M injx m2x2 0 0pM wJ

The Lagrange multipliers are ususally referred to as dual variables in linear programming. The key point is their interpretation as the shadow prices of the input constraints. At the optimal solution, the value of A.*, p*, or p' gives the increase in revenue the firm would earn if it acquired a little bit more of the respective inputs and allocated that optimally between the outputs. The last three conditions also tell us that if a shadow price is positive, all of the corresponding input is used up, while if at the optimum, there is some amount of an input unused; its shadow price is zero.

It is always useful to use a diagram to obtain a sense of the solution possibilities. In figure 15.4 we see that there may be as many as seven The lines denoted R are iso-revenue lines; this means that they show U|, .ii )-pairs that generate the same revenue, and so they satisfy

R = p i.ri 4- p2Xi for given R. 7"hey therefore have slope -p\/p2. and at given prices, the higher the line, the greater the revenue. The shaded area is the feasible set in each case, and we have assumed that the exact form of the constraints is such that each could be binding. If one constraint coincided with another or lay entirely outside another, then it could be dropped and the number of solution possibilities would consequently fall. The constraint L" has slope M" has slope —m \/m2, and

R° has slope —ri/r2. No point above any constraint line is feasible. The feasible set is the intersection of the sets of points lying on or below a constraint line. It is assumed in the figure that

11 m2 r2

15 p MANY VARIARI fp. ANn CONSTRAINTS 691

(g)

Figure 15.4 Solution possibilities in the linear-programming problem

Then, which of the solution possibilities results depends on the value of p\/p2 relative to the slopes of the constraint lines.

In figure 15.4 (a), we have p\/p2 < l\/h- and so the solution is at point a. with .vi > 0 and v," = 0. Note also that only the labor constraint is binding, so V > 0 while n' = p' = 0.

In figure 15.4 (b). we have p,/p2 — Ulh-The highest possible revenue line coincides with the labor constraint. In that case any point on the segment [a. h\ is optimal. The labor constraint is binding, so A* > 0. However, note that small variations in either of the two other constraints cannot change the maximized value of revenue, and so again /<* = p* = 0.

In figure 15.4(c), we have/|//2 < p\/p2 < wi/wj. There is a unique optimum at I), with x | > 0. JC, > 0. In addition both labor and machine-time constraints are binding, and so A* > 0. p* > 0 while p' = 0.

In figure 15.4(d), we have p\/p2 =»t|/»ij. Any point on the segment [b, cl is optimal, and so .rf >0, Jtj > 0. The machine-time constraint is certainly binding, and so p' > 0. However, small variations in the other two constraints leave maximum revenue unchanged, and so V = p* = 0.

In figure 15.4 (e), we have ni\/m2 < p\/p2 <r\/r2, and there is a unique solution at c widt .c* > 0, x* > 0. The machine-time and raw-material constraints are both binding, and so p" > 0, p% > 0 but X* = 0.

In figure 15.4 (f). we have p\/p2 =r|/>2, and so any point on the segment |c, d\ is optimal. The raw-material constraint is certainly binding, and so p" > 0. hut small variations in the other two constraints leave maximized revenue unchanged, and so A.* -- p" - 0.

In figure 15.4 (g), we have p\/p2 > r{/r2, and so there is a unique solution with .t* > 0, x* = 0. Only the raw-material constraint is binding, and so p* > 0 and k' = p* = 0.

One notable feature of the solutions is that the optimum could always be taken to be at a comer point of the upper boundary of the feasible set. a point such as a. b, c, or d. This is what greatly facilitates numerical solution of linear problems: it is necessary simply to evaluate the objective function at corner points, rather than over the entire feasible set. However, here we are interested in the economic, rather than the computational, aspects of the solution.

A second notable feature is that there were never more than two binding constraints at any solution: at least one constraint was always nonbinding. This is a consequence of the fact that there are two variables in the problem. Thus consider the hist three K-T conditions above and note that if ai most two variables can be strictly positive, then three constraints cannot be nontrivially binding, because that would give three equations in only two unknowns.

Consider now the first of the K-T conditions, which relates to the optimal outputs. Suppose that both v* > (• and .v* > 0. and (hat only the raw-material constraint is nonbinding so that we have p" = 0. Then these conditions take the form

P\ - A.*/1 - p.*mj = 0 p2 - A"l2 - M*"': = 0

Recall that a' and p" are interpreted as the shadow prices of the inputs. Then these conditions have the interpretation of "marginal revenue = marginal cost' conditions. To see this, note that I, is the amount of labor used in producing one unit of v, so that A 7, is the imputed cost, valued at the shadow price of labor, of the amount of labor used id produce one unit of a, . Likewise //;, is the amount of machine time used per unit of a, . and so p'm, is the imputed cost of the machine time used per unit of ,v(. Therefore A 7, -f p'm, is the imputed cost of a unit of.*,, while p, is the marginal revenue of a,.

Notice that whatever may be the actual prices the firm pays for the inputs, these unit costs are evaluated at the shadow prices because these are the appropriate measures of the marginal opportunity costs of the inputs to the firm. In particular, note that the cost of raw material plays no part in the unit cost calculation. Because p* — 0, this input is not relatively scarce to the firm.

Reluming to the K-T conditions, we see that

implies that, c* = (J. This says that if the marginal revenue falls short ol its uml cost evaluated at the appropriate shadow prices, then the good should nol be produced All the resources should be allocated to the other good for which the equality in this condition will hold.

Finally, note that multiplying through the first condition bv ,rj\ and the second by xf, and adding gives

PiX* + p2x\ = A*(/|X|* + hx'J + n*(m I*" + m2x\ i = VL* + i.rMn

Therefore, nol only do the shadow prices give the marginal value or marginal-opportunity cost of each input, but they do so in such a way that the entire revenue of the firm is imputed to the inputs: we could regard as the share of revenue that can be imputed to labor, and /i \<V/° the share imputed to machine time, and these shares exactly exhaust available revenue.

A slight paradox here is that this makes it seem as though raw materials are valueless to the firm, which is clearly not true in an absolute sense, because raw material is used (in the amount n r," + r2x2 < R°) and output could not be produced without it. The point is that at the margin the stock of raw material is valueless, because the firm has more of it than is optimal to use. The result above then shows that the sum of the costs of the inputs evaluated at their shadow prices, just equals the total revenue. This is essentially a result of the constant-returns-to-scale assumption underlying the linear model.

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