Chapter 6
Slide 1
The Technology of Production
nProduction Function:
lA mathematical representation that Indicates the highest output that a firm can produce for every
specified combination of inputs given the state of technology (during time these conditions may
change).
lShows what is technically feasible when the firm operates efficiently.
lInputs= resources such labour, capital equipment, raw materials
nThe production function for two inputs, for a given technology:
n Q = f(K,L)  or Q ≤ f(K,L)
n          Q = quantity of output, K = quantity of capital employed, L = quantity of labour used

Chapter 6
Slide 2


Chapter 6
Slide 3
The Effect of Technological Improvement
Labor per
time period
Output
per
time
period
50
100
0
2
3
4
5
6
7
8
9
10
1
A
O1
C
O3
O2
B
Labor productivity
can increase if there
are improvements in
technology, even though
any given production
process exhibits
diminishing returns to
labor.

Chapter 6
Slide 4
The Technology of Production
nInput requirements function:
lShows the minimum amount of input (e.g. Labour) required to produce a given amount of output Q
l
nExample:
n Q = L1/2  then L=Q2
n
nto produce 7 units of output we need 49 units of labour

Chapter 6
Slide 5
nLabor Productivity: AP = Total Output/Total Labor Input
nLabor Productivity and the Standard of Living: Consumption can increase only if productivity
increases.
nDeterminants of Productivity
uStock of capital
uTechnological change
n
Production with One Variable Input (Labor)

Chapter 6
Slide 6
Amount Amount Total Average Marginal
of Labor (L) of Capital (K) Output (Q) Product Product
Production with One Variable Input (Labor)
n 0 10 0 --- ---
n 1 10 10 10 10
n 2 10 30 15 20
n 3 10 60 20 30
n 4 10 80 20 20
n 5 10 95 19 15
n 6 10 108 18 13
n 7 10 112 16 4
n 8 10 112 14 0
n 9 10 108 12 -4
n 10 10 100 10 -8

Chapter 6
Slide 7


Chapter 6
Slide 8
Total product functions
nSingle-intput production function (e.g. labour)
nIncreasing marginal returns to labour: the region along the total product function where output
rises with additional labour at an increasing rate
nDiminishing marginal returns to labour: the region along the total product function where output
rises with additional labour at an increasing rate
nDiminishing total returns to labour: the region along the total product function where output
rises with additional labour at an increasing rate
n
n

Chapter 6
Slide 9
1)With additional workers, output (Q) increases, reaches a maximum, and then decreases.
2)The average product of labor (AP), or output per worker, increases and then decreases.
Production with One Variable Input (Labor)

Chapter 6
Slide 10


Chapter 6
Slide 11
n3) The marginal product (MP) of labor or output of the additional worker increases rapidly
initially and then decreases and becomes negative..
Production with One Variable Input (Labor)

Chapter 6
Slide 12


Chapter 6
Slide 13
Production function with more than one input
nhow easily a firm can substitute among the inputs within its production
ntwo inputs: labor and capital
nthe quantity of output Q depends on the quantity of labor L and the quantity of capital K employed
ntotal product hill - a three-dimensional graph that shows the relationship between the quantity of
output and the quantity of the two inputs employed by the firm.

Chapter 6
Slide 14
Production Function for Food
n1 20 40 55 65 75
n2 40 60 75 85 90
n3 55 75 90 100 105
n4 65 85 100 110 115
n5 75 90 105 115 120
Capital Input 1 2 3 4 5
Labor Input

Chapter 6
Slide 15


Chapter 6
Slide 16
Isoquants
nTo illustrate economic trade-offs, it helps to reduce the three-dimensional graph of the
production function (the total product hill) to two dimensions
nObservations:
n 1) For any level of K, output increases with more L.
n 2) For any level of L, output increases with more K.
n 3) Various combinations of inputs produce the same output.
nIsoquants: Curves showing all possible combinations of inputs that yield the same output

Chapter 6
Slide 17


Chapter 6
Slide 18
Production with Two Variable Inputs (L,K)
Labor per year
1
2
3
4
1
2
3
4
5
5
Q1 = 55
The isoquants are derived
from the production
function for output of
of 55, 75, and 90.
A
D
B
Q2 = 75
Q3 = 90
C
E
Capital
per year
The Isoquant Map

Chapter 6
Slide 19
Isoquants - exercise
nConsider the production function whose equation is given by the formula
(a)What is the equation of the isoquant corresponding to Q  20?
(b)For the same production function, what is the general equation of an isoquant, corresponding to
any level of output Q?

Chapter 6
Slide 20


Chapter 6
Slide 21


Chapter 6
Slide 22
Isoquants
nThe backward-bending and upward-sloping regions of the isoquants make up the uneconomic region of
production. In this region, the marginal product of one of the inputs is negative (diminishing
total returns). nA cost-minimizing firm would never produce in the uneconomic region.

