Linear Programming Short-run decision making model –Optimizing technique –Purely mathematical Product prices and input prices fixed Multi-product production CRS Fixed input combinations

Basic Applications Optimal process selection Optimal product mix Satisfying minimum product requirements Long-run capacity planning

Solution Methods Graphical Method –Establish feasible region –Optimum on boundary of feasible region Primal Dual

Conventional Economic Analysis Built on production functions Assumed technically efficient processes determined and used Real world –Several feasible processes –Difficult to identify which one is most efficient

Technology Composed as finite number of processes –Uses inputs in fixed proportions to produce output Each process described by a set of technical coefficients (a 1, a 2,..., a m ) –Shows amount of 1 st through m th input needed to produce one unit of output

Steps Toward Solving a LP problem Express objective function as an equation Express constraints as inequalities Graph the inequality constraints and define feasible region Graph objective function as a isoprofit or isocost lines Find optimal solution

Profit Maximization Firm has fixed amounts of inputs available in short run –Referred to as constraints –Each unit of output yields a certain amount of profit Determine the activity level of each process that will maximize profit Formulate problem, solve graphically, then solve mathematically

Cost Minimization Formulate problem Solve graphically Solve algebraically

The Dual Problem and Shadow Prices Every primal problem (LP) has a dual Profit max primal has a cost min dual Solutions to dual are shadow prices –Change in value of objective function per unit change in constraint –Imputed or marginal value, or worth of input –Means of pricing output of one division that is input for another division –Means of pricing services of non-profit firms