production. costs problem 6 on p.194. outputfcvctcafcavcatcmc 010,000--- 100200 125 300133.3 400150...

Production. Costs Problem 6 on p.194.

Output FC VC TC AFC AVC ATC MC

0 10,000 ---

100 200

200 125

300 133.3

400 150

500 200

600 250



0 10,000 ---

100 10,000 200

200 10,000 125

300 10,000 133.3

400 10,000 150

500 10,000 200

600 10,000 250



0 10,000 ---

100 10,000 20,000 200

200 10,000 125

300 10,000 133.3

400 10,000 150

500 10,000 200

600 10,000 250



0 10,000 ---

100 10,000 20,000 200

200 10,000 25,000 125

300 10,000 40,000 133.3

400 10,000 60,000 150

500 10,000 100,000 200

600 10,000 150,000 250



0 10,000 ---

100 10,000 10,000 20,000 200

200 10,000 15,000 25,000 125

300 10,000 30,000 40,000 133.3

400 10,000 50,000 60,000 150

500 10,000 90,000 100,000 200

600 10,000 140,000 150,000 250



0 10,000 0 ---

100 10,000 10,000 20,000 200

200 10,000 15,000 25,000 125

300 10,000 30,000 40,000 133.3

400 10,000 50,000 60,000 150

500 10,000 90,000 100,000 200

600 10,000 140,000 150,000 250



0 10,000 0 10,000 ---

100 10,000 10,000 20,000 200

200 10,000 15,000 25,000 125

300 10,000 30,000 40,000 133.3

400 10,000 50,000 60,000 150

500 10,000 90,000 100,000 200

600 10,000 140,000 150,000 250



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 200

200 10,000 15,000 25,000 50 125

300 10,000 30,000 40,000 33.3 133.3

400 10,000 50,000 60,000 25 150

500 10,000 90,000 100,000 20 200

600 10,000 140,000 150,000 16.7 250



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 100 200

200 10,000 15,000 25,000 50 75 125

300 10,000 30,000 40,000 33.3 100 133.3

400 10,000 50,000 60,000 25 125 150

500 10,000 90,000 100,000 20 180 200

600 10,000 140,000 150,000 16.7 233.3 250



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 100 200

200 10,000 15,000 25,000 50 75 125

300 10,000 30,000 40,000 33.3 100 133.3

400 10,000 50,000 60,000 25 125 150

500 10,000 90,000 100,000 20 180 200

600 10,000 140,000 150,000 16.7 233.3 250

MC = cost of making an extra unit



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 100 200

200 10,000 15,000 25,000 50 75 125

300 10,000 30,000 40,000 33.3 100 133.3

400 10,000 50,000 60,000 25 125 150

500 10,000 90,000 100,000 20 180 200

600 10,000 140,000 150,000 16.7 233.3 250

Q

TCMC



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 100 200

200 10,000 15,000 25,000 50 75 125

300 10,000 30,000 40,000 33.3 100 133.3

400 10,000 50,000 60,000 25 125 150

500 10,000 90,000 100,000 20 180 200

600 10,000 140,000 150,000 16.7 233.3 250

Q

VC

Q

TCMC



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 100 200 100

200 10,000 15,000 25,000 50 75 125

300 10,000 30,000 40,000 33.3 100 133.3

400 10,000 50,000 60,000 25 125 150

500 10,000 90,000 100,000 20 180 200

600 10,000 140,000 150,000 16.7 233.3 250

Q

VC

Q

TCMC



0 10,000 0 10,000 --- --- ---

100 10,000 10,000 20,000 100 100 200 100

200 10,000 15,000 25,000 50 75 125 50

300 10,000 30,000 40,000 33.3 100 133.3

400 10,000 50,000 60,000 25 125 150

500 10,000 90,000 100,000 20 180 200

600 10,000 140,000 150,000 16.7 233.3 250

Q

VC

Q

TCMC



0 10,000 0 10,000 --- --- --- ---

100 10,000 10,000 20,000 100 100 200 100

200 10,000 15,000 25,000 50 75 125 50

300 10,000 30,000 40,000 33.3 100 133.3 150

400 10,000 50,000 60,000 25 125 150 200

500 10,000 90,000 100,000 20 180 200 400

600 10,000 140,000 150,000 16.7 233.3 250 500

Q

VC

Q

TCMC

If cost is given as a function of Q, then For example:

TC = 10,000 + 200 Q + 150 Q2

MC = ?

dQ

TCdMC

)(

Profit is believed to be the ultimate goal of any firm. If the production unit described in the problem above can sell as many units as it wants for P=$360, what is the best quantity to produce (and sell)?


