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M. Mahdi Mesbahi
Managerial EconomicsTHE COBB-DOUGLAS PRODUCTION FUNCTION
Definition of the Production Function
A Production Function relates
the maximum quantity of output
(Q) that can be produced from
given amounts of inputs (K,L)
Q= f(K,L)
Page 2 of 9
A Production Function relates
the maximum quantity of output
(Q) that can be produced from
given amounts of inputs (K,L)
Q= f(K,L)
Created by M. Mahdi Mesbahi
Definition of the Cobb-Douglas P.F.
Q= aKa Lb
K=Capital
L=Labor
LK
Q
Page 3 of 9
Short Run:
K=cte.Sec
tionShort Run:
K=cte.
Created by M. Mahdi Mesbahi
Definition of the Cobb-Douglas P.F.
Q= aKa Lb
K=Capital
L=Labor
Page 4 of 9
Short Run:
K=cte.0 5 10 15 20 25 30 35
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
L
Q
(
xK
a)
Created by M. Mahdi Mesbahi
The Problem:maximum of the Average Product
APL= K.3 L-.2
APL= Q/L Q= APL.L = K.3 L(-.2+1)
Q= K.3 L.8
MPL= dQ/dL= (K.3 L.8) =0.8 K.3 L(.8-1)
MPL = 0.8K.3 L-.2
Page 5 of 9
ddL
Created by M. Mahdi Mesbahi
The Problem:maximum of the Average Product
Q= K.3 L.8
APL= K.3 L-.2
MPL = 0.8K.3 L-.2
APL= K.3 L-.2
MPL = 0.8(K.3 L-.2)
MPL = 0.8 APL
Page 6 of 9Created by M. Mahdi Mesbahi
The Problem:maximum of the Average Product
Q= K.3 L.8
APL= K.3 L-.2
MPL = 0.8K.3 L-.2
APL maximum
APL = MPL
MPL = 0.8 APL
APL = 0.8 APL
APL = 0 K.3 L-.2=0
L=∞Page 7 of 9
whenwhen
Created by M. Mahdi Mesbahi
The Problem:maximum of the Average Product
Q= K.3 L.8
APL= K.3 L-.2
MPL = 0.8K.3 L-.2
APL maximum
APL = MPL
MPL = 0.8 APL
APL = 0.8 APL
APL = 0 K.3 L-.2=0
L=∞Page 8 of 9
whenwhen
Created by M. Mahdi Mesbahi
Marginal
Production
Average Production
Max
Max
The Problem:maximum of the Average Product
Q= K.3 L.8
APL= K.3 L-.2
MPL = 0.8K.3 L-.2
L Q=K0.3L0.8 APL=K0.3L-0.2 MPL=0.8K0.3L-0.2
0.000001 1.58489E-05 15.84893192 12.679145540.00001 1E-04 10 80.0001 0.000630957 6.309573445 5.0476587560.001 0.003981072 3.981071706 3.1848573640.01 0.025118864 2.511886432 2.0095091450.1 0.158489319 1.584893192 1.2679145540.2 0.275945932 1.379729661 1.1037837290.3 0.381677891 1.272259637 1.0178077090.4 0.480449774 1.201124434 0.9608995470.5 0.574349177 1.148698355 0.9189586840.6 0.664539806 1.107566343 0.8860530750.7 0.751758647 1.073940924 0.8591527390.8 0.836511642 1.045639553 0.8365116420.9 0.919166119 1.021295688 0.817036551 1 1 0.82 1.741101127 0.870550563 0.6964404513 2.408224685 0.802741562 0.6421932494 3.031433133 0.757858283 0.6062866275 3.623898318 0.724779664 0.5798237316 4.192962713 0.698827119 0.5590616957 4.743276394 0.677610913 0.5420887318 5.278031643 0.659753955 0.5278031649 5.799546135 0.644394015 0.515515212
10 6.309573445 0.630957344 0.50476587620 10.98560543 0.549280272 0.43942421730 15.19487052 0.506495684 0.405196547
0
∞Page 9 of 9
Short Run:
K=cte.
Created by M. Mahdi Mesbahi
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
QAPLMPL
L
Q ,
APL
, M
PL
(xK
0.3
)
The Problem:maximum of the Average Product
∞
Page 10 of 9Created by M. Mahdi Mesbahi
