginal cost curve cuts the average cost and the average variable cost curve at their minimum points,Why?
The average variable cost curve and the marginal cost curve take the same value for the flrst unit of output,Why?
The area underneath the marginal cost curve is equal to variable costs,Why?
Production | Firms 12
Long Run Cost Curves
When all factors are variable long run average costs must be no larger than short run average costs.
The long run average cost curve is the lower envelope of all the short run average cost curves.
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0
Costs
y 0
Costs
y
LRAC
SRAC
y? y?
LRAC
SRMC
LRMC
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Notice the minima of the SRAC curves do not intercept the LRAC curve except at the minimum,Why?
The second graph shows the connections between long run marginal cost and the other cost curves.
The LRMC curve is simply the collection of points from the \optimal" SRMC curves at each output level y?.
The Production Set
Production requires inputs | factors of production | for example,labour and capital equipment.
A production set is the set of outputs feasible given a particular combination of inputs,The production function
describes the maximum output given a particular combination of inputs,Both are illustrated below.
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.......................................................................................................................................0
Output = y
Input = x 0
x2
x1
Production set
y = f(x)= Production function
Isoquant
y = y
An isoquant maps all the combinations of the (two) inputs that that are just su–cient to produce a given output.
Technology is assumed to be monotonic and convex,producing isoquants like those above,Monotonicity (or free
disposal) says that more of one input will result in at least as much output,Convexity says that if two difierent
combinations of inputs produce output y,then a weighted average will produce at least y.
Production | Firms 2
Marginal Products
When one of the factors is increased a small amount,￠x1,output rises by ￠y,The marginal product is:
MP1(x1;x2)= ￠y￠x
1
= f(x1 +￠x1;x2)?f(x1;x2)￠x
1
… @f@x
1
Marginal product gives the slope of the production function y = f(x1;x2).
How much additional input 2 is required to just continue producing exactly y units of output,given a decrease in
the amount of input 1? The answer is the technical rate of substitution (TRS),If output remains constant,then:
￠y = MP1(x1;x2)￠x1 +MP2(x1;x2)￠x2 =0
This equation can be solved for the change in input 2 divided by the change input 1,It is the slope of the isoquant.
TRS= ￠x2￠x
1
=?MP1(x1;x2)MP
2(x1;x2)
The \law" of diminishing marginal product states that marginal product will decrease as the amount of the input
used is increased,keeping all other inputs constant,Why is this considered a \law"?
Diminishing TRS is also assumed | it follows from the convexity of the isoquant.
Production | Firms 3
The Long and Short Run
All factors can be varied in the long run,At least one is flxed in the short run.
Suppose input 2 is land,and can only be varied in the long run,The short run production function is written
y = f(x1;x2),It looks identical to the earlier function | getting atter as x1 increases,Why?
When all inputs are increased by a common factor k what happens to production? Returns to scale can be:
1,Constant,Output also increases by a factor of k | f(kx1;kx2)= kf(x1;x2).
2,Decreasing,Output increases by less than k | f(kx1;kx2) < kf(x1;x2).
3,Increasing,Output increases by more than k | f(kx1;kx2) > kf(x1;x2).
Production | Firms 4
Short Run Proflt Maximisation
Proflt is total revenue minus total cost,If x2 is flxed at x2 in the short run and the prices of output and the two
inputs are given by p,w1 and w2 respectively,then short run proflt is,… = py?w1x1?w2x2.
Isoproflt lines are drawn by rearranging to give y as a function of x1,Hence y =(…+w2x2)=p+(w1=p)x1.
.......
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....................
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....................
....................
.
..............
0
y
x1
y = f(x1;x2)
Isoproflt lines
(…+w2x2)=p
x?1
y?,,,,,,,,,,,,,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The flrm is assumed to maximise proflts,Pushing the isoproflt lines upward increases proflt,However,they cannot
go higher than the production set as that output combination would be infeasible,Hence;
max… =maxx
1
pf(x1;x2)?w1x1?w2x2 =) w1p = MP1(x?1;x2)
The slope of the isoproflt line is equal to the slope of the production function,This means the value of the marginal
product is set equal to the price of the input,pMP1(x?1;x2)= w1 | a natural condition,Why?
Production | Firms 5
Comparative Statics
The analysis so far can be used to show what happens when various prices change,The slope of the isoproflt line is
w1=p and so is afiected whenever w1 or p change.
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.......
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y
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x1
y = f(x1;x2) y = f(x1;x2)?
x1
y?
x01
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In the flrst graph,w1 increases to w01 and the result is a fall in the factor demand for input 1 and a fall in y.
