Chapter 9:Production

Chapter OutlineThe Production FunctionProduction In The Short RunProduction In The Long RunReturns To ScaleObjective of the Firm –maximize profits subject to cost minimization in Cost-Benefit Analysis.Note: Chaps. 9 and 10 – develop the cost side1. Chapter 9 – Theory of Production –how L and K and other inputs combine to produce Q.2. Chapter 10 – how costs vary with output.

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Figure 9.2: The Production Function

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Production function: the relationship that describes how inputs like capital and labor are transformed into output.

Mathematically,Q = F (K, L)

K = CapitalL = Labor

Evolving state of technology

The Production Function – time

Long run: the shortest period of time required to alter the amounts of all inputs used in a production process. Q = F (K, L), L & K are variable

Short run: the longest period of time during which at least one of the inputs used in a production process cannot be varied. Q = F (L), K is fixed but L is variable.

Variable input: an input that can be varied in the short run {L}

Fixed input: an input that cannot vary in the short run{K} 9-3

Figure 9.3: A Specific Short-Run Production Function

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Short-run Production FunctionThree properties:

1.It passes through the origin2.Initially, the addition of variable inputs augments

output an increasing rate3.beyond some point, additional units of the

variable input give rise to smaller and smaller increments in output.

Assume: K= K0 = 1K increases from K =1 to K=3 orQ= 2* (K=3)*L

Figure 9.4: Another Short-Run Production Function

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Short-run Production FunctionLaw of diminishing returns: if other inputs are fixed (SR), the increase in output from an increase in the variable input must eventually decline.

Figure 9.5: The Effect of Technological Progress in Food

Production

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Increase in technological progress

(1808)

(2008)

Note: The Law of Diminishing Returns applies even if there is technological progress. The production has more than kept pace with population growth.

Figure 9.6: The Marginal Product of a Variable Input

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Short-run Production FunctionTotal product curve: a curve showing the amount

of output as a function of the amount of variable input (Q).

Marginal product: change in total product due to a 1-unit change in the variable input: MPL = ∆TPL/∆L= ∆Q/∆L = slope of the PF

Average product: total output divided by the quantity of the variable input: APL =TPL/L= Q/L

Figure 9.7: Total, Marginal, and Average Product Curves

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1. The APL is the slope of the line joining the origin to a point on the TPL or short-run production function. See R1, R2, and R3

2. APL at L =2 is slope of R1 = 14/2 = 73. APL at L=4 and L=8 with R2 indicates the same

slope for L=4 and 8 = 43/4 =86/8 =10.754. R3 intersects TPL only at L=6 where APL =72/6=12

a. Between L=1 and L=6, MPL>APL

b. Between L=6 and L=8, MPL<APL

c. Beyond L=8, MPL <0 though APL >0

d.Only and only at L=6, is MPL=APL.

Relationship between Total, Marginal and Average Product

Curves

When the marginal product curve lies above the average product curve, the average product curve must be rising (MPL >APL →→↑APL)

When the marginal product curve lies below the average product curve, the average product curve must be falling (MPL <APL →→↓APL)

The two curves intersect at the maximum value of the average product curve MPL =APL →APL is at maximum)

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Worked Problem: Production Function

 

Question: Sketch graph a standard short-run production function, and identify on it the points where the average product peaks, the marginal product peaks, the marginal product reaches zero, and the average and marginal product intersect. Key: Make sure the average product peaks at the output where the ray from the origin is tangent to the total product curve and where the marginal product passes through it. The marginal product must peak at the output where the inflection point is on the total product curve, and the marginal product reaches zero when the total product peaks. Peak of APL(should be

here)

MPL =0

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Isoquant- the set of all input combinations that yield a given level of output.

Marginal rate of technical substitution (MRTS): the rate at which one input can be exchanged for another without altering the total level of output(i.e. along isoquant, ∆Q=0)

Long-run Production function: Q =F(K, L) where both K and L are now flexible.

Flexibility in the LR is a very important means to controlling costs, critical to profit maximization.

Suppose Q =F(K, L) =2KL= 16. Problem –how to diagram 3 variables into a 2-dimensional space! Solve for K=8/L and draw isoquants for various levels L and K that yield Q =32, 64 and so forth.

