- Decision makers should adopt a new activity if and only if the activity’s
direct benefit exceeds its opportunity cost. We measure this opportunity
cost by the sum of the resources used in the activity valued at their
respective shadow prices.
Nuts and Bolts
- Formulating linear programs requires identifying the relevant decision
variables, specifying the objective function, and writing down the
relevant constraints as mathematical inequalities. - Solving linear programs requires identifying the binding constraints and
solving them simultaneously for the optimal values of the decision variables. - For two-variable problems, the optimal solution can be found by
graphing the feasible region (framed by the binding constraint lines)
and superimposing contours of the objective function. The optimal
corner is found where the highest contour (or, for minimization
problems, the lowest contour) touches the feasible region. The optimal
corner determines which constraints are binding. - The shadow price of a constraint is found by changing the right-hand side
of the inequality by a unit, solving the binding constraints for the decision
variables, and recomputing the objective function. The shadow price is
simply the change between the new and old values of the objective.
Questions and Problems
- Explain whether LP techniques can be used in each of the following
economic settings.
a. There are increasing returns to scale in production.
b. The objective function and all constraints are linear, but the number
of decision variables exceeds the number of constraints.
c. The firm faces a downward-sloping linear demand curve. (To sell
more output, it must lower its price.)
d. The firm can vary the amounts of two basic chemicals in producing a
specialty chemical, but, for quality control reasons, the relative
proportions of chemicals must be between 40/60 and 60/40. - Which of the following formulations can be solved via the LP method?
a. Maximize: x 2y, subject to: x y 2 and 3x y 4.
b. Maximize: xy, subject to: x y 2 and 3x y 4.
c. Maximize: x 2y, subject to: x y 2 and 3x y 4.
d. Maximize: x 2y, subject to: x y 2 and 3x y 8.
e. Maximize: x 2y, subject to: x y 2 and x/(x y) .7.
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