107
2
costs. To express the shadow prices, or marginal rates of return,
as the MVP of capital the following transformation was made.
Letting,
R = revenue
C = total production cost exclusive of interest (that is, all
inputs expressed in terms of an aggregate capital input)
i = interest rate
SP = r = shadow price or net rate of return,
then,
R C(l+i) = SP = r (6-0
C(1+i)
Equation (6-1) expresses the rate of return to capital with
interest costs included, as calculated in the linear programming solu
tion. By simplifying and combining terms (Equations 6-2 through 6-4)
AVP can be expressed in terms of the shadow price, or rate of return.
This value for AVP does not have interest costs included.
R_ 1 = r (6-2)
c( 1+0
R_ = r + 1 (6-3)
C(l+i)
R = AVP = (r+1) (1 + 0 = r( 1 + 0 + i + 1 (6-4)
C
In a linear production function which passes through the origin
the MVP = AVP of an input.
Let Y = bX
thus, AP = Y = b
X
2
In this case, shadow price may be considered as a net rate of
return- in that it represents the increase in profita net value per
unit of input.