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Constrained OptimizationLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The following was implemented in Maple by Marcus Davidsson (2009)
davidsson_marcus@hotmail.com
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">1) Introduction to Budget Constraints</Font></Text-field>JSFHJSFHJSFHWe start by noting that our general budget constrain is given by: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(); # input placeholderwhere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn is is the quantity of capital, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn is the quantity of labor, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiUEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiS0YnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GOVEnbm9ybWFsRic= is the price of capital and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiUEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GOVEnbm9ybWFsRic= is the price of laborThe above equation says that income LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiSUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn must be equal to the sum of capital and labor expenditures. We assume that we have L on the y-axis which means that the equation for L would be given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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print(); # input placeholder which can be written 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(); # input placeholderwhere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRJyYjOTIxO0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUklbXN1YkdGJDYlLUYvNiZRIlBGJ0YyL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGQ0ZFRjJGQ0ZFLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkZDRkUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlQvJSliZXZlbGxlZEdGN0YyRjg= is the intercept and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JJW1zdWJHRiQ2JS1JI21pR0YkNiZRIlBGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYyNiZRIktGJ0Y1RjhGO0Y1RjhGOy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYjNiYtRi82JUYxLUYjNiYtRjI2JlEiTEYnRjVGOEY7RjVGOEY7RkNGNUY4RjsvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlQvJSliZXZlbGxlZEdRJmZhbHNlRidGNS9GPFEnbm9ybWFsRic= is the slope and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== is the x-variableWe can now plot our budget 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%;Note that the combinations of K and L that are underneath the budget constraint are affordable combinations of K and L.
Note that the combinations of K and L that are above the budget constraint are non-affordable combinations of K and L.
Note also that the point where the budget line intercepts the y-axis (intercept) given by 160 in the graph above can be interpret as the number
of units of labor (y-axis) we would get if we buy labor for all our income 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
Note also that the point where the budget line intercepts the x-axis given by 20 in the graph above can be interpret as the number
of units of Capital (x-axis) we would get if we buy Capital for all our income 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We now note what happens when we increase income 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%;We can see that the slope remains the same but the intercept becomes larger which means that we can afford to consume a larger
amount of L and K since the budget line shifts outwardsWe now note what happens when we decrease income 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%;We can see that the slope remains the same but the intercept becomes smaller which means that we have to consume a smaller
amount of L and K since the budget line shifts downwards.We now note what happens when we increase the price of capital 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%;We can see that the amount of labor remains the same L=160 but the budget constraint shifts in on the K-axis (x-axis). Since the price of capital increases the consumer can afford a smaller amount of capital We now note what happens when we decrease the price of capital LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiUEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiS0YnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZkcC1JI21pR0YkNiZRKHJlc3RhcnRGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi9RIjtGJ0YvRjJGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0ZWLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUYsNiZRJXdpdGhGJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2Ji1GIzYmLUYsNiZRJnBsb3RzRidGL0YyRjVGL0YyRjVGL0YyRjUtRjk2L1EiOkYnRi9GMkY1RjwvRkBGPkZBRkNGRUZHRkkvRkxGUEZORlFGaW4tRl1vNiYtRiM2Ji1GLDYmUSpwbG90dG9vbHNGJ0YvRjJGNUYvRjJGNUYvRjJGNUZkb0ZRLUZSNiZGVEZXRlovRmduUSVhdXRvRictRjk2L1EifkYnRi9GMkY1RjxGZ29GQUZDRkVGR0ZJRksvRk9GTUZRLUYsNiZRJyYjOTIxO0YnRi9GMkY1LUY5Ni9RKiZjb2xvbmVxO0YnRi9GMkY1RjxGZ29GQUZDRkVGR0ZJRmhvRk4tSSNtbkdGJDYmUSQ4MDBGJ0YvRjJGNUZkb0ZkcEZkcC1JJW1zdWJHRiQ2JS1GLDYmUSJQRidGL0YyRjUtRiM2Ji1GLDYmUSJLRidGL0YyRjVGL0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZbