Production Inputs and Cost Supply Analysis

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Production, Inputs, Cost, Output , Price and Profit. Marginal

analysis

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Slides based on Chapter 7 and 8 Economics Principles and Policy by Baumol and BlinderChapter 13 Principles of Economics by N. Gregory MankiwQuantitative Methods for Business by Waters, Donald

Production, Inputs, Cost, Output , Price and Profit: Marginal Analysis

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Short-run versus Long-run Costs: What Makes an Input Variable?

Production and Input Choice, with One Variable Input

Multiple Input Decisions: The Choice of Optimal Input Combinations

Cost and Its Dependence on Output

Contents

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Economies of Scale Appendix: Production Indifference Curves

Contents (continued)

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Short-run versus Long-run Costs

The Economic Short Run vs the Long Run

Short run

• a period of time during which some of the firm’s cost commitments will not have ended.

• In the short run, output can change but production processes are fixed.

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The Economic Short Run vs the Long Run Long run

a period of time long enough for all of the firm’s commitments to come to an end.

In the long run, all inputs can be varied and production processes can be changed.

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Fixed Costs (FC) and Variable Costs (VC)In the production of commodities and services the firms have Fixed Cost and Varible Costs. Fixed costs are for instance the buildings and the machinery. Variable costs are for instance labour force and raw material involved in the production.Fixed costs = costs that cannot be changedVariable costs = costs that can be changedIn the short run, some costs are fixed. In the long run, all costs are variable.

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Production and Input choice, with 1 Variable Input

Total physical product (TPP) is the amount of output that can be produced as one input changes, with all other inputs held constant

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Total Physical Product for Al’s Building Company

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Total physical product with Different Quantities of Carpenters

TPP

G

F E

D

C

B

A

5

Quantity of Carpenters per Year

7 6 4 3 2 1

40

32 35

30

25

20

15

10

5

0

Gar

ages

per

Yea

rT

ota

l Ou

tpu

t in

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Production and Input Choice, with 1 Variable Input

Average Physical Product (APP) = measures the output produced per unit of input. Is the total physical product (TPP) divided by the quantity of input. (APP=TPP/X where X is que quantity of input)

Marginal physical product (MPP) = output per input (is the increase in total output that results from an additional unit increase in the input, holding the amounts of all other inputs constant)

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Al’s Product Schedule

Copyright © 2003 South-Western/Thomson Learning. All rights reserved.

The MPP of the fourth carpenter is the total output (garages) when Al (the owner) employes 4 carpenters minus the total output when he employes 3 carpenters (32-24=8)

The MPP of the fourth carpenter is the total output (garages) when Al (the owner) employes 4 carpenters minus the total output when he employes 3 carpenters (32-24=8)

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Al’s Marginal Physical Product (MPP) Curve -

Copyright © 2003 South-Western/Thomson Learning. All rights reserved.

6

MPP

Negative marginal returns

Diminishing marginal returns

Increasing marginal returns

Number of Carpenters

7 5 4 3 2 1

14

12

10

8

6

4

2

0

–2

–4

–6

MP

P i

n

Gar

ages

pe

r Y

ear

0

This graph of Marginal Physical Product shows how much additional output (garages per year) Al gets from each additional carpenter he employes.

This graph of Marginal Physical Product shows how much additional output (garages per year) Al gets from each additional carpenter he employes.

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The law of dimishing marginal returns states than an increase in an input (holding the amounts of all other constants) ultimately leads to lower marginal returns to the expanding input ( one input additional output created by each additional unit of the input)

The law of dimishing marginal returns explains the shape of the marginal physical product (MPP) curve.

The so-called law is simply based on some observed facts; it is not a theorem deduced analytically.

MPP and the “Law” of Diminishing Marginal Returns

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MRP (Marginal revenue product) = marginal physical product (MPP) output price

It means that the MRP of an input is the additional revenue that the producer earns from the increased sales when it uses an additional unit of the input.

The amount of an input is optimal when marginal revenue product (MRP) = (P) price of the input.

It means when the marginal revenue product of an input exceeds its price it pays the firm to use more of that input.

When the marginal revenue product of the input is less than its price it pays the firm to use less.

Marginal Physical Product (MPP) and the “Law” of Diminishing Marginal Returns

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The optimal quantity of input and dimishing returns

When the marginal physical product (MPP)of input begins to decline, the money value of that product falls, as well; that is the marginal revenue product (MRP)also declines.

