Intermediate microeconomics - s u

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

● Inputs (factors of production): land, labour, capital (physical and

financial).

● 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)

2017-01-27 Adam Jacobsson, Department of Economics 4

Production set and production function with one input


Output

Input

Productionset

Productionfunctiony=f(x)

Assumptions about the technology’s properties● Monotonicity: If the amount of one input

increases, output will increase or remain unchanged.

● Free disposal: The firm can costlessly dispose of any inputs.

● 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= 𝜆𝑋 + 1 − 𝜆 𝑍, 𝜆 ∈ [0,1]


h2

An isoquant consists of all combinations of inputs that are

just sufficient to produce a given quantity of output.


Input2

Input1h1𝑥1 z1

𝑥2

z2Isoquanty’>y

Isoquanty

X

Z

ConvexityimpliesthatusinginputsH=(h1,h2)leadstoanoutputatleastaslargeasy.

Marginal product (MP)

● For any given combination of inputs, the MP measures how much

output changes in relation to an change in the amount of input i:

𝑀𝑃𝑖 𝑥1, 𝑥2 =𝜕𝑦𝜕𝑥𝑖 =

𝜕𝑓(𝑥1, 𝑥2)𝜕𝑥𝑖

● 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.


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?

𝑑𝑦 =𝜕𝑓(𝑥1, 𝑥2)

𝜕𝑥1𝑑𝑥1 +

𝜕𝑓(𝑥1, 𝑥2)𝜕𝑥2

𝑑𝑥2 = 0

= 𝑀𝑃1𝑑𝑥1 + 𝑀𝑃2𝑑𝑥2 = 0


TRS, continued

⟹ 𝑇𝑅𝑆 𝑥1, 𝑥2 =𝑑𝑥2𝑑𝑥1

= −𝑀𝑃1 𝑥1, 𝑥2𝑀𝑃2 𝑥1, 𝑥2

● 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.

● Assumption about decreasing TRS: the slope of the isoquant decreases in absolute terms (that is, it becomes less negative).


Short run and long run

● Fixed factor: can only be used in a fixed amount. The

firm cannot abstain from the input even if nothing is

produced.

● Variable factor: can be used in different amounts. The

firm can abstain from the input if nothing is produced.

● 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.


Short and long run, continued

● Short run: One or more inputs are fixed (for example physical capital like factory buildings).

● 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.


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)


Example!

2. Profit maximization

● Profit = revenues – costs.

● In economics the concept of profit implies that all inputs and outputs

are valued according to their opportunity costs.

● Hence, an input has to be valued according to its best alternative use

instead of being valued according to its acquisition value.– The cost of a machine is measured in terms of what it would cost

to rent during the time it is used.– 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 machine’s selling price after production.


● A firm that produces n goods by using m inputs makes the following profit:

Π =M𝑝𝑖𝑦𝑖

O

PQR

−M𝑤𝑗𝑥𝑗

U

VQR

Where pi is the price of good i and wj is the price of input j and 𝑦𝑖 = 𝑓(𝑥1, 𝑥2, … , 𝑥U[R, 𝑥𝑚)


Short run profit maximization

Assume two inputs, where input 2 is fixed, i.e. x2=x]2.The optimization problem:

max_R

Π𝑠 = 𝑝𝑓(𝑥1, x]2) − 𝑤1𝑥1 − 𝑤2x]2FOC: 𝜕Π𝑠

𝜕𝑥1= 𝑝

𝜕𝑓(𝑥R∗, x]2)𝜕𝑥1

− 𝑤1 = 0

⇔ 𝑝𝑀𝑃1(𝑥R∗, x]2) = 𝑤1

The market value of the marginal product of the input has to equal the price of this input (assuming decreasing 𝑀𝑃1).


Isoprofit linesProfit is given by: Πc = 𝑝𝑦 − 𝑤1𝑥1 − 𝑤2x]2By solving for y we obtain isoprofit lines:

𝑦 =Πc

𝑝 +𝑤2

𝑝 x]2 +𝑤1

𝑝 𝑥1

The slope can also be obtained in the following way:𝑑Πc = 𝑝𝑑𝑦 − 𝑤1𝑑𝑥1 = 0

⇒𝑑𝑦𝑑𝑥R

e fghQi =𝑤1

𝑝Theslope of theisoprofit lines expresshow much outputchanges inresponse toachange intheamount of input,giventhat profitremains constant.


Add𝑤1𝑥1 + 𝑤2x]2toLHS&RHSanddividebyp!

Slope

Profit maximization in the short run


Output,𝑦

Input1

Productionset

Productionfunctiony=f(x1,x]2)

IsoprofitlinewithslopelR

m

Πc

𝑝 +𝑤2

𝑝 x]2

y*

x1*

Πc↑

Profitmax Attheprofitmaxpointthefollowingistrue:lR

m= 𝑀𝑃1(𝑥R∗, x]2)

Long run profit maximizationNo input is fixed now!

