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Long Run Production Function
Theory of the Firm:
Production and Cost in the Long Run In the long run all inputs are variable!
Two or more variable input factors
Q=Q(K;L)
Herbert Stocker Different combinations of inputs can produce the
herbert.stocker@uibk.ac.at same output;
Insitute of International Studies Input substitution: degree to which one input can
University of Ramkhamhaeng be substituted for another.
&
Department of Economics
University of Innsbruck Some processes may not be conducive to
substitution.
Long Run Production Function Input Substitution
Input Substitution:
Labor-intensive method: process that uses large Factors Influencing Input Substitution:
amounts of labor relative to other inputs. Technology
Prices of inputs
Incentives facing a given producer, e.g.
competitiveness of markets,
Capital-intensive method: process that uses large Nonprofit organizations or firms with market power
may have fewer incentives to minimize production
amounts of capital equipment relative to other costs.
inputs. X-inefficiency: inefficiency that may result in firms
having fewer incentives to minimize costs.
1
Description of Technology Isoquants
The Production Function again describes the Input substitution can easily be modeled and
maximum output that can be obtained with any illustrated with isoquants.
combination of inputs. This can be shown as a table
Isoquants: show all combinations of inputs that
→ K → produce a constant level of output.
0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1.00 1.62 2.16 2.64 3.09 3.51 3.90 4.29 4.66 5.01 Isoquants (constant out-
2 0 1.23 2.00 2.66 3.25 3.80 4.32 4.81 5.28 5.73 6.17
3 0 1.39 2.26 3.00 3.67 4.29 4.87 5.43 5.96 6.47 6.97 put) correspond to indif-
↓ 4 0 1.52 2.46 3.27 4.00 4.68 5.31 5.92 6.50 7.06 7.60 ference curves (constant 10
L 5 0 1.62 2.63 3.50 4.28 5.00 5.68 6.33 6.95 7.55 8.12 8
↓ 6 0 1.71 2.78 3.69 4.52 5.28 6.00 6.68 7.34 7.97 8.58 y
6
10
7 0 1.79 2.91 3.87 4.73 5.53 6.28 7.00 7.69 8.35 8.99 utility) in the theory of the 4
8
2
8 0 1.87 3.03 4.03 4.92 5.76 6.54 7.29 8.00 8.69 9.35 0
6
9 0 1.93 3.14 4.17 5.10 5.96 6.78 7.55 8.29 9.00 9.69 household. 0
0
x2
2
2
4
10 0 2.00 3.24 4.31 5.27 6.16 6.99 7.79 8.55 9.29 10.0 They are just like contour 4
4
x1
x1
6
6
2
8
8
or more simply as a function Q = Q(K;L) lines on a map! 10
10
0
Cobb-Douglas Production Function Isoquants
Isoquants
Product curves
for K
L
higher
output
10 10
8 8
10
→ 10 y
6
U ) 6 4
8
Q( 2
4 0
6
8 0
0
x2
2 2
2
4
4
4
Output0 6 x1
x1
6
6
2
00 → 8
8
)Q2 10
10
0
4 K
22 ( K
Production- 44
function 66 2 Capital
Lab
Q1Q1 or (
L) → 88
1010 0
2
Isoquants and Factor Substitution Technology
K Q
0
Well-behaved Technologies:
Monotonic: more inputs produce more output.
KUSA bc Convex: sometimes averages produce more than
extremes.
Wecan’t take monotonic transformations (like
KAfrica bc with utility functions) any more!
LUSA LAfrica L
The same output (e.g. 100 km road within one year)
can be produced with different factor intensities.
Marginal Product
Marginal Product: (MP )
1
MP is how much extra output you get from
Properties of Technology 1
increasing the input of factor 1 holding factor 2
fixed.
1 Marginal Product MP ≡∂Q; MP ≡∂Q
L ∂L K ∂K
2 Marginal Rate of Substitution
Diminishing marginal product:
3 Returns to Scale
more and more of a single input produces more
output, but at a decreasing rate → law of
diminishing returns.
The ‘law of diminishing returns’ is an important
property of almost all technologies!
3
Marginal Rate of Technical Substitution Marginal Rate of Technical Substitution
(MRTS) The MRTS can also be expressed as the ratio of
The MRTS is a measure for how easily input two marginal products:
factors can be substituted, holding output ∆K MP
MRTS=− = L
constant. ∆L MP
K
The MRTS is the slope of an isoquant
∆K Why? Remember that output is kept constant along a
MRTS=− isoquant, therefore ∆Q = 0.
∆L The total derivative of the production function is
∆Q=MP∆L+MP ∆K
The minus sign is added to make MRTS a positive L K
number, since ∆K=∆L, the slope of an isoquant, is Combining:
negative. ∆K MP
0 = ∆Q =MP ∆L+MP ∆K ⇒ − = L
it shows how many units of K are necessary to replace L K ∆L MP
one unit of L when output is kept constant. K
Diminishing MRTS MRTS
Diskrete changes: Infinitesimal small
Diminishing marginal rate of technical K changes:
substitution: the MRTS decreases the more K
bc
intensively a factor is used in production.
The isoquant becomes flatter, the further we ∆K
move along an axis.
equivalent to convexity. ∆K bc
Note the difference between diminishing MP and ∆K bc
diminishing MRTS.
∆L ∆L ∆L L L
4
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