The country of Utopia has a newly appointed Minister of International Trade. She has decided that Utopia's welfare can be served best in the upcoming year by maximizing the net dollar value of Utopia's exports (i.e., the dollar value of the exports minus the cost of the materials imported to produce the exports). The following information is relevant to this decision.
• Utopia produces only three products: steel, machinery, and trucks. For the coming year, the minister feels Utopia can sell all it can produce of these three products on the export market at the existing world market prices of $900 per ton of steel, $2500 per machine, and $3000 per truck.
• To produce one ton of steel, it takes 0.05 machines, 0.08 trucks, 0.5 person-years of labor, and imported materials costing $300. Utopia's steel mills have the capacity to produce up to 300,000 tons per year.
• To produce one machine, it takes 0.75 tons of steel, 0.12 trucks, 5 person-years of labor, and imported materials costing $150. Utopia's machinery plants have the capacity to produce up to 50,000 machines per year.
• To produce one truck, it takes 1 ton of steel, 0.1 machine, 3 person-years of labor, and imported materials costing $500. Utopia's truck plants have the capacity to produce up to 550,000 trucks per year.
- The pool of labor in Utopia is equivalent to 1,200,000 person-years.
The minister plans to issue a self-sufficiency edict, declaring that Utopia cannot import steel, machinery, or trucks. She would like to determine the optimal production quantities and optimal export quantities for steel, machinery, and trucks when that edict is in force.
- Find the optimal export plan for Utopia's economy, under self-sufficiency.