
The Production Function

Production In The Short Run

Production In The Long Run

Returns To Scale
9-2

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

 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

Assume: K= K
0
= 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: MP
L

= ∆TP
L
/∆L= ∆Q/∆L
Average product: total output divided by the quantity of the variable input:
AP
L
=TP
L
/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 MRTS
K,L
C. 10
= MP
L
/MP
K
 2 = MP
L
/10  MP
L
=20
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

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 AP
L
( should be here )


the set of all input combinations that yield a given level of output.

the rate at which one input can be exchanged for another
9-13

Question: A daily production function for calculators is
Q = 12L 2 - L 3 . Show all your work for the following questions. a) What is the marginal product equation for labor?
b) What is the AP
L function?
Key: a) dQ/dL = MP
L
= 24L - 3L 2 b) Q/L= AP
L
= (12L 2 – L 3 )/L = 12L – L 2

Figure 9.8: Part of an Isoquant
Map for the Production Function
Figure 9.9: The Marginal Rate of Technical Substitution
More is preferred to less
Actual output
ΔQ
0
=0 =
ΔKMP
K
=
Δ KMP
ΔLMP
| ΔK/ΔL| = MP
L
L
K
+ ΔLMP
/MP
L
K
=MRTS
L,K the slope.
is
9-15

Question: In the production function
Q =10L 1/2 K 1/2 , 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-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

9-19

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 = 90L 2 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

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

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 = K ½ L ½
MP
L
= 1/2K 1/2 L -1/2
MP
K
=1/2K -1/2 L 1/2
MP
L
/MP
K
= K/L
9-25

Figure A9.7: Isoquant Map for the Leontief
Production Function: Q = min (2 K ,3 L )
9-26