1. A firm produces two products X and Y. Product X requires 2 hour of labor , 1

Question

1. A firm produces two products X and Y. Product X requires 2 hour of labor , 1 hour of machine and 1 unit of primary raw material. Product Y requires 1 hour of labor, 2 hours of machining, and 1 unit of primary raw material. There are 20 labor hours available for production, and 24 hours of machine time. Only 14 units of the primary raw material is given for this period. A unit of product X gives $3,000 profit, and Product Y $4,000.

a) What combination of X and Y units produced and sold can maximize the firm's profit this period.

b) Determine the maximum profit in dollars.

Explanation / Answer

BY GIVEN INFORMATION

iF X PRODUCES 6 UNITS TO EARN A PROFIT OF (6*3000 = 18000) THEN IT NEEDS TO CONSUME

12 LABOUR HRS; 6 MACHINE HRS & 6 PRIME RAW MATERIAL .(PRM)

TOTAL AVALIABLITY OF THESE FACTORS IS 20 LABOR HRS; 24 MACHINE HRS & 14 PRIMARY RAW MATERIAL HENCE FACOTRS LEFT WITH FIRM Y IS 8 LABOR HRS; 16 MACHINE HRS & 8 PRM

HENCE ATMOST 8 UNITS OF Y CAN BE PRODUCED THERE FORE WE HAVE A PROFIT OF ($8*4000) = $32000

FOR (X,Y) = (6,6) WE HAVE A PROFIT OF $ 50,000.

SIMILARLY FOR PAIRS (7,6) & (5,7) WE HAVE PROFITS OF $ 45000 & $48000 HENCE IF WE DIFFERENTIATE THE PROFIT FUNCTION WITH RESPECT TO QUANTITIES PRODUCED OF X & Y WE HAVE IT IS MXIMUM AT (6,6).

EVEN IF WE PRODUCE ONLY Y UNITS WE HAVE A PROFIT OF 48000 AS MAXIMUM OF 14 Y UNITS COULD BE PRODUCED.

MAXIMUM PROFIT IS 50000 USD FOR 6 UNITS OF X & 6 UNTS OF Y

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