### The Production Function

```Chapter 9
Production
Chapter Outline
The Production Function
Production In The Short Run
Production In The Long Run
Returns To Scale
9-2
Figure 9.2: The Production Function
Production function: the
relationship that describes
how inputs like capital
and labor are transformed
into output.
Mathematically,
Q = F (K, L)
K = Capital
L = Labor
Evolving state of technology
9-3
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}
9-4
Figure 9.3: A Specific Short-Run
Production Function
Assume: K= K0 = 1
Short-run Production Function
Three properties:
1.It passes through the origin
2.Initially the addition of variable inputs augments output an
increasing rate
3.beyond some point additional units of the variable input
give rise to smaller and smaller increments in output.
9-5
Figure 9.4: Another Short-Run Production Function
Short-run Production Function
Law of diminishing returns: if other inputs are fixed,
the increase in output from an increase in the variable
input must eventually decline.
9-6
Figure 9.5: The Effect of Technological
Progress in Food Production
Increase in technological progress
(2008)
(1808)
9-7
Figure 9.6: The Marginal Product of a
Variable Input
Short-run Production Function
Total 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
Average product: total output divided by the quantity of the variable input:
APL =TPL/L= Q/L
9-8
Worked Out Problem (after Slide 15)!
Question: Suppose that at a firm's current level of production
the marginal product of capital is equal to 10 units, while the
marginal rate of technical substitution between capital and
labor is 2. Given this, we know the marginal product of labor
must be:
Key:
A. 5
B. 20 since MRTSK,L = MPL/MPK  2 = MPL/10  MPL =20
C. 10
D. It is not possible to say with the information given in the
problem
Figure 9.7: Total, Marginal, and Average Product
Curves
9-10
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
 When the marginal product curve lies below the
average product curve, the average product curve
must be falling.
 The two curves intersect at the maximum value of
the average product curve.
9-11
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)
Production In The Long Run
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.
9-13
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?
b) What is the APL function?
Key:
a) dQ/dL = MPL= 24L - 3L2
b) Q/L= APL= (12L2 – L3)/L = 12L – L2
Figure 9.8: Part of an Isoquant Map for
the Production Function
Figure 9.8: Part of an Isoquant
Figure 9.9: The Marginal Rate
Map for the Production Function of Technical Substitution
More is
preferred to less
Actual output
ΔQ0 =0 = Δ KMPK+ ΔLMPL
-ΔKMPK = ΔLMPL
|ΔK/ΔL| = MPL/MPK =MRTSL,K is
the slope.
9-15
Worked Out Problem
Question: 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:
Figure 9.10: Isoquant Maps for Perfect
Substitutes and Perfect Complements
9-17
Returns To Scale –Long Run Concept
Returns to Scale--The relationship between scale 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.
9-18
Figure 9.11: Returns to Scale Shown on
the Isoquant Map
9-19
Figure A9.1: Effectiveness vs. Use: Lobs
and Passing Shots
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 ∂P/∂L = 70 – 180L=0 or L* =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)
9-20
Figure A9.2: The Optimal Proportion
of Lobs
9-21
Figure A9.3: At the Optimizing Point, the Likelihood of
Winning with a Lob is Much Greater than of Winning with
a Passing Shot
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%)
9-22
Figure A9.4: The Production Mountain
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)
9-23
Figure A9.5: The Isoquant Map Derived from
the Production Mountain
9-24
Figure A9.6: Isoquant Map for the Cobb-Douglas
Production Function: Q = K½L½
MPL= 1/2K1/2L-1/2
MPK=1/2K-1/2L1/2
MPL/MPK = K/L
9-25
Figure A9.7: Isoquant Map for the Leontief
Production Function: Q = min (2K,3L)
9-26
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