The optimization problem:max_R,_o

Π = 𝑝𝑓(𝑥1, 𝑥2) − 𝑤1𝑥1 − 𝑤2𝑥2

FOC:

pgp_R

= 𝑝 pq(_r∗,_s∗)

p_R− 𝑤1 = 0 (1)

pgp_o

= 𝑝 pq(_r∗,_s∗)

p_o− 𝑤2 = 0 (2)

Rearrange (1) & (2)!

⇔ 𝑝𝑀𝑃1(𝑥R∗, 𝑥o∗) = 𝑤1 &

𝑝𝑀𝑃2(𝑥R∗, 𝑥o∗) = 𝑤2


● Hence, the value of the marginal product of each input should equal its price.

● From conditions (1) and (2) the optimal solutions 𝑥R∗and𝑥o∗ can be obtained.

● By varying p, w1 and w2 we obtain the factor demand functions 𝑥R∗(𝑝, 𝑤1, 𝑤2) and 𝑥o∗ 𝑝, 𝑤1, 𝑤2 !


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 (𝑥o∗).

𝑝𝑀𝑃1(𝑥R∗, 𝑥o∗) = 𝑤1


Factorpriceinput1(=w1)

Input1𝑥R∗

Factordemandcurveforinput1

pMP1(x1,𝑥o∗)

3. Cost minimization

An isocost curve consists of inputs 1 and 2, x1 and x2, for which costs are constant (=C).

𝑤1𝑥1 + 𝑤2𝑥2=𝐶Or (solving for 𝑥2) : 𝑥2 =

| lo− lR

lo𝑥1


Slope

𝑥o∗

The cost minimization problem, continued


Input2

Input1𝑥R∗

Isoquant𝑓(𝑥1, 𝑥2)=𝑦} withslopeTRS=−~�R _R,_o

~�o _R,_o

Isocostlineswithslope-lR

lo

𝑥2 =𝐶𝑤2

−𝑤1

𝑤2𝑥1

Attheoptimum:-lR

lo=TRS

The cost minimization problem, continuedMinimize costs to attain a given production level 𝑦]

min_R,_o

𝑤1𝑥1 + 𝑤2𝑥2𝑠. 𝑡. 𝑓 𝑥1, 𝑥2 = 𝑦]

Set up the Lagrangian:𝑀 𝑥1, 𝑥2, 𝜇 = 𝑤1𝑥1 + 𝑤2𝑥2 − 𝜇 𝑓 𝑥1, 𝑥2 − 𝑦]

FOC:p~p_r

= 𝑤1 − 𝜇∗pq _r∗,_s∗

p_r= 0 (i)

p~p_s

= 𝑤2 − 𝜇∗pq _r∗,_s∗

p_s= 0 (ii)

p~p�= − 𝑓 𝑥R∗, 𝑥o∗ − 𝑦] = 0 (iii)


Rearrange (i) and (ii):

𝑤1 = 𝜇∗ pq _r∗,_s∗

p_r(i)

𝑤2 = 𝜇∗ pq _r∗,_s∗

p_s(ii)

Divide (i) by (ii):

𝑤1

𝑤2=

𝜕𝑓 𝑥R∗, 𝑥o∗𝜕𝑥R

𝜕𝑓 𝑥R∗, 𝑥o∗𝜕𝑥o

=𝑀𝑃1𝑀𝑃2�

[�� _r∗,_s∗


Byadding𝜇∗ pq _r∗,_s∗

p_r

and𝜇∗ pq _r∗,_s∗

p_sto

bothRHSandLHSrespectively.

● The optimal solutions 𝑥R∗ 𝑤1, 𝑤2, y and𝑥o∗ 𝑤1, 𝑤2, yare the conditional factor demand equations.

● Note the difference between these demand equations and the ones we got from profit maximization: 𝑥R∗ 𝑤1, 𝑤2, 𝑝 and𝑥o∗ 𝑤1, 𝑤2, 𝑝.

● The cost function:

𝑐 𝑤1, 𝑤2, 𝑦 = 𝑤1𝑥R∗ 𝑤1, 𝑤2, y + 𝑤2𝑥o∗ 𝑤1, 𝑤2, y

measures the minimal cost to produce y given factor prices w1 and w2.


4. Cost functions and returns to scale

● Assume constant returns to scale (CRS).

● Solve the cost minimization problem for y=1.

● We then obtain the unit cost function c(w1,w2,1).

