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

TC = 10,000 + 200 Q + 1.5 Q2

 MC = ?

dQ

TCdMC

)(

Q

VC

Q

TCMC

While dealing with discrete data,

Marginal cost

Production. Costs 

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

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

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

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

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

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 =

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 …

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 =

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

Amount of input used

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)

Profit is believed to be the ultimate goal of any firm.  In Problem 2, what number of units of capital maximizes the firm’s profit? PK = $75 PL = $15 POUTPUT = $2

The aggregate approach:K L Q VC FC Profit0 20 01 20 502 20 1503 20 3004 20 4005 20 4506 20 475

Profit is believed to be the ultimate goal of any firm.  In Problem 2, what number of units of capital maximizes the firm’s profit? PK = $75 PL = $15 POUTPUT = $2

The aggregate approach:K L Q VC FC Profit0 20 0 01 20 50 752 20 150 1503 20 300 2254 20 400 3005 20 450 3756 20 475 450

Profit is believed to be the ultimate goal of any firm.  In Problem 2, what number of units of capital maximizes the firm’s profit? PK = $75 PL = $15 POUTPUT = $2

The aggregate approach:K L Q VC FC Profit0 20 0 0 3001 20 50 75 3002 20 150 150 3003 20 300 225 3004 20 400 300 3005 20 450 375 3006 20 475 450 300

Profit is believed to be the ultimate goal of any firm.  In Problem 2, what number of units of capital maximizes the firm’s profit? PK = $75 PL = $15 POUTPUT = $2

The aggregate approach:K L Q VC FC Profit0 20 0 0 300 - 3001 20 50 75 300 - 2252 20 150 150 300 - 1503 20 300 225 300 754 20 400 300 300 2005 20 450 375 300 2256 20 475 450 300 200

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

The marginal approach:To find the profit maximizing amount of input (part d), we need to compare 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

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 50 1002 20 150 100 2003 20 300 150 3004 20 400 100 2005 20 450 50 1006 20 475 25 50

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 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

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 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

What if the cost of labor (the fixed cost in this case) changes? How does the profit maximization point change?  

K L Q VC FC Profit0 20 0 0 300 - 3001 20 50 75 300 - 2252 20 150 150 300 - 1503 20 300 225 300 754 20 400 300 300 2005 20 450 375 300 2256 20 475 450 300 200

What if the cost of labor (the fixed cost in this case) changes? How does the profit maximization point change?  

K L Q VC FC Profit0 20 0 0 300 600 - 300 1 20 50 75 300 600 - 2252 20 150 150 300 600 - 1503 20 300 225 300 600 754 20 400 300 300 600 2005 20 450 375 300 600 2256 20 475 450 300 600 200

What if the cost of labor (the fixed cost in this case) changes? How does the profit maximization point change?  

K L Q VC FC Profit0 20 0 0 300 600 - 300 - 6001 20 50 75 300 600 - 225 - 5252 20 150 150 300 600 - 150 - 4503 20 300 225 300 600 75 - 2254 20 400 300 300 600 200 - 1005 20 450 375 300 600 225 - 756 20 475 450 300 600 200 - 100

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.)

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?

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 Cost of ‘input’

Bottled Frappucino, 9.5oz 80

Coca-Cola, 12oz 40

Mountain Dew, 12oz 60

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 Cost of ‘input’

Bottled Frappucino, 9.5oz 80 $2.00

Coca-Cola, 12oz 40 $0.80

Mountain Dew, 12oz 60 $1.00

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 Cost of ‘input’ Mg/$

Bottled Frappucino, 9.5oz 80 $2.00

Coca-Cola, 12oz 40 $0.80

Mountain Dew, 12oz 60 $1.00

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 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

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 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

Profit maximization in different market structures

In the Elasticity topic, we discussed a firm facing downward-sloping demand – if a company wants to attract more customers, it has to lower its price.

Tonight, we considered examples where the price the firm could get for each unit of out put did not depend on the number of units produced.

Which way is right?

Which way is more realistic therefore more relevant? 

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. P

Q

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

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

Q

TREvery unit sells at the same price so…

Slope equals price

How about the Marginal Revenue graph?

Q

MREvery 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 =

1. Aggregate:

Profit = TR – TC = 160 Q – (100 + 40 Q + 5 Q2)

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 QMR = MC

160 = 40 + 10 Q

120 = 10 Q

Q = 12

What if the market is NOT perfectly competitive?

(This happens if some or all of the attributes of perfect competition are not present. For example:

- The firm in question is large (takes up a large portion of the market);

- The firm produces a good that consumers perceive as different from the others;

- Searching for the best deal is costly for consumers;

Etc.)

For now, we will consider Monopolistic competition and Monopoly

Many small firms - One LARGE firm

Fairly easy entry and exit - Entry is very costlyor impossible

As a firm in either of these markets raises its price, the quantity it is able to sell drops –

Demand curve is downward sloping!

In both markets, a firm can vary its price to some extent

Demand curve of an individual firm:

Q

P

Total Revenue:

Q

TR

Marginal Revenue as a function of quantity:

Q

MR- Starts out positive;- Gradually decreases;- Hits zero when TR = MAX;- Goes on into the negative

range.

In fact, when demand is linear, so is MR

Q

MR

P

MR and demand curves share the same vertical intercept but MR decreases twice as fast.

The profit maximization story told graphically:In aggregate terms:

Q

TR

TC

Profit

In marginal terms:

Q

MR

P

Profit-maximizing quantity

Profit-maximizing price

Note that for a profit-maxing firm with market power price is always higher than the MC of the last item produced

The more market power a firm has, the greater that difference(“profit margin”,

“markup”)

MC

Recall also that profit is maximized where MR=MC……whereas revenue is maximized where MR=0

Q

MR

P

MC

Profit-max quantity

Revenue-max quantity

Profit-maximizing price

Revenue-maximizing price

Recall also that profit is maximized where MR=MC……whereas revenue is maximized where MR=0

Q

MR

P

MC

Profit-max quantity

Revenue-max quantity

Profit-maximizing price

Revenue-maximizing price

For an imperfectly competitive firm, profit is always maximized at a smaller output quantity (therefore at a higher price) than revenue is maximized.

Analytically:

TC = 100 + 40 Q + 5 Q2 , And the demand facing the firm is given by

QD = 25 – 0.1 P

What is the profit maximizing quantity AND price?