The producer always profits by expanding input use until dimishing returns set in and reduce the MRP to the price of the input. The firm has employed the proper amount of input only when dimishing returns reduce the marginal revenue product of the input to the level of its price, because then the firm will be wasting no opportunity to add to its total profit.

Thus, the optimal quantity of an input is that at which MRP equals its price. IN SYMBOLS: MRP= P of input (marginal revenue product equals Price of input)

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COST CURVE AND INPUT QUANTITIES

How much to produce? The quantity of output that is most desiderable for firm depends on the way in which costs change when output varies. Such information is typically displayed in the form of cost curves.

We need three cost curves: TOTAL COST CURVE, AVERAGE COSTS CURVE, AND THE MARGINAL COST CURVE

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TOTAL COST, AVERAGE COST AND MARGINAL COST

For any given output TOTAL COST is defined as a total input quantities per prices of the input needed .

For any given output AVERAGE COST is defined as a total cost divided by quantity produced.

MARGINAL COST is defined as the increase in total cost that results from the production of an additional unit of output.

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TOTAL COST, MARGINAL COST AND AVERAGE COST (AN EXAMPLE)

Suppose, for example, we know the total cost of making a number of units of a product. We can divide this TOTAL COST by the number of units to get the AVERAGE COST. But it is often more useful to look at the MARGINAL COST which is defined as the cost of making one extra unit.

Suppose we have already made 100 units at a TOTAL COST of 50.000 euro. The AVERAGE COST is 50.000/100 =500. But the MARGINAL COST is the cost of making the 101st unit. Because all investment in equipment may already have been recovered, and we have a lot of experience in making the product, this MARGINAL COST might be considerably lower than the AVERAGE COST.

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A BIT OF CALCULUS…Marginal Cost and Total Cost

The marginal cost MC of a product is defined as the additional cost of making one extra unit.

In making this extra unit the total cost will increase by MC, so the marginal cost is the rate at which the total cost is changing

If we know the equation for the total cost curve we can differenciate it to get the marginal cost

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A bit of calculus…

Then

TOTAL COST (TC)=y (it means, it is a function that usually can be obtained from analysis of empirical data)

MARGINAL COST (MC)=dy/dx (derivative of y divided by derivative of x)

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WORKED EXAMPLE

The total cost of making a quantity X of product is calculated as TC= 2x2 + 4x + 500.

What are the expressions for TOTAL, FIXED, VARIABLE MARGINAL AND AVERAGE COSTS?

What are these costs if 500 units of the product are made?

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WORKED EXAMPLE: solution

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WORKED EXAMPLE: solution

Fixed Cost: remain unchanged (500)

VC=2×500²+ 4×500=502.000

MC =4×500+4=2004

AC= 2×500+4+500:500=1005

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Multiple Input DecisionsSubstitutability: The Choice of Input Proportions

Firms usually have a variety of techological options and can substitute one input for another.

Given a target level of production, a firm that cuts down on the use of one input (say labour) will normally have to increase its use of another input (say machinery)The combination of inputs that represents the least costly way to product the desired level of output depends on the relative prices of the various inputs.

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Multiple Input Decisions

The Marginal Rule for Optimal Input Proportion A firm can reduce the cost of producing a given

output by using less of some input A and making up for it by using more of another input B, whenever the ratio of Marginal Physical Product of A to the price of A is less than the ratio of the Marginal Physical Product of B to the price of B, that is, whenever MPPa/Pa is less than MPPb/PB

(MPP it means Marginal Physical Product and P means price)

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Rule for optimal input proportions = the ratio of marginal physical product to price should be the same for all inputs

MPPa/Pa = MPPb/Pb Where a and b means input a and input b

If the ratio is higher for one input (for instance the price of raw material, the price of machinery…) more of that input should be used, and less of the others, until the ratios are equal.

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Changes in Input Prices and Optimal Input Proportions input price ratio of marginal physical

product to price To maximize profits, the firm should switch away

from that input until its marginal physical product rises enough to equalize the ratios again.

