IntermediatemicroeconomicsLecture 3: Production theory.
Varian, chapters 19-24
Part 1: Profit maximization
1. Technologya) Production quantity and production functionb) Marginal product and technical rate of substitutionc) Short run and long rund) Returns to scale
2. Profit maximizationa) Profitb) Profit maximization in the short run and in the long
runc) Profit maximization and returns to scale
Adam Jacobsson, Department of Economics2017-01-27 2
Part 2: Cost minimization and supply
3. Cost minimization
4. Cost functions and returns to scale
5. Sunk costs
6. Cost curves
7. Firm supply in the short run
8. Profit and producer surplus
9. Firm supply in the long run
10. Market supply
Adam Jacobsson, Department of Economics2017-01-27 3
1. Technology
β Inputs (factors of production): land, labour, capital (physical and
financial).
β The technological constraints are described by the production set:
Definition 1: The production set contains all possible combinations of
outputs and inputs.
Definition 2: The production function f measures the maximum possible
output of good y for any given amount of inputs (x1,x2):
y=f(x1,x2)
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Production set and production function with one input
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Output
Input
Productionset
Productionfunctiony=f(x)
Assumptions about the technologyβs propertiesβ Monotonicity: If the amount of one input
increases, output will increase or remain unchanged.
β Free disposal: The firm can costlessly dispose of any inputs.
β Convexity: If it is possible to produce y with inputs X=(x1,x2)or Z=(z1,z2),then it is possible to produce y with inputs H= ππ + 1 β π π, π β [0,1]
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h2
An isoquant consists of all combinations of inputs that are
just sufficient to produce a given quantity of output.
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Input2
Input1h1π₯1 z1
π₯2
z2Isoquantyβ>y
Isoquanty
X
Z
ConvexityimpliesthatusinginputsH=(h1,h2)leadstoanoutputatleastaslargeasy.
Marginal product (MP)
β For any given combination of inputs, the MP measures how much
output changes in relation to an change in the amount of input i:
πππ π₯1, π₯2 =ππ¦ππ₯π =
ππ(π₯1, π₯2)ππ₯π
β Assumption about decreasing MP: MP for one input decreases as the
amount of this input increases, given that the amounts of all other
inputs remain constant.
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Technical rate of substitution (TRS)
For a given combination of inputs: Which change in the amounts of inputs is consistent with an unchanged output level?
ππ¦ =ππ(π₯1, π₯2)
ππ₯1ππ₯1 +
ππ(π₯1, π₯2)ππ₯2
ππ₯2 = 0
= ππ1ππ₯1 + ππ2ππ₯2 = 0
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TRS, continued
βΉ ππ π π₯1, π₯2 =ππ₯2ππ₯1
= βππ1 π₯1, π₯2ππ2 π₯1, π₯2
β The TRS equals the slope of the isoquant, that is, how much less of input 2 is needed if the firm uses one more unit of input 1 and output is fixed.
β Assumption about decreasing TRS: the slope of the isoquant decreases in absolute terms (that is, it becomes less negative).
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Short run and long run
β Fixed factor: can only be used in a fixed amount. The
firm cannot abstain from the input even if nothing is
produced.
β Variable factor: can be used in different amounts. The
firm can abstain from the input if nothing is produced.
β Quasifixed factor: is needed in a fixed amount
independent of how much is produced, but if nothing is
produced the firm can abstain from this input.
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Short and long run, continued
β Short run: One or more inputs are fixed (for example physical capital like factory buildings).
β Long run: All inputs can be varied freely, for example by setting up new production facilities (factories for example). The firm can also choose zero inputs to produce zero output.
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Returns to scaleIf the amounts of all inputs are scaled up by a factor t>1, by
how much does output increase?
Constant returns to scale (CRS):
Output is also scaled up by a factor t:
f(tx1,tx2)=tf(x1,x2)
Increasing returns to scale (IRS):
Output is scaled up by more than a factor t:
f(tx1,tx2)>tf(x1,x2)
Decreasing returns to scale (DRS):
Output is scaled up by less than a factor t:
f(tx1,tx2)<tf(x1,x2)
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Example!
2. Profit maximization
β Profit = revenues β costs.
β In economics the concept of profit implies that all inputs and outputs
are valued according to their opportunity costs.
β Hence, an input has to be valued according to its best alternative use
instead of being valued according to its acquisition value.β The cost of a machine is measured in terms of what it would cost
to rent during the time it is used.β If there is no well-functioning machine market: the cost of use is
then the price of the machine at the beginning of the production minus the machineβs selling price after production.
