Production, Costs, and Profit

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

Production. Costs Problem 6 on p.194.

Output FC VC TC AFC AVC ATC MC 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

QVC

QTCMC

∆∆

=∆∆

=

If cost is given as a function of Q, then For example: TC = 10,000 + 200 Q + 150 Q2 MC =

dQTCdMC )(

=

0 + 200 + 300 Q


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 Profit 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

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

P=$360

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


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

The marginal approach to decision making

Costs (–)

Revenue (+)

Least profit Most profit

The cost-benefit approach to decision making (an “artistic” view)

Costs (–)

Revenue (+)

Least profit Most profit

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


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 The revenue they generate exceeds the costs of producing them, and stopping right before the unit for which

MR=MC

MR>MC MR<MC

Note that in this problem, 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

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


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



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



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



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 marginal 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 marginal 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, MP Range of diminishing returns

The growth in total product slows down Marginal product of input starts decreasing

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 = $0.80 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, MP This 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)

K L Q VC FC TC TR Profit 0 20 0 0 300 300 0 - 300 1 20 50 75 300 375 100 - 275 2 20 150 150 300 450 300 - 150 3 20 300 225 300 525 600 75 4 20 400 300 300 600 800 200 5 20 450 375 300 675 900 225 6 20 475 450 300 750 950 200 7 20 475 525 300 825 950 125

Aggregate approach:

Back to problem 2, p.194.

K L Q MPK 0 20 0 --- 1 20 50 50 2 20 150 100 3 20 300 150 4 20 400 100 5 20 450 50 6 20 475 25

The marginal approach compares the benefit from a change (in this case, a change in the amount of capital) to the cost of than change. First, calculate the additional output (marginal product of capital, MPK):

K L Q MPK VMPK r 0 20 0 --- --- 1 20 50 50 100 75 2 20 150 100 200 75 3 20 300 150 300 75 4 20 400 100 200 75 5 20 450 50 100 75 6 20 475 25 50 75

> > > > > < STOP

Next, convert MP, the benefit expressed in terms of units of output, into VMP, or “Value of MP”, expressed in dollars... … and compare it to the price the firm is paying for each unit of capital, or the “rental rate”, r.

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 one after another. 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! (Surprise?)

Note also that the starting point of the DMR range is NOT the profit maximizing point!!! DMR deal exclusively with the input, or the cost, side.

Discussion of profit involves both costs and revenue.

Here is how this works: -The firm is increasing output by gradually increasing the use of variable inputs; - At first, the marginal product of a unit of input may be increasing – GOOD, keep going! - Then you enter the range of DMR (MP starts to decrease) – you should keep going a bit further; - Finally, MP drops to the level below the cost of that unit of input. This is an indication that you have reached the profit-max point. STOP.

In the problems we did above, the price the firm could get for each unit of output did not depend on the number of units produced. This is typical of a market structure called “perfect competition”.

Traditionally, economics textbooks distinguish four types of markets, or of market structures. They differ in the degree of market power an individual firm has: • Perfect competition • Monopolistic competition • Oligopoly • Monopoly

“Market power” also known as “pricing power” is defined in the managerial literature as the ability of an individual firm to vary its price while still remaining profitable or as the firm’s ability to charge the price above its MC.

the least market power

the most market power

Perfect competition The features of a perfectly competitive market are:

•Large number of competing firms;

•Firms are small relative to the entire market;

•Products different firms make are identical;

•Information on prices is readily available.

As a result, the price is set by the interaction of supply and demand forces, and an individual firm can do nothing about the price.

This is the story of any small-size firm that cannot differentiate itself from the others.

Entire market Individual firm Q, mln lb Q, thousand lb

$1

P P

(Individual firm’s demand is perfectly elastic ).

What does a Total Revenue (TR) graph look like for such a firm, if plotted against quantity produced/sold?

Q

TR Every unit sells at the same price so…

Slope equals price

How about the Marginal Revenue graph?

Q

MR Every unit sells at the same price so…

MR = P

The profit maximization story told graphically: In aggregate terms:

Q

TR

TC

FC

max capacity

Profit

max profit

In marginal terms:

Q

MR

MC

max profit

Doing the same thing mathematically:

TC = 100 + 40 Q + 5 Q2 ,

And the market price is $160,

What is the profit maximizing quantity (remember, price is determined by the market therefore it is given)?

Just like in the case with tabular data, there are two approaches.

1. Aggregate: Profit = TR – TC = 160 Q – (100 + 40 Q + 5 Q2) = =120 Q – 100 – 5 Q2

A function is maximized when its derivative is zero; Specifically, when it changes its sign from ( + ) to ( – )

d(Profit)/dQ = 0 120 – 10 Q = 0 Q = 12

2. Marginal (looking for the MR = MC point)

MR = Price = $160

MC = d(TC)/dQ ; TC = 100 + 40 Q + 5 Q2

MC = 40 + 10 Q MR = MC 160 = 40 + 10 Q 120 = 10 Q Q = 12

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 capital-labor-output 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: Gary 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?

Think of the following analogy: Gary 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 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 Gary’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 while producing the desired target output will always make the firm better off!

The textbook refers to “wage” as the shortcut name for the cost of each unit of labor; “rent” – the cost of each unit of capital

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

rentMP

wageMP KL >

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

rentMP

wageMP KL <

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

rentMP

wageMP KL =

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

Production, Costs, and Profit

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