In the second graph,p increases to p0 and the result is a rise in the demand for input 1 and a rise in the output y.
Changing w2 has no efiect upon the slope and so nothing changes in the short run | except proflt levels.
Production | Firms 6
Long Run Proflt Maximisation
In the long run all factors are variable,The flrm solves the following problem:
maxx
1;x2
pf(x1;x2)?w1x1?w2x2 =) pMP1(x?1;x?2)= w1 and pMP1(x?1;x?2)= w2
Given the optimal amount of input 2 (x?2) the function pMP1(x1;x?2)= w1 deflnes the inverse factor demand curve
which is the relationship between optimal factor demand and factor price,derived from the previous slide.
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pMP1(x1;x?2)= w1
0
w1
x1
The factor demand curve is always downward sloping,There are no \Gifien" production inputs,Why not?
A proflt maximising flrm with constant returns to scale makes zero proflt,Why?
Production | Firms 7
Cost Minimisation
An equivalent way to think about the flrm’s production decision is cost minimisation.
minx
1;x2
w1x1 +w2x2 subject to f(x1;x2)? y
That is,the flrm minimises their cost of producing at least an output of y,Isocost curves connect all the input
combinations of equal cost and are linear.
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y = y
Isocost lines w1x1 +w2x2 = c
The second graph is an isoquant showing the constraint in the above equation,f(x1;x2)? y at all points above it.
The similarity with consumer theory is striking,The slope of the isocost lines is?w1=w2.
Production | Firms 8
The TRS Condition
The isocost line is moved downward as far as it will go giving another tangent condition.
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The isoquant slope (the TRS) is set equal to the isocost slope giving the TRS condition:
TRS=?MP1(x
1;x?2)
MP2(x?1;x?2) =?
w1
w2
This is exactly what proflt maximisation had revealed,The two methods are equivalent.
The conditional factor demands are the optimal factor demands in terms of the amount of output the flrm wishes to
produce and the factor prices,x1(w1;w2;y) and x2(w1;w2;y).
Production | Firms 9
The Cost Function
The (long run) cost function is the minimum cost required to produce a certain output,It solves the problem:
c(y)= minx
1;x2
(w1x1 +w2x2) subject to f(x1;x2)? y
It can be written in terms of the conditional factor demands,c(y)= w1x1(w1;w2;y)+w2x2(w1;w2;y).
The short run cost function is the solution to the same problem when one of the factors is flxed,e.g,x2 = x2.
Constant costs that have to be paid no matter which level of output is produced are called flxed costs,Constant
costs that have to paid no matter which level of output is produced as long as it is not zero are called quasi-flxed
costs,All other costs are called variable costs,For example:
c(y)=
8<
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1+k+y2 if y > 0
1+y2 if y =0
The above example has flxed costs of 1,quasi-flxed costs of k and variable costs of y2.
There are no flxed costs in the long run,although there could be quasi-flxed costs.
Production | Firms 10
Average Cost
Costs can always be written as c(y)= cv(y)+F where cv(y) is variable and includes any quasi-flxed costs.
Average cost is the minimum cost per unit of producing a total output y,Hence:
AC(y)= c(y)y = cv(y)y + Fy = AVC(y)+AFC(y)=Average variable cost+Average flxed cost
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AFC slopes downward,AVC will eventually slope upward,So AC is usually drawn with a quadratic shape.
Production | Firms 11
Marginal Cost
Marginal cost measures the change in cost given a small change in output,So:
MC(y)= ￠c(y)￠y = c(y+￠y)?c(y)￠y = ￠cv(y)￠y
The last equality follows because flxed costs do not change as output changes,by deflnition.
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MC
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Variable cost
The marginal cost curve cuts the average cost and the average variable cost curve at their minimum points,Why?
The average variable cost curve and the marginal cost curve take the same value for the flrst unit of output,Why?
The area underneath the marginal cost curve is equal to variable costs,Why?
Production | Firms 12
Long Run Cost Curves
When all factors are variable long run average costs must be no larger than short run average costs.
The long run average cost curve is the lower envelope of all the short run average cost curves.
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Costs
y 0
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y
LRAC
SRAC
y? y?
LRAC
SRMC
LRMC
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Notice the minima of the SRAC curves do not intercept the LRAC curve except at the minimum,Why?
The second graph shows the connections between long run marginal cost and the other cost curves.
The LRMC curve is simply the collection of points from the \optimal" SRMC curves at each output level y?.