● 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:

𝑐 𝑤1, 𝑤2, 𝑦 = 𝑦𝑐 𝑤1, 𝑤2,1

That is, costs are proportional to y when we have CRS.


● If we have IRS, costs increase less than proportionately:

𝑐 𝑤1, 𝑤2, 𝑦 < 𝑦𝑐 𝑤1, 𝑤2,1

● If we have DRS, costs increase more than

proportionately:

𝑐 𝑤1, 𝑤2, 𝑦 > 𝑦𝑐 𝑤1, 𝑤2,1

● Define average costs:𝐴𝐶 𝑦 =

𝑐 𝑤1, 𝑤2, 𝑦𝑦

● For y>1:

𝐴𝐶 𝑦 > 𝑐 𝑤1, 𝑤2,1 if DRS

𝐴𝐶 𝑦 = 𝑐 𝑤1, 𝑤2,1 if CRS

𝐴𝐶 𝑦 < 𝑐 𝑤1, 𝑤2,1 if IRS


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.


6. Cost curves

● The total cost for producing y is given by:𝑐 𝑦 = 𝑐𝑣 𝑦 + 𝐹

Where cv(y)is the variable cost for producing y, and F is the fixed cost.● The average cost is given by:

𝐴𝐶 𝑦 =𝑐 𝑦𝑦 =

𝑐𝑣 𝑦𝑦

��|(�)

+𝐹𝑦⏟

��|(�)


● The marginal cost is given by:

𝑀𝐶 𝑦 =𝜕𝑐 𝑦𝜕𝑦 =

𝜕𝑐𝑣 𝑦𝜕𝑦

● For y=0 we have:𝑀𝐶 0 = 𝐴𝑉𝐶(0)


● How is the average variable cost affected by changes in the scale of production?

𝜕𝐴𝑉𝐶(𝑦)𝜕𝑦 =

𝜕(𝑐𝑣 𝑦𝑦 )

𝜕𝑦 =

Remember the following rule: if we have f(x)g(x), then 𝜕(𝑓 𝑥 𝑔 𝑥 )

𝜕𝑥 = 𝑓� 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔� 𝑥

Also, ��

can be written as𝑐𝑣 𝑦q(�)

𝑦[R��(_)

=𝜕𝑐𝑣 𝑦𝜕𝑦 𝑦[R − 𝑐𝑣 𝑦 𝑦[o =

=𝜕𝑐𝑣 𝑦𝜕𝑦

1𝑦 − 𝑐𝑣 𝑦

1𝑦2 =


Factor out R�:

=1𝑦𝜕𝑐𝑣 𝑦𝜕𝑦 −

𝑐𝑣 𝑦𝑦 =

=1𝑦 𝑀𝐶(𝑦) − 𝐴𝑉𝐶(𝑦)

For a given y the following applies:

● If MC<AVC:AVCdecreases.● If MC=AVC:AVCis constant.● If MC>AVC:AVCincreases.


● How is the average cost affected by changes in the scale of production?

𝐴𝐶 𝑦 =𝑐𝑣(𝑦)𝑦 +

𝐹𝑦

𝑑𝐴𝐶(𝑦)𝑑𝑦 =

𝜕𝐴𝑉𝐶𝜕𝑦 +

𝜕𝐴𝐹𝐶𝜕𝑦 =

=𝜕(𝑐𝑣 𝑦𝑦 )

𝜕𝑦 +𝜕(𝐹𝑦)

𝜕𝑦 =

=𝜕𝑐𝑣 𝑦𝜕𝑦

1𝑦 − 𝑐𝑣 𝑦

1𝑦2 −

𝐹𝑦2 =


Factor out R�:

=1𝑦𝜕𝑐𝑣 𝑦𝜕𝑦~|

−𝑐𝑣 𝑦 + 𝐹

𝑦��|��|

�|

=1𝑦 𝑀𝐶 𝑦 − 𝐴𝐶(𝑦)

For a given y the following applies:● If MC<AC: AC decreases.● If MC=AC: AC is constant.● If MC>AC: AC increases.


MC, AVC and AC


MC,AVC,AC

y

MC

AVC(0)=MC(0)

AC AVC

TheMC curvecrossestheAVC andAC curvesattheirlowestpoints!SeepreviousconditionsrelatingMC,AVCandAC!

Cost in the short and long run● Let k (a fixed input like capital – previously we called this x]2) be fixed in the short run.

● The cost function in the short run is given by 𝑐𝑠 𝑦, 𝑘 .

● The cost function in the long run is given by 𝑐 𝑦 .