Chapter 6
Slide 23
Marginal rate of technical substitution (MRTS)
nIf we would like to invest in sophisticated robotics we would naturally be interested in the
extent to which it can replace humans with robots nHow many robots will it need to invest in to
replace the labor power of one worker? nThe “steepness” of an isoquant determines the rate at which
the firm can substitute between labor and capital in its production process= MRTS
n

Chapter 6
Slide 24
Marginal rate of technical substitution (MRTS)
nThe rate at which the quantity of capital can be decreased for every one-unit increase in the
quantity of labor, holding the quantity of output constant, or nThe rate at which the quantity of
capital must be increased for every one-unit decrease in the quantity of labor, holding the
quantity of output constant nAs we move down along the isoquant, the slope of the isoquant
increases (becomes less negative), which means that the MRTSL,K gets smaller and smaller =>
diminishing marginal rate of technical substitution
n

Chapter 6
Slide 25


Chapter 6
Slide 26


Chapter 6
Slide 27
Marginal rate of technical substitution (MRTS)
nWhen the production function offers limited input substitution opportunities, the MRTSL,K changes
substantially as we move along an isoquant. In this case, the isoquants are nearly L-shaped, nWhen
the production function offers abundant input substitution opportunities, the MRTSL,K changes
gradually as we move along an isoquant. In this case, the isoquants are nearly straight lines

Chapter 6
Slide 28
Elasticity of substitution
na numerical measure that describe the firm’s input substitution opportunities nmeasures how
quickly the marginal rate of technical substitution of labor for capital changes as we move along
an isoquant = the percentage change in the capital–labor ratio for each 1 percent change in MRTSL,K
as we move along an isoquant:

Chapter 6
Slide 29


Chapter 6
Slide 30
Elasticity of substitution
nConsider a production function whose equation is given by the formula:
n
n
na) Show that the elasticity of substitution for this production function is exactly equal to 1, no
matter what the values of K and L are.

Chapter 6
Slide 31
Special production functions
nLINEAR PRODUCTION FUNCTION (PERFECT SUBSTITUTES) nFIXED-PROPORTIONS PRODUCTION FUNCTION  (PERFECT
COMPLEMENTS)
nCOBB-DOUGLAS PRODUCTION FUNCTION
nCONSTANT ELASTICITY OF SUBSTITUTION PRODUCTION FUNCTION
n

Chapter 6
Slide 32
Linear Production Function
nthe marginal rate of technical substitution of one input for another may be constant nA production
function of the form Q=aL + bK, where a and b are positive constants. nA linear production function
is a production function whose isoquants are straight lines nthe slope of any isoquant is constant,
and the marginal rate of technical substitution does not change as we move along the isoquant.
nMRTSL,H=0, this means that the elasticity of substitution for a linear production function
must be infinite

Chapter 6
Slide 33


Chapter 6
Slide 34
Fixed-proportions production function
nA production function where the inputs must be combined in fixed proportions e.g. isoquants for
the production of water, where the inputs are atoms of hydrogen (H) and atoms of oxygen (O).
nAdding more hydrogen to a fixed number of oxygen atoms gives us no additional water molecules;
neither does adding more oxygen to a fixed number of hydrogen atoms.
nThus, the quantity Q of water molecules that we get is given by:
n
n
nthe elasticity of substitution is zero

Chapter 6
Slide 35


Chapter 6
Slide 36
Cobb-Douglas Production Function
nintermediate between a linear production function and a fixed-proportions production function. nA
production function of the form Q=ALαKß where Q is the quantity of output from L units of labor and
K units of capital and where A, α and ß are positive constants. nWith the Cobb–Douglas production
function, capital and labor can be substituted for each other. nthe elasticity of substitution
along a Cobb–Douglas production function is always equal to 1

Chapter 6
Slide 37


Chapter 6
Slide 38
Returns to scale
nTheme - how increases in all input quantities affect the quantity of output the firm can produce.
nWhen inputs have positive marginal products, a firm’s total output must increase when the
quantities of all inputs are increased simultaneously
n

Chapter 6
Slide 39
Returns to scale
nincreasing returns to scale. In this case, a proportionate increase in all input quantities
results in a greater than proportionate increase in output. nconstant returns to scale. In this
case, a proportionate increase in all input quantities results in the same proportionate increase
in output. ndecreasing returns to scale. In this case, a proportionate increase in all input
quantities results in a less than proportionate increase in output.

Chapter 6
Slide 40
Firm Size and Output: Increasing Returns to Scale
Labor (hours)
Capital
(machine
hours)
10
20
30
Increasing Returns to scale: The isoquants move
closer together
5
10
2
4
0
A

Chapter 6
Slide 41
Firm Size and Output: Constant Returns to Scale
Labor (hours)
Capital
(machine
hours)
Constant Returns:
Isoquants are                              equally spaced
10
20
30
15
5
10
2
4
0
A
6

Chapter 6
Slide 42
Firm Size and Output: Decreasing Returns to Scale
Labor (hours)
Capital
(machine
hours)
Decreasing Returns:
Isoquants get further
apart
10
20
30
5
20
2
8
0
A