Reference no: EM13887984
A ?rm has two processes for producing a good, X. Each process uses ?ve chemicals: A, B, C, D, and E. The ?rst process, when operated at unit level, produces 3 tons of X and uses 5, 4, 3, 2, and 1 tons of chemicals A, B, C, D, and E, respectively. If the process is operated at level a, where a > 0, it uses 5a , 4a , 3a , 2a, and a tons of the chemicals, respectively, and produces 3a tons of X. The second process, when operated at unit level, produces 4 tons of X and uses 1, 2, 3, 4, and 5 tons, respectively, of chemicals A, B, C, D, and E. If this process is operated at level a, where a > 0, it uses a , 2a , 3a , 4a, and 5a tons, respectively, of the chemicals and produces 4a tons of X. The company has 10 tons of each chemical available. The price of X is $100 per ton.
(a) What is the maximum amount of X the ?rm can produce? (Hint: The requirement that no more than 10 units of the nth resource be used places a linear constraint on the outputs, y1 and y2, produced by the two processes. Calculate and graph these constraints for each resource. The answer should then be obvious.)
(b) At this optimum, how much is produced by each process?
(c) What is the maximum price the ?rm is willing to pay for additional marginal amounts of each chemical? (Hint: Calculate the Kuhn- Tucker coef?cients corresponding to each resource constraint.)
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