● 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:

𝑐 𝑦 ≤ 𝑐𝑠 𝑦, 𝑘

Let 𝑘∗ = 𝑘(𝑦∗)be the optimal value of k for 𝑦∗ (i.e. for some given value of y). Hence, for 𝑘∗we

have 𝑐 𝑦∗ = 𝑐𝑠 𝑦∗, 𝑘∗


AC and MC in the short run


SAC,SMC

y

SAC1SMC1

SAC2SMC2 SAC3

SMC3

SAC4SMC4

SAC5SMC5

𝑦cR∗ 𝑦co∗ 𝑦c¦∗ 𝑦c§∗ 𝑦c¨∗

Onebakery

Twobakeries

Threebakeries

Fourbakeries

Fivebakeries

AC and MC in the short and long run


SAC,SMC,LAC,LMC

y

SAC3SMC3

LACLMC

7. Firm supply in the short run

● The firm’s decision about how much to produce is constrained by:– Technology (the cost function)– Market conditions

● Assume perfect competition (many firms):– Price is taken as given (does not depend on

the firm’s choice of output).– No strategic interaction!


The firm’s output decision in a market with perfect competition● The optimization problem:

max�

Π(𝑦) = 𝑅(𝑦) − 𝑐 𝑦

FOC:𝑑Π(𝑦)𝑑𝑦 =

𝜕𝑅(𝑦∗)𝜕𝑦

~�(�∗)

−𝜕𝑐 𝑦∗

𝜕𝑦~| �∗

= 0, 𝑜𝑟

𝑀𝑅 𝑦∗ = 𝑀𝐶(𝑦∗)● Since we have R(y)=py in a market with perfect competition:

𝜕𝑅(𝑦∗)𝜕𝑦 = 𝑝

● The FOC under perfect competition can thus be expressed as:𝑝 = 𝑀𝐶(𝑦∗)


● If p>MC, the firm can increase profits by increasing supply.

● If p<MC, the firm can increase profits by decreasing supply.

● Note that p=MC is a necessary, but not a sufficient condition for profit maximization.

● It has to be profitable to produce something!– If p=MC<AVC, the firm cannot cover its

variable costs. Therefore this part of the MC-curve is not part of the firm’s supply curve.

– The firm’s supply curve is thus given by the part of the MC-curve that lies above AVC, i.e. where MC≥AVC.


The firm’s supply curve in the short run


MC,AVC,AC

y

MC

AVC(0)=MC(0)

AC AVC

Shortrunsupplycurve

8. Profit and producer surplus● Profit = Revenue minus total costs

Π = 𝑝𝑦 − 𝑐𝑣 𝑦 − 𝐹

● Producer surplus = revenues minus variable costs

𝑃𝑆 = 𝑝𝑦 − 𝑐𝑣 𝑦

● Production should cease if the PS is negative, i.e. if variable costs

exceed revenues:

𝑃𝑆 < 0 ⟺ 𝑝𝑦 − 𝑐𝑣 𝑦 < 0 ⟺ 𝑝𝑦 < 𝑐𝑣 𝑦

Since 𝑐𝑣 𝑦 = 𝐴𝑉𝐶 𝑦 𝑦 we thus obtain the following shutdown condition:

𝑝𝑦 < 𝐴𝑉𝐶 𝑦 𝑦, or

𝑝 < 𝐴𝑉𝐶 𝑦


● If the market price of output is lower than the average variable cost, production ceases.

● However, there is an interval of prices,𝐴𝑉𝐶 𝑦 ≤ 𝑝 < 𝐴𝐶 𝑦 ,

for which producer surplus is positive, but profit is negative.

– 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.


9. Firm supply in the long run

● In the long run all inputs can be varied.

● In the long run it is also possible to shut down production.

● Profits must therefore be non-negative:Π = 𝑝𝑦 − 𝑐 𝑦 ≥ 0, 𝑜𝑟

𝑝 ≥𝑐 𝑦𝑦 = 𝐿𝐴𝐶 𝑦

i.e. price must be at least as large as long run average costs.● Thefirm’slongrunsupplycurveisthereforegivenbythesectionofthe

LMC thatliesabovetheLAC.


The firm’s supply curve in the long run


LMC,LAC

y

LMC

LACmin=LAC(ymin)

LAC

----Longrunsupplycurve

ymin

Levelofproductionwithminimallongrununitcost,LACmin.

10. Market supply (Industry supply)● Market supply with n firms is given by:

𝑆 𝑝 =M𝑆P(𝑝)O

PQRWhere 𝑆P(𝑝) is firm i’s supply at output price p.● 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.

● In the long run, firms that use the technology of profitable firms will enter the market, given “free entry”, putting downward pressure on the market price.

● 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.


Market supply curve in the long run


Price

Quantity

LACmin

ymin 3ymin 4ymin2ymin

S1

S1+ S2

S1+ S1 +S1

Supplyoffirm1

Supplyoffirms1&2

Supplyoffirms1,2&3

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