0 10,000 0 10,000 --- --- --- ---

100 10,000 10,000 20,000 100 100 200 100

200 10,000 15,000 25,000 50 75 125 50

300 10,000 30,000 40,000 33.3 100 133.3 150

400 10,000 50,000 60,000 25 125 150 200

500 10,000 90,000 100,000 20 180 200 400

600 10,000 140,000 150,000 16.7 233.3 250 500

Profit is believed to be the ultimate goal of any firm. If the production unit described in the problem above can sell as many units as it wants for P=$360, what is the best quantity to produce (and sell)?

Output FC VC TC

0 10,000 0 10,000

100 10,000 10,000 20,000

200 10,000 15,000 25,000

300 10,000 30,000 40,000

400 10,000 50,000 60,000

500 10,000 90,000 100,000

600 10,000 140,000 150,000

Doing it the “aggregate” way,by actually calculating the profit:

Output FC VC TC TR

0 10,000 0 10,000

100 10,000 10,000 20,000

200 10,000 15,000 25,000

300 10,000 30,000 40,000

400 10,000 50,000 60,000

500 10,000 90,000 100,000

600 10,000 140,000 150,000


P=$360

Output FC VC TC TR

0 10,000 0 10,000 0

100 10,000 10,000 20,000 36,000

200 10,000 15,000 25,000 72,000

300 10,000 30,000 40,000 108,000

400 10,000 50,000 60,000 144,000

500 10,000 90,000 100,000 180,000

600 10,000 140,000 150,000 216,000


P=$360

Output FC VC TC TR Profit

0 10,000 0 10,000 0

100 10,000 10,000 20,000 36,000

200 10,000 15,000 25,000 72,000

300 10,000 30,000 40,000 108,000

400 10,000 50,000 60,000 144,000

500 10,000 90,000 100,000 180,000

600 10,000 140,000 150,000 216,000


P=$360


0 10,000 0 10,000 0 –10,000

100 10,000 10,000 20,000 36,000 16,000

200 10,000 15,000 25,000 72,000 47,000

300 10,000 30,000 40,000 108,000 68,000

400 10,000 50,000 60,000 144,000 84,000

500 10,000 90,000 100,000 180,000 80,000

600 10,000 140,000 150,000 216,000 66,000


P=$360

Alternative: The Marginal ApproachThe firm should produce only units that are worth producing, that is, those for which the selling price exceeds the cost of making them.


0 10,000 0 10,000 --- --- --- ---

100 10,000 10,000 20,000 100 100 200 100

200 10,000 15,000 25,000 50 75 125 50

300 10,000 30,000 40,000 33.3 100 133.3 150

400 10,000 50,000 60,000 25 125 150 200

500 10,000 90,000 100,000 20 180 200 400

600 10,000 140,000 150,000 16.7 233.3 250 500

< 360

> 360

Principle (Marginal approach to profit maximization): If data is provided in discrete (tabular) form, then profit is maximized by producing all the units for which and stopping right before the unit for which

Principle (Marginal approach to profit maximization): If data is provided in discrete (tabular) form, then profit is maximized by producing all the units for which MR > MCand stopping right before the unit for which MR < MC In our case, price of output stays constant throughout therefore MR = P (an extra unit increases TR by the amount it sells for)

If costs are continuous functions of QOUTPUT, then profit

is maximized where

Principle (Marginal approach to profit maximization): If data is provided in discrete (tabular) form, then profit is maximized by producing all the units for which MR > MCand stopping right before the unit for which MR < MC In our case, price of output stays constant throughout therefore MR = P (an extra unit increases TR by the amount it sells for)

If costs are continuous functions of QOUTPUT, then profit

is maximized where MR=MC

What if FC is $100,000 instead of $10,000? How does the profit maximization point change?


