107
costs. To express the shadow prices, or marginal rates of return,2 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. (6-1)
C(1+i)
Equation (6-I) expresses the rate of return to capital with
interest costs included, as calculated in the linear programming solution. 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(I+i)
R = r + 1 (6-3)
C(l+i)
R = AVP = (r+l) (I+i) = r(l+i) + 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
2
In this case, shadow price may be considered as a net rate of return-in that it represents the increase in profit--a net value per unit of input.