4. (20 points) You are starting a small business to sell toys. The demand for toys...

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4. (20 points) You are starting a small business to sell toys. The demand for toys are unlimit and far exceeds your production capacity. The selling price for one toy is $10 and you are price takers in the toy market (i.e. your production will not affect the market price). The annual production depends solely on labor working hours: annual production = 6 x N Where N denotes the annual labor working hours. Moreover, in the labor market, the labor's working hours depend directly on hourly wage you pay: N=-2500+ 1000 x w Where w denotes the hourly wage. In the first year of your business, you have a budget of $25000 as labor cost. Your goal is to maximize your annual profit revenue - labor cost. a. (6 points)How do you formulate the annual revenue function from selling toys in terms of hourly wage w? b. (6 points)How do you formulate the total labor cost function in terms of w? c. (8 points)Write down a non-linear program for this optimization problem. Appendix. Random Number Table 37 63 28 02 82 57 68 28 69 02 36 49 71 98 94 90 36 06 49 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00 84 57 07 74 35 24 03 29 60 74 85 90 73 59 55 17 60 05 94 03 11 27 79 90 87 92 41 09 25 36 77 99 32 10 75 21 95 90 94 38 97 71 72 49 78 23 67 89 85 29 21 25 73 69 34 85 76 56 59 23 78 71 72 90 57 01 98 57 31 95 11 96 52 62 87 33 69 27 21 60 95 89 68 48 17 89 34 09 93 50 44 51 50 33 50 95 13 44 34 62 64 39 55 29 30 64 49 44 30 16 88 32 18 50 62 57 34 56 62 31 15 40 90 34 51 95 26 14 90 30 36 24 69 82 51 74 30 35 36 85 01 55 92 64 09 85 50 48 61 18 85 23 08 54 17 12 80 69 24 84 92 16 49 59 27 88 21 62 69 64 48 31 12 73 02 68 00 16 16 46 13 85 45 14 46 32 13 49 66 62 74 41 86 98 92 98 84 54 33 40 81 02 01 78 82 74 97 37 45 31 94 99 42 49 27 64 89 42 66 83 14 74 27 76 03 33 11 97 59 81 72 00 64 61 13 52 93 09 96 33 52 78 13 06 28 30 94 23 37 39 74 11 46 82 59 94 25 34 32 23 17 01 58 73 74 05 82 82 30 34 87 01 4. (20 points) You are starting a small business to sell toys. The demand for toys are unlimit and far exceeds your production capacity. The selling price for one toy is $10 and you are price takers in the toy market (i.e. your production will not affect the market price). The annual production depends solely on labor working hours: annual production = 6 x N Where N denotes the annual labor working hours. Moreover, in the labor market, the labor's working hours depend directly on hourly wage you pay: N=-2500+ 1000 x w Where w denotes the hourly wage. In the first year of your business, you have a budget of $25000 as labor cost. Your goal is to maximize your annual profit revenue - labor cost. a. (6 points)How do you formulate the annual revenue function from selling toys in terms of hourly wage w? b. (6 points)How do you formulate the total labor cost function in terms of w? c. (8 points)Write down a non-linear program for this optimization problem. Appendix. Random Number Table 37 63 28 02 82 57 68 28 69 02 36 49 71 98 94 90 36 06 49 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00 84 57 07 74 35 24 03 29 60 74 85 90 73 59 55 17 60 05 94 03 11 27 79 90 87 92 41 09 25 36 77 99 32 10 75 21 95 90 94 38 97 71 72 49 78 23 67 89 85 29 21 25 73 69 34 85 76 56 59 23 78 71 72 90 57 01 98 57 31 95 11 96 52 62 87 33 69 27 21 60 95 89 68 48 17 89 34 09 93 50 44 51 50 33 50 95 13 44 34 62 64 39 55 29 30 64 49 44 30 16 88 32 18 50 62 57 34 56 62 31 15 40 90 34 51 95 26 14 90 30 36 24 69 82 51 74 30 35 36 85 01 55 92 64 09 85 50 48 61 18 85 23 08 54 17 12 80 69 24 84 92 16 49 59 27 88 21 62 69 64 48 31 12 73 02 68 00 16 16 46 13 85 45 14 46 32 13 49 66 62 74 41 86 98 92 98 84 54 33 40 81 02 01 78 82 74 97 37 45 31 94 99 42 49 27 64 89 42 66 83 14 74 27 76 03 33 11 97 59 81 72 00 64 61 13 52 93 09 96 33 52 78 13 06 28 30 94 23 37 39 74 11 46 82 59 94 25 34 32 23 17 01 58 73 74 05 82 82 30 34 87 01

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