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β A firm that produces n goods by using m inputs makes the following profit:
Ξ =Mπππ¦π
O
PQR
βMπ€ππ₯π
U
VQR
Where pi is the price of good i and wj is the price of input j and π¦π = π(π₯1, π₯2, β¦ , π₯U[R, π₯π)
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Short run profit maximization
Assume two inputs, where input 2 is fixed, i.e. x2=x]2.The optimization problem:
max_R
Ξ π = ππ(π₯1, x]2) β π€1π₯1 β π€2x]2FOC: πΞ π
ππ₯1= π
ππ(π₯Rβ, x]2)ππ₯1
β π€1 = 0
β πππ1(π₯Rβ, x]2) = π€1
The market value of the marginal product of the input has to equal the price of this input (assuming decreasing ππ1).
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Isoprofit linesProfit is given by: Ξ c = ππ¦ β π€1π₯1 β π€2x]2By solving for y we obtain isoprofit lines:
π¦ =Ξ c
π +π€2
π x]2 +π€1
π π₯1
The slope can also be obtained in the following way:πΞ c = πππ¦ β π€1ππ₯1 = 0
βππ¦ππ₯R
e fghQi =π€1
πTheslope of theisoprofit lines expresshow much outputchanges inresponse toachange intheamount of input,giventhat profitremains constant.
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Addπ€1π₯1 + π€2x]2toLHS&RHSanddividebyp!
Slope
Profit maximization in the short run
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Output,π¦
Input1
Productionset
Productionfunctiony=f(x1,x]2)
IsoprofitlinewithslopelR
m
Ξ c
π +π€2
π x]2
y*
x1*
Ξ cβ
Profitmax Attheprofitmaxpointthefollowingistrue:lR
m= ππ1(π₯Rβ, x]2)
Long run profit maximizationNo input is fixed now!
The optimization problem:max_R,_o
Ξ = ππ(π₯1, π₯2) β π€1π₯1 β π€2π₯2
FOC:
pgp_R
= π pq(_rβ,_sβ)
p_Rβ π€1 = 0 (1)
pgp_o
= π pq(_rβ,_sβ)
p_oβ π€2 = 0 (2)
Rearrange (1) & (2)!
β πππ1(π₯Rβ, π₯oβ) = π€1 &
πππ2(π₯Rβ, π₯oβ) = π€2
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β Hence, the value of the marginal product of each input should equal its price.
β From conditions (1) and (2) the optimal solutions π₯Rβandπ₯oβ can be obtained.
β By varying p, w1 and w2 we obtain the factor demand functions π₯Rβ(π, π€1, π€2) and π₯oβ π, π€1, π€2 !
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w1
The inverse factor demand curve of input 1 measures what
the price of input 1 must be for a given quantity of input 1 to
be demanded, given the optimal choice of input 2 (π₯oβ).
πππ1(π₯Rβ, π₯oβ) = π€1
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Factorpriceinput1(=w1)
Input1π₯Rβ
Factordemandcurveforinput1
pMP1(x1,π₯oβ)
3. Cost minimization
An isocost curve consists of inputs 1 and 2, x1 and x2, for which costs are constant (=C).
π€1π₯1 + π€2π₯2=πΆOr (solving for π₯2) : π₯2 =
| loβ lR
loπ₯1
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Slope
π₯oβ
The cost minimization problem, continued
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Input2
Input1π₯Rβ
Isoquantπ(π₯1, π₯2)=π¦} withslopeTRS=β~οΏ½R _R,_o
~οΏ½o _R,_o
Isocostlineswithslope-lR
lo
π₯2 =πΆπ€2
βπ€1
π€2π₯1
Attheoptimum:-lR
lo=TRS
The cost minimization problem, continuedMinimize costs to attain a given production level π¦]
min_R,_o
π€1π₯1 + π€2π₯2π . π‘. π π₯1, π₯2 = π¦]
Set up the Lagrangian:π π₯1, π₯2, π = π€1π₯1 + π€2π₯2 β π π π₯1, π₯2 β π¦]
FOC:p~p_r
= π€1 β πβpq _rβ,_sβ
p_r= 0 (i)
p~p_s
= π€2 β πβpq _rβ,_sβ
p_s= 0 (ii)
p~pοΏ½= β π π₯Rβ, π₯oβ β π¦] = 0 (iii)
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Rearrange (i) and (ii):
π€1 = πβ pq _rβ,_sβ
p_r(i)
π€2 = πβ pq _rβ,_sβ
p_s(ii)
Divide (i) by (ii):
π€1
π€2=
ππ π₯Rβ, π₯oβππ₯R
ππ π₯Rβ, π₯oβππ₯o
=ππ1ππ2οΏ½
[οΏ½οΏ½οΏ½ _rβ,_sβ
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Byaddingπβ pq _rβ,_sβ
p_r
andπβ pq _rβ,_sβ
p_sto
bothRHSandLHSrespectively.
β The optimal solutions π₯Rβ π€1, π€2, y andπ₯oβ π€1, π€2, yare the conditional factor demand equations.
