The document discusses production functions and their relationships. It shows that a production function relates the maximum quantity of output (Q) that can be produced from given amounts of inputs (capital K and labor L). The production function is represented as Q=f(K,L). It then derives and graphs the equations for total product (Q), average product (APL), and marginal product (MPL) based on the Cobb-Douglas production function of Q=K^0.3 L^0.8. It finds that average product is maximized when average product equals marginal product.

1. M. Mahdi Mesbahi

2. 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)
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Created by M. Mahdi Mesbahi

3. Q= aK L
K=Capital
Q
L=Labor
Short Run: L
K=cte. K
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Created by M. Mahdi Mesbahi

4. 4.5
Q= aK L 4
3.5
3
K=Capital (xK )
2.5
2
L=Labor
Q
1.5
1
0.5
Short Run: 0
0 10 20 30
K=cte.
L
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Created by M. Mahdi Mesbahi

5. APL= K.3 L-.2
APL= Q/L Q= APL.L = K.3 L(-.2+1)
Q= K.3 L.8
MPL= Q/ L = d (K.3 L.8) = 0.8 K.3 L(.8-1)
dL
MPL = 0.8K.3 L-.2
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Created by M. Mahdi Mesbahi

6. Q= K.3 L.8
APL= K.3 L-.2 APL= K.3 L-.2
MPL = 0.8K.3 L-.2 MPL = 0.8(K.3 L-.2)
MPL = 0.8 APL
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Created by M. Mahdi Mesbahi

7. Q= K.3 L.8 APL maximum
when
APL= K.3 L-.2 APL = MPL
MPL = 0.8K.3 L-.2 APL = 0.8 APL
MPL = 0.8 APL APL = 0 K.3 L-.2=0
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L= ∞
Created by M. Mahdi Mesbahi

8. Q= K.3 L.8 APL maximum
when
APL= K.3 L-.2 APL = MPL
MPL = 0.8K.3 L-.2 APL = 0.8 APL Max
Max
MPL = 0.8 APL APL = 0 K.3 L-.2=0
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L= ∞
Created by M. Mahdi Mesbahi

9. L Q=K0.3L0.8 APL=K0.3L-0.2 MPL=0.8K0.3L-0.2
0.000001 1.58489E-05 15.84893192 12.67914554
Q= K.3 L.8 0.00001
0.0001
1E-04
0.000630957 6.309573445
10
5.047658756
8
APL= K.3 L-.2
0 0.001
0.01
0.1
0.2
0.003981072
0.025118864
0.158489319
0.275945932
3.981071706
2.511886432
1.584893192
1.379729661
3.184857364
2.009509145
1.267914554
1.103783729
0.3 0.381677891 1.272259637 1.017807709
0.4 0.480449774 1.201124434 0.960899547
MPL = 0.8K.3 L-.2
0.5 0.574349177 1.148698355 0.918958684
0.6 0.664539806 1.107566343 0.886053075
0.7 0.751758647 1.073940924 0.859152739
0.8 0.836511642 1.045639553 0.836511642
0.9 0.919166119 1.021295688 0.81703655
1 1 1 0.8
2 1.741101127 0.870550563 0.696440451
3 2.408224685 0.802741562 0.642193249
4 3.031433133 0.757858283 0.606286627
Short Run: 5
6
3.623898318
4.192962713
0.724779664
0.698827119
0.579823731
0.559061695
7 4.743276394 0.677610913 0.542088731
K=cte. 8 5.278031643 0.659753955 0.527803164
∞
9 5.799546135 0.644394015 0.515515212
10 6.309573445 0.630957344 0.504765876
20 10.98560543 0.549280272 0.439424217
Page 9 of 9 30 15.19487052 0.506495684 0.405196547
Created by M. Mahdi Mesbahi

10. 5
4.5
4
3.5
Q , APL , MPL
3
Q
(xK0.3)
2.5
2
APL
MPL
1.5
1
0.5
0
∞
-1 0 1 2 3 4 5 6
L
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Created by M. Mahdi Mesbahi

11. Present like Professionals
Created by:
Mahdi Mesbahi
mahdi.mesbahi@gmail.com