0 10,000 0 10,000 0 –10,000

100 10,000 10,000 20,000 36,000 16,000

200 10,000 15,000 25,000 72,000 47,000

300 10,000 30,000 40,000 108,000 68,000

400 10,000 50,000 60,000 144,000 84,000

500 10,000 90,000 100,000 180,000 80,000

600 10,000 140,000 150,000 216,000 66,000



0 100,000 0 10,000 0 –10,000

100 100,000 10,000 20,000 36,000 16,000

200 100,000 15,000 25,000 72,000 47,000

300 100,000 30,000 40,000 108,000 68,000

400 100,000 50,000 60,000 144,000 84,000

500 100,000 90,000 100,000 180,000 80,000

600 100,000 140,000 150,000 216,000 66,000



0 100,000 0 100,000 0 –10,000

100 100,000 10,000 110,000 36,000 16,000

200 100,000 15,000 115,000 72,000 47,000

300 100,000 30,000 130,000 108,000 68,000

400 100,000 50,000 150,000 144,000 84,000

500 100,000 90,000 190,000 180,000 80,000

600 100,000 140,000 240,000 216,000 66,000



0 100,000 0 100,000 0–100,000

100 100,000 10,000 110,000 36,000 –74,000

200 100,000 15,000 115,000 72,000 –43,000

300 100,000 30,000 130,000 108,000 –22,000

400 100,000 50,000 150,000 144,000 –6,000

500 100,000 90,000 190,000 180,000 –10,000

600 100,000 140,000 240,000 216,000 –24,000

Fixed cost does not affect the firm’s optimal short-term output decision and can be ignored while deciding how much to produce today.

Principle:

Consistently low profits may induce the firm to close down eventually (in the long run) but not any sooner than your fixed inputs become variable ( your building lease expires, your equipment wears out and new equipment needs to be purchased,

you are facing the decision of whether or not to take out a new loan, etc.)

Sometimes, it is more convenient to formulate a problem not through costs as a function of output but through output (product) as a function of inputs used.

Problem 2 on p.194.

“Diminishing returns” – what are they?

In the short run, every company has some inputs fixed and some variable. As the variable input is added, every extra unit of that input increases the total output by a certain amount; this additional amount is called “marginal product”. The term, diminishing returns, refers to the situation when the marginal product of the variable input starts to decrease (even though the total output may still keep going up!)

Total output, or Total Product, TP

Amount of input used


Marginal product, MPRange of diminishing returns

K L Q MPK

0 20 01 20 502 20 1503 20 3004 20 4005 20 4506 20 475

Calculating the marginal product (of capital) for the data in Problem 2:

K L Q MPK

0 20 0 ---1 20 50 502 20 1503 20 3004 20 4005 20 4506 20 475


K L Q MPK

0 20 0 ---1 20 50 502 20 150 1003 20 300 1504 20 400 1005 20 450 506 20 475 25


In other words, we know we are in the range of diminishing returns when the marginal product of the variable input starts falling, or, the rate of increase in total output slows down.(Ex: An extra worker is not as useful as the one before him)

Implications for the marginal cost relationship:

Worker #10 costs $8/hr, makes 10 units. MCunit =



Worker #10 costs $8/hr, makes 10 units. MCunit = $0.80

Worker #11 costs $8/hr, makes …




Worker #11 costs $8/hr, makes 8 units. MCunit =




Worker #11 costs $8/hr, makes 8 units. MCunit = $1




Worker #11 costs $8/hr, makes 8 units. MCunit = $1

In the range of diminishing returns, MP of input is falling and MC of output is increasing

Marginal cost, MC

Amount of output


Marginal product, MPThis amount of output corresponds to this amount of input

When MP of input is decreasing, MC of output is increasing and vice versa.