Production In The Long Run

Figure 9.8: Part of an Isoquant Map for the Production Function

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Figure 9.8: Part of an Isoquant Map for the Production Function

Figure 9.9: The Marginal Rateof Technical Substitution

More is preferred to less

Actual output

Along the isoquant, ΔQ0 =0 = Δ KMPK+ ΔLMPL i.e., -ΔKMPK = ΔLMPL

|ΔK/ΔL| = MPL/MPK =MRTSL,K is the slope.

Worked Problem

Question: A daily production function for calculators is Q = 12L2- L3. Show all your work for the following questions. a) What is the marginal product equation for labor (MPL)?b) What is the APL function? Key:a) dQ/dL = MPL= 24L - 3L2

b) Q/L= APL= (12L2 – L3)/L = 12L – L2

The APL and MPL curves can be drawn for various amounts of L. Note: for L =0, APL =MPL = 0

Worked Out Problem -LR

Question: Suppose that at a firm's current level of production the marginal product of capital (MPK) is equal to 10 units, while the marginal rate of technical substitution between capital and labor (MRTSL,K) is 2. Given this, we know the marginal product of labor (MPL) must be: Key:A. 5B. 20 since MRTSK,L = MPL/MPK 2 = MPL/10 MPL =20C. 10D. It is not possible to say with the information given in the problem

Worked Out ProblemQuestion: In the production function Q =10L1/2K1/2, calculate the slope of the isoquant when the entrepreneur is producing efficiently with 9 laborers and 16 units of capital. (Hint: The slope of the isoquant = the ratio of the marginal product of labor to the marginal product of capital.) Key: MPL=dQ/dL =1/2 10L-1/2K1/2=5L-1/2 K1/2

MPK=dQ/dK =1/2 10L1/2K-1/2=5L1/2 K-

1/2

 

 

Figure 9.10: Isoquant Maps for Perfect Substitutes and Perfect

Complements

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OutputTyping = Min{Typists, Typewriters}QGas = f(Texaco + Amoco)

Returns To Scale –A Long Run Concept

Returns to Scale--The relationship between scale or size and efficiency, ceteris paribus

Increasing returns to scale: the property of a production process whereby a proportional increase in every input yields a more than proportional increase in output.

Constant returns to scale: the property of a production process whereby a proportional increase in every input yields an equal proportional increase in output.

Decreasing returns to scale: the property of a production process whereby a proportional increase in every input yields a less than proportional increase in output.

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Figure 9.11: Returns to Scale Shown on the Isoquant Map

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Figure A9.1: Effectiveness vs. Use: Lobs and Passing

Shots

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Define P = % of total points won in a tennis match.F(L) = % of points from lobs as a function of times lobbed (L) and

G(L) = % of points from passing shots as a function of times in passing (L).

P = LF(L) + (1- L)G(L) and from the Figure, F(L) = 90 – 80L and G(L) =30 +10L.Plugging these into the above P expression yields P = 30 -70L -90L2 and dP/dL=MPL = 70 – 180L=0 or L* =70/180= 0.389– the value of L that maximizes points won.Note: plug L* into P= 30 + 70 (0.389) -90*(0.389)2 = 43.62 (next

slide)

Question:What is the best proportion of lobs and passing shots to use when an opponent comes to the net?L= lobs

(APL)

Figure A9.2: The Optimal Proportion

of Lobs

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MPL=

The P function = percentage of total points =weighted average of G(L) and F(L)

Figure A9.3: At the Optimizing Point, the Likelihood of Winning with a Lob is Much Greater than of Winning with a Passing

Shot

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Note that at L* =0.389, the likelihood of winning with lobs (58.9%) is greater than that of winning with passing shots (33.9%)Percentage of points won

with a lob =F(L)

Percentage of points won with a passing shot =G(L)

MPL= = APL

Figure A9.4: The Production Mountain

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A geometric derivation of Isoquant.Start with a 3-dimensional graph (Q, L, and K) and fix output

at Q0 and slice segments AB, CD and so forth and project them on a two-dimensional space L, and K.

This action gives rise to an Isoquant Map (Production) similar to an IC map or Utility Curve map (Consumption)

Figure A9.5: The Isoquant Map Derived from the Production

Mountain

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Figure A9.6: Isoquant Map for the Cobb-Douglas Production Function: Q

= K½L½

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MPL= 1/2K1/2L-1/2

MPK=1/2K-1/2L1/2

MPL/MPK = K/L

Figure A9.7: Isoquant Map for the Leontief Production Function: Q =

min (2K,3L)

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