β Note the difference between these demand equations and the ones we got from profit maximization: π₯Rβ π€1, π€2, π andπ₯oβ π€1, π€2, π.
β The cost function:
π π€1, π€2, π¦ = π€1π₯Rβ π€1, π€2, y + π€2π₯oβ π€1, π€2, y
measures the minimal cost to produce y given factor prices w1 and w2.
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4. Cost functions and returns to scale
β Assume constant returns to scale (CRS).
β Solve the cost minimization problem for y=1.
β We then obtain the unit cost function c(w1,w2,1).
β If we produce y>1 units, CRS implies that we have to scale up the amounts of inputs by y. Thus, costs will be scaled up by y:
π π€1, π€2, π¦ = π¦π π€1, π€2,1
That is, costs are proportional to y when we have CRS.
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β If we have IRS, costs increase less than proportionately:
π π€1, π€2, π¦ < π¦π π€1, π€2,1
β If we have DRS, costs increase more than
proportionately:
π π€1, π€2, π¦ > π¦π π€1, π€2,1
β Define average costs:π΄πΆ π¦ =
π π€1, π€2, π¦π¦
β For y>1:
π΄πΆ π¦ > π π€1, π€2,1 if DRS
π΄πΆ π¦ = π π€1, π€2,1 if CRS
π΄πΆ π¦ < π π€1, π€2,1 if IRS
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Costofproducingthefirstunit
5. Sunk costs
Definition 1. A sunk cost is a payment that cannot be
recovered.
Example:
A firm uses SEK 100 000 to purchase furniture. At the end of
the year the furniture can be sold at a price of 80 000.
The sunk cost is the reduction in value, that is, 20 000.
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6. Cost curves
β The total cost for producing y is given by:π π¦ = ππ£ π¦ + πΉ
Where cv(y)is the variable cost for producing y, and F is the fixed cost.β The average cost is given by:
π΄πΆ π¦ =π π¦π¦ =
ππ£ π¦π¦
οΏ½οΏ½|(οΏ½)
+πΉπ¦β
οΏ½οΏ½|(οΏ½)
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β The marginal cost is given by:
ππΆ π¦ =ππ π¦ππ¦ =
πππ£ π¦ππ¦
β For y=0 we have:ππΆ 0 = π΄ππΆ(0)
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β How is the average variable cost affected by changes in the scale of production?
ππ΄ππΆ(π¦)ππ¦ =
π(ππ£ π¦π¦ )
ππ¦ =
Remember the following rule: if we have f(x)g(x), then π(π π₯ π π₯ )
ππ₯ = ποΏ½ π₯ π π₯ + π π₯ ποΏ½ π₯
Also, οΏ½οΏ½ οΏ½οΏ½
can be written asππ£ π¦q(οΏ½)
π¦[RοΏ½οΏ½(_)
=πππ£ π¦ππ¦ π¦[R β ππ£ π¦ π¦[o =
=πππ£ π¦ππ¦
1π¦ β ππ£ π¦
1π¦2 =
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Factor out RοΏ½:
=1π¦πππ£ π¦ππ¦ β
ππ£ π¦π¦ =
=1π¦ ππΆ(π¦) β π΄ππΆ(π¦)
For a given y the following applies:
β If MC<AVC:AVCdecreases.β If MC=AVC:AVCis constant.β If MC>AVC:AVCincreases.
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β How is the average cost affected by changes in the scale of production?
π΄πΆ π¦ =ππ£(π¦)π¦ +
πΉπ¦
ππ΄πΆ(π¦)ππ¦ =
ππ΄ππΆππ¦ +
ππ΄πΉπΆππ¦ =
=π(ππ£ π¦π¦ )
ππ¦ +π(πΉπ¦)
ππ¦ =
=πππ£ π¦ππ¦
1π¦ β ππ£ π¦
1π¦2 β
πΉπ¦2 =
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Factor out RοΏ½:
=1π¦πππ£ π¦ππ¦~|
βππ£ π¦ + πΉ
π¦οΏ½οΏ½|οΏ½οΏ½οΏ½|
οΏ½|
=1π¦ ππΆ π¦ β π΄πΆ(π¦)
For a given y the following applies:β If MC<AC: AC decreases.β If MC=AC: AC is constant.β If MC>AC: AC increases.
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MC, AVC and AC
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MC,AVC,AC
y
MC
AVC(0)=MC(0)
AC AVC
TheMC curvecrossestheAVC andAC curvesattheirlowestpoints!SeepreviousconditionsrelatingMC,AVCandAC!
Cost in the short and long runβ Let k (a fixed input like capital β previously we called this x]2) be fixed in the short run.
β The cost function in the short run is given by ππ π¦, π .
β The cost function in the long run is given by π π¦ .
