Some Simple Production Mechanics

Lecture V

Single Product Primal Optimization

Profit Maximization

1 2 1 1 2 2

11 1 1

2 12

22 2

max

0

0

Y

Y

Y

p x x w x w x

Yp wx x wx x

Y wp wx x

11 1 2 1

1

11 1

2

111 1

2

1111

2

1111

* 1 11 1 2

2

0 0

, ,

Y Y

Y

Y

Y

Y Y

Yp w p x x wx

wp x ww

wp x ww

wp xw

wx p w w pw

1111

11 12

2 2

1 11 1111 1

2

11* 12 1 2

1

1 111 1 111 11 1 11

, ,

Y

Y

Y Y

w wx pw w

wpw

x p w w pw

11

2w

By convention

11 11

* 11 1 2

1 2

11 11

* 12 1 2

2 1

, ,

, ,

Y Y

Y Y

x p w w pw w

x p w w pw w

Supply Curve

1

1 11* 1

1 21 2

11 11

1

1 2

1 11

1 2

11

1 2

, ,Y Y

Y

Y

Y

Y p w w pw w

pw w

pw w

pw w

1

Profit Function

1 1

* 11 2

1 2

11 11

11

1 2

11 11

12

1 2

, ,Y Y Y

Y

Y

p w w p pw w

w pw w

w pw w

Cost Minimization

1 1 2 2

1 2

min

. .

w x w x

s t x x Y

1 1 2 2 1 2

11 1 2 1

1 1

12 1 2 2

2 2

1 2

0 0

0 0

0

L w x w x Y x x

L Yw x x wx xL Yw x x wx xL Y x x

Taking the ratio of the first two first-order conditions yields:

1 1 2 1

2 1 2

2

12 1

2

Yx w x wY w x wx

wx xw

11 1

2

11

2

1* 21 1 2

1

1* 12 1 2

2

0

0

, ,

, ,

wY x xw

wY xw

wx Y w w Yw

wx Y w w Yw

Substituting these results into the cost function yields:

11 1

2 11 2 1 2

1 2

, , w wC Y w w wY w Yw w

Note that the profit function can be derived from the cost function:

1 2

1 2*1 2

max , ,

, ,, , : 0

Y

Y Y

p Y C Y w w

C Y w wY p w w Y p

Y

Multiproduct Primal Functions

As discussed in class, the production function can be extended to multiple outputs. However, closed form functions are somewhat limited.

In this section, I want to briefly discuss the theoretical application of the multproduct primal function within the context of a planting problem.

Specifically, assume that there exists a multivariate production function, f(y,x), where y is a vector of (two) outputs, and x is a vector of (two) inputs.

The profit function for this formulation can be formulated as

1 1 2 2 1 1 2 2max

. . , 0p y p y w x w xs t f y x

1 1 2 2 1 1 2 2 ,L p y p y w x w x f y x

1 21 1 2 2

1 21 2

1 21 1 2 2

1 21 2

. .0;

0; 0

. . .0; 0

0; 0

f fL Lp py y y y

L Ly yy y

f f fL w wx x x x

L Lx xx x

Taken together, these conditions imply that the value of marginal product of each input equals the input price, if positive quantities of each output are produced and positive quantities of inputs are used

These conditions admit three possible solutions. First, if only y1 is produced:

1 11 1

2 22 2

.0, 0

.0, 0

fL p yy y

fL p yy y

Second, only y2 could be produced:

1 11 1

2 22 2

.0, 0

.0, 0

fL p yy y

fL p yy y

Third, both outputs could be produced:

1 11 1

2 22 2

.0, 0

.0, 0

fL p yy y

fL p yy y

1 21

2 1

2

.

.

fp yy

fp yy

1y

2y