Ridge Lines

8/12/2019 Ridge Lines

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Production


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Production

an activity that creates value


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Inputs for Production

raw materials, labor, land, capital, &entrepreneurial or managerial talent.

Capital includes tools, machinery,equipment, & physical facilities.


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Production Function

Q = f(X 1,X2,X3,X4,,X n)where Q is the quantity of output that can

be produced with amounts of inputs,X1,X2,X3,X4,,X n.


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Long run

time period long enough for all inputs to bechanged


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Fixed input

an input whose quantity can not be changedin the short run


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Variable input

an input whose quantity can be changed in a

short period of timeExamples: labor, raw materials


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We can look at the productivity of a variableinput given a fixed level of fixed input.


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Graphical Interpretation of MP

The continuous MP is the slope ofthe total product curve at aparticular point.

The discrete MP is the slope of theline segment connecting 2 points onthe total product curve.

Q = TP

X


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Example: Q = 21X + 9X 2 X3

X Q = TP

0 0

1 292 70


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Example: Q = 21X + 9X 2 X3

AP = Q / X = (21X + 9X 2 X3) / X

= 21 + 9X X2

X = 1: AP = 29

X = 2: AP = 35


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In this graph, we see that initially, AP is increasing

X

Q = TP

0


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Total Product,Marginal Product, & Average Product

Curves

Q = TP

XMP

AP

X

AP

MP

incr margreturns

pt of dim

marg returns

Diminishing marginal

returns set in whenMP starts to fall (butis still positive).

The TP curve gets

flatter as the slope ofTP falls.


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Q = TP

XMP

AP

X

AP

MP

pt of dimavg returns

Diminishing averagereturns set in when

AP starts to fall.(Remember that APis the slope of theline from the origin tothe point on the TPcurve.)


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Q = TP

XMP

AP

X

AP

MP

marginal returnsbecome negative

dim totalreturns

Diminishing totalreturns set in when

the TP curve turnsdownward and MPbecomes negative.


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Isoquant

a curve showing all possible efficientcombinations of input that are capable of

producing a certain quantity of output

(Note: iso means same, so isoquant meanssame quantity)


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Isoquants for different output levels

50

100

125

Quantity of capitalused per unit of time

Quantity of laborused per unit of time

As you move in a northeasterlydirection, the amount of outputproduced increases, along withthe amount of inputs used.


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If you move out from the origin along a ray withconstant slope, the input combinations have a

constant capital-labor ratio.

50100

125


Quantity of labor used per unit of time

140

15 24 36 45

15

12

8

5

Each of the indicated pointsuses one-third as muchcapital as labor.


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It is possible for an isoquant to havepositively sloped sections.



In these sections, youreincreasing the amounts ofboth inputs, but output isnot increasing, because

the marginal product ofone the inputs is negative.


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The lines connecting the points where the isoquantsbegin to slope upward are called ridge lines.



ridge lines


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No profit-maximizing firm will operate at a pointoutside the ridge lines, since it can produce the

same output with less of both outputs.

Quantity of labor usedper unit of time

Quantity ofcapital used perunit of time

L1 L2

K2

K1

Notice, for example, thatsince points A & B are onthe same isoquant, theyproduce the sameamount of output.However, point B is amore expensive way toproduce since it usesmore capital & morelabor.

B A


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Marginal rate of technical substitution(MRTS)

The slope of the isoquantThe rate at which you can trade off inputsand still produce the same amount of output.For example, if you can decrease theamount of capital by 1 unit while increasingthe amount of labor by 3 units, & stillproduce the same amount of output, themarginal rate of technical substitution is 1/3.


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What is the MRTS or slope of the isoquant?

Q 2

Quantity of capital

used per unit oftime

Quantity of laborused per unit of time

K A

KB

L A LB

A

B

Q 1

C

Consider 2 points A & B on the same isoquant.

Lets divide the movement between A & B into 2 parts, from A to C, & from C to B.Moving from A to C, Q = ( Q/ K) K .Moving from C to B, Q = ( Q/ L) L .Moving from A to B, Q = ( Q/ K) K + ( Q/ L) L

= MPK K + MP

L L .

Since A & B are on the same isoquant, Q = 0.So, MP K K + MP L L = 0 .

MP K K = - MP L L .

K/ L = - MP L/MP K

slope ofisoquant


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How does output respond to changes in

scale in the long run?

Three possibilities:

1. Constant returns to scale2. Increasing returns to scale3. Decreasing returns to scale


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Constant returns to scale

Doubling inputs results in double the output.


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Increasing returns to scaleDoubling inputs results in more than doublethe output.One reason this may occur is that a firmmay be able to use production techniquesthat it could not use in a smaller operation.


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Decreasing returns to scale

Doubling inputs results in less than doublethe output.One reason this may occur is the difficultyin coordinating large organizations (morepaper work, red tape, etc.)


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Graphs of Constant, Increasing, &Decreasing Returns to Scale

Cons tan t Returns toScale : isoquants foroutput levels 50, 100,150, etc. are evenly

spaced.

Capital

Labor

150

100

50

150

Capital

Labor

50100

150

Capital

Labor

10050

Increas ing Returns toScale : isoquants foroutput levels 50, 100,150, etc. get closer &

closer together.

Decreas ing Return sto Scale : isoquantsfor output levels 50,100, 150, etc. become

more widely spaced.


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Methods of estimating production functions

1. using statistical analysis of time series orcross-sectional data.

2. based on experimentation or experiencewith day-to-day operations.


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Ridge Lines

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