β The cost of producing y in the short run is at least as large as the cost of producing y in
the long run, since k can always be adjusted in the long run:
π π¦ β€ ππ π¦, π
Let πβ = π(π¦β)be the optimal value of k for π¦β (i.e. for some given value of y). Hence, for πβwe
have π π¦β = ππ π¦β, πβ
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AC and MC in the short run
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SAC,SMC
y
SAC1SMC1
SAC2SMC2 SAC3
SMC3
SAC4SMC4
SAC5SMC5
π¦cRβ π¦coβ π¦cΒ¦β π¦cΒ§β π¦cΒ¨β
Onebakery
Twobakeries
Threebakeries
Fourbakeries
Fivebakeries
AC and MC in the short and long run
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SAC,SMC,LAC,LMC
y
SAC3SMC3
LACLMC
7. Firm supply in the short run
β The firmβs decision about how much to produce is constrained by:β Technology (the cost function)β Market conditions
β Assume perfect competition (many firms):β Price is taken as given (does not depend on
the firmβs choice of output).β No strategic interaction!
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The firmβs output decision in a market with perfect competitionβ The optimization problem:
maxοΏ½
Ξ (π¦) = π (π¦) β π π¦
FOC:πΞ (π¦)ππ¦ =
ππ (π¦β)ππ¦
~οΏ½(οΏ½β)
βππ π¦β
ππ¦~| οΏ½β
= 0, ππ
ππ π¦β = ππΆ(π¦β)β Since we have R(y)=py in a market with perfect competition:
ππ (π¦β)ππ¦ = π
β The FOC under perfect competition can thus be expressed as:π = ππΆ(π¦β)
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β If p>MC, the firm can increase profits by increasing supply.
β If p<MC, the firm can increase profits by decreasing supply.
β Note that p=MC is a necessary, but not a sufficient condition for profit maximization.
β It has to be profitable to produce something!β If p=MC<AVC, the firm cannot cover its
variable costs. Therefore this part of the MC-curve is not part of the firmβs supply curve.
β The firmβs supply curve is thus given by the part of the MC-curve that lies above AVC, i.e. where MCβ₯AVC.
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The firmβs supply curve in the short run
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MC,AVC,AC
y
MC
AVC(0)=MC(0)
AC AVC
Shortrunsupplycurve
8. Profit and producer surplusβ Profit = Revenue minus total costs
Ξ = ππ¦ β ππ£ π¦ β πΉ
β Producer surplus = revenues minus variable costs
ππ = ππ¦ β ππ£ π¦
β Production should cease if the PS is negative, i.e. if variable costs
exceed revenues:
ππ < 0 βΊ ππ¦ β ππ£ π¦ < 0 βΊ ππ¦ < ππ£ π¦
Since ππ£ π¦ = π΄ππΆ π¦ π¦ we thus obtain the following shutdown condition:
ππ¦ < π΄ππΆ π¦ π¦, or
π < π΄ππΆ π¦
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β If the market price of output is lower than the average variable cost, production ceases.
β However, there is an interval of prices,π΄ππΆ π¦ β€ π < π΄πΆ π¦ ,
for which producer surplus is positive, but profit is negative.
β In this case production is not shut down despite the fact that a loss has occured, because revenues exceed variable costs. The producer gets some revenue to pay at least a part of the fixed costs.
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9. Firm supply in the long run
β In the long run all inputs can be varied.
β In the long run it is also possible to shut down production.
β Profits must therefore be non-negative:Ξ = ππ¦ β π π¦ β₯ 0, ππ
π β₯π π¦π¦ = πΏπ΄πΆ π¦
i.e. price must be at least as large as long run average costs.β Thefirmβslongrunsupplycurveisthereforegivenbythesectionofthe
LMC thatliesabovetheLAC.
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The firmβs supply curve in the long run
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LMC,LAC
y
LMC
LACmin=LAC(ymin)
LAC
----Longrunsupplycurve
ymin
Levelofproductionwithminimallongrununitcost,LACmin.
10. Market supply (Industry supply)β Market supply with n firms is given by:
π π =MπP(π)O
PQRWhere πP(π) is firm iβs supply at output price p.β In the short run, market supply consists both of firms
that make a loss and of firms that make profits. In the long run, however, firms can adjust fixed inputs. Firms making a loss will quit the market.
β In the long run, firms that use the technology of profitable firms will enter the market, given βfree entryβ, putting downward pressure on the market price.
β If there are sufficiently many firms in the long run, the equilibrium market price will be close to the minimal unit cost, LACmin. Profits of firms will then go to zero.
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Market supply curve in the long run
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Price
Quantity
LACmin
ymin 3ymin 4ymin2ymin
S1
S1+ S2
S1+ S1 +S1
Supplyoffirm1
Supplyoffirms1&2
Supplyoffirms1,2&3