Therefore the range of diminishing returns can be identified by looking at either of the two graphs.(Diminishing marginal returns set in at the max of the MP graph, or at the min of the MC graph)

Back to problem 2, p.194.To find the profit maximizing amount of input (part d), we will once again use the marginal approach, which compares the marginal benefit from a change to the marginal cost of than change.More specifically, we compare VMPK, the value of marginal product of capital, to the price of capital, or the “rental rate”, r.

K L Q MPK VMPK r0 20 0 ---1 20 50 502 20 150 1003 20 300 1504 20 400 1005 20 450 506 20 475 25


K L Q MPK VMPK r0 20 0 --- ---1 20 50 50 1002 20 150 100 2003 20 300 150 3004 20 400 100 2005 20 450 50 1006 20 475 25 50


K L Q MPK VMPK r0 20 0 --- ---1 20 50 50 100 752 20 150 100 200 753 20 300 150 300 754 20 400 100 200 755 20 450 50 100 756 20 475 25 50 75

>>>>>< STOP

Why would we ever want to be in the range of diminishing returns?

Consider the simplest case when the price of output doesn’t depend on how much we produce. Until we get to the DMR range, every next worker is more valuable than the previous one, therefore we should keep hiring them. Only after we get to the DMR range and the MP starts falling, we should consider stopping.Therefore, the profit maximizing point is always in the diminishing marginal returns range!

Surprised?

Cost minimization

(Another important aspect of being efficient.)

Suppose that, contrary to the statement of the last problem, we ARE ABLE to change not just the amount of capital but the amount of labor as well. (Recall the distinction between the long run and the short run.)

Given that extra degree of freedom, can we do better?(In other words, is there a better way to allocate our budget to achieve our production goals?)

In order not to get lost in the multiple possible (K, L, Q) combinations, it is useful to have some of them fixed and focus on the question of interest.

In our case, we can either:• Fix the total budget spent on inputs and see if

we can increase the total output; or,

• Fix the target output and see if we can reduce the total cost by spending our money differently.

Think of the following analogy:

Sam needs his 240 mg of caffeine a day or he will fall asleep while driving, and something bad will happen.He can get his caffeine fix from several options listed below:

Option Caffeine, mg

Bottled Frappucino, 9.5oz 80

Coca-Cola, 12oz 40

Mountain Dew, 12oz 60

Which one should he choose?



Option Caffeine, mg Cost of ‘input’

Bottled Frappucino, 9.5oz 80

Coca-Cola, 12oz 40

Mountain Dew, 12oz 60



Option Caffeine, mg Cost of ‘input’

Bottled Frappucino, 9.5oz 80 $2.00

Coca-Cola, 12oz 40 $0.80

Mountain Dew, 12oz 60 $1.00



Option Caffeine, mg Cost of ‘input’ Mg/$

Bottled Frappucino, 9.5oz 80 $2.00

Coca-Cola, 12oz 40 $0.80

Mountain Dew, 12oz 60 $1.00



Option Caffeine, mg Cost of ‘input’ Mg/$

Bottled Frappucino, 9.5oz 80 $2.00 40

Coca-Cola, 12oz 40 $0.80 50

Mountain Dew, 12oz 60 $1.00 60

Next, think of caffeine as Sam’s ‘target output’ (what he is trying to achieve)and drinks as his inputs, which can to a certain extent be substituted for each other.

The same principle holds for any production unit that is trying to allocate its resources wisely:

In order to achieve the most at the lowest cost possible, a firm should go with the option with the highest MPinput/Pinput ratio.

Note that following this principle will make the firm better off regardless of the demand it is facing!

If

then - reduce the amount of capital; - increase the amount of labor.

rent

MP

wage

MP KL

If

then - reduce the amount of labor; - increase the amount of capital.

rent

MP

wage

MP KL

If

then inputs are used in the right proportion. No need to change anything.

rent

MP

wage

MP KL

As you do that,- MPL will decrease;- MPC will increase;- the LHS will get smaller,- the RHS bigger

production. costs problem 6 on p.194. outputfcvctcafcavcatcmc 010,000--- 100200 125 300133.3 400150...

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