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Input Quantities and Total, Average, and Marginal Cost CurvesTotal cost = the total cost (including opportunity

cost) of producing any level of output when inputs are optimally employed

Average cost = total cost per unit of outputMarginal cost = increase in total cost from

producing an additional unit of outputLETS SEE WHAT HAPPENS WHEN WE DRAW

THE CURVEs of Total Cost, Average Cost and Marginal Cost curves (Al’s firm)

Cost and Its Dependence on Output …I repeat again…

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Al’s (Variable) Cost Schedules

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(a) Al’s Total Cost Curve

TC

(a)

To

tal C

os

t p

er

Yea

r

(th

ou

san

ds

$)

Quantity of Garages

10 8 6 4 2

200 180 160 140 120 100 80 60 40 20

0

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(b) Al’s Average Cost Curve

C

D

AC

(b)

Av

era

ge

Co

st p

er

Ga

rag

e

(th

ou

san

ds

$)

Quantity of Garages

10 8 6 4 2

30

25

20

15

10

5

0

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(c) Al’s Marginal Cost Curve

(c)

Mar

gin

al C

ost

per

Ad

ded

Gar

age

(t

ho

usa

nd

s $)

Quantity of Garages

10 8 6 4 2

50 45 40 35 30 25 20 15 10 5

0

MC

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Total cost = total fixed cost + total variable cost

Total fixed cost: are constant over all levels of output.

Input Quantities and Total, Average Cost, and Marginal Cost Curves

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Fixed Costs: Total

TFC

(a)

To

tal F

ixed

Co

st

per

Yea

r (t

ho

usa

nd

s o

f $

)

Output 10 8 6 4 9 7 5 3 1 2

14

12

10

8

6

4

2

0

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Average fixed cost = total fixed cost per unit of output

Average fixed cost falls as output rises

Input Quantities and Total, Average Cost, and Marginal Cost Curves

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Al’s Fixed Costs

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Fixed Costs: Average

AFC

(b)

Ave

rag

e F

ixed

Co

st p

er G

arag

e (t

ho

usa

nd

s $)

Output 10 8 6 4 9 7 5 3 1 2

14

12

10

8

6

4

2

0

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A typical average cost curve declines at first because average fixed costs decline.

It then reaches a minimum and begins to rise because the law of decreasing marginal returns.

The Average Cost Curve in the Short and Long Run

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Costs differ in the short and long runs, because in the long run, more adjustments can be made.

The long-run average cost curve shows the lowest possible short-run average cost corresponding to each output level.

The Average Cost Curve in the Short and Long Run

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Economies of Scale

The law of diminishing returns holds in the case of the expansion of a single input, holding other inputs constant.

Returns to scale are relevant when all inputs increase at the same rate.

Economies of scale affect operations in many modern industries. Where the exists they give larger firms cost advantages over smaller ones and thereby foster large firm sizes. Automobile production and telecommunications are two good examples of industries with important economies of scale.

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Short-run and Long-Run Average Cost Curves

V B

S

Ave

rag

e C

ost

per

Po

un

d o

f C

hic

ken

$0.40 0.35

Output in Pounds of Chicken

100 40 0

U L

W

G T

The producer has a choice of 2 sizes of chicken coop, a small one with the Average Curve SL and a big one with the Average Curve BG. These are the short-run curves that apply as long as the farm is stuck with its chosen coop.

In the long run, however, it can pick any point on the orange lower boundary of these curves. This lower boundary STG is the long-run Average Cost Curve

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Economies of Scale

Production is said to involve ECONOMIES OF SCALE, also referred to as to as INCREASING RETURNS TO SCALE, if, when all input quantities are doubled, the quantity of output is more than doubled

Then a) Economies of scale = output rises faster than the

common rate of growth of all the inputs. b) Economies of scale = increasing returns to scale C) Economies of scale long-run declining average

cost curves

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Economies of Scale: the shapes of the long-rung average cost curve

Production functions with economies of scale lead to long-run average cost curves (AC) that decline as output expands (graph a)

Next graph represents the shapes that the long-run average cost curve can take.

Remember that AC(average cost)=TC(total cost)/Q (output).

Sometimes the firm have constant returns of scale that leads to a long run Average Cost curve flat (graph b) . Also is possible that the firm has decreasing returns of scale when long-run average cost rise as output expands (graph c)

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3 Possible Shapes for the Long-Run AC Curve

Lo

ng

-Ru

n A

ve

rag

e C

os

t

(c) Quantity of Output

Decreasing returns to scale

Lo

ng

-Ru

n A

ve

rag

e C

os

t

(b) Quantity of Output

Constant returns to scale

Lo

ng

-Ru

n A

ve

rag

e C

os

t

(a) Quantity of Output

Increasing returns to scale

AC

AC

AC

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All points on the analytical cost curve (used in economic analysis) refer to the same period of time.

An historical cost curve, showing the actual relationship between cost and output at different periods of time, is probably not a good indicator of the analytical cost curve.

Historical Costs versus Analytical Costs Curves

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Declining HCC (historical cost curve) ACC (analytical cost curve)

B

A

2002 analytical cost curve


Historical cost curve

$100

75

50

25

0

Co

st p

er U

nit

Quantity of Output

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Declining HCC (historical cost curve) with /U-shaped Analytical ACC

A B



Historical cost curve

$100

75

50

25

0

Co

st

pe

r U

nit

Quantity of Output

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Real business situations are more complex than those outlined in this lecture, and the quality of the available information is less precise.

Yet when managers are doing their jobs well and the market is functioning smoothly, these models are a good approximation to the real world.

Cost Minimization in Theory and Practice

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Some conclusions A firms total cost curve shows its lowest possible

cost of producing any given level of output. This curve is derived from the input combination that the firm uses to produce any given output and the prices of the input.

The marginal physical product of an input is the increase in total output resulting from a 1-unit increase in that input, holding the quantities of all other inputs constant.

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The law of dimishig maginal returns states that if a firm increases the amount of one input (holding all other input quantities constant) the marginal physical product of the expanding input will eventually begin to decline.

To maximize profits, a firm must purchase an input up to the point at which dimishing returns reduce the input’s marginal revenue product to equal its price.

Cost Minimization in Theory and Practice

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Some conclusions The long rung is a period sufficiently long for the

firm’s plant to require replacement and for all of its current contractual commitments to expire. The short run is any period briefer than that.

TC (Total Cost) =TFC (Total Fixed Cost) +TVC (Total Variable Cost)

AC (Average Cost) =AFC (Average Fixed Cost) +AVC (Average Variable Cost)

It is possible to produce the same quantity of output in a variety of ways by substituting more of one input for less of another. Firms normally seek the least costly way to produce any given output.

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Some conclusions

A firm that wants to minimize costs will select input quantities at which the ratios of the marginal physical product of each input to its price are equal for all inputs.

If a doubling of all the firm’s inputs just doubles its output, the firm is said to have constant returns to scale.

If a doubling of all inputs leads to more than twice as much output, it has increasing returns to scale (or economies of scale).

If a doubling of inputs produces less than a doubling of output the firm has decreasing returns of scale.

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….and a bit of calculus more…i know I am a bad guy

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This figure shows a graph of the equation y= ax² +bx+c. This is a continuous function with a clear minimum. If you look at the gradient at point A it is clearly negative, showing that y is falling in value as x increase.

At point B the gradient is positive, showing that y is increasing in value as x increases. The most interesting point comes between these at point C where the gradient is zero, in other words the tangent to the curve is parallel with the x axis. This only happens at one specific point and this is the minimum valuee of the graph.

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WORKED EXAMPLE:

What can you say about the minimum of y=2x²-4x+10?

We can diferénciate y=2x²-4x+10 to find the gradient at any point. Then: dy/dx= 2×2×x-4=4x-4For a minimum value of y this gradient is equal to zero. In other words:4x-4=0 or x=1Substituting x=1 into the equation for y gives the minimum value:y=2x²-4x+10=2×1²-4×1+10=8So the minimum value of the graph is y=8 which occurs when x=1

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WORKED EXAMPLE 1

If y=4x²+3x – 2, what are a) the derivative, b) the second derivative?

Solution:

If y=4x² +3x-2 then differenciating in the usually

way gives the first derivative dy/dx= 8x+3

If dy/dx= 8x+3 then differenciating in the usual way gives the second derivative d²y/dx²=8

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WORKED EXAMPLE 2The total cost of a manufacturing process is found to be 3x²-12x +30 where x is the number of units produced in hundreds each week. What production level minimizes the total cost?

We know that y= 3x²-12x+`30. This has a gradient given by dy/dx=6x-12. There is a turning point when this gradient has a value 0: that is 6x-12=0 or x=2. At this point the function has the value

Y=3x²-12x+30=3×2²-12×2+30=18. The second derivative

d²y/dx²=6. This is positive, so the turning point has to bea minimum. Then the total cost of production is a

minimum of 18 when production is 200 units a week.

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Production Inputs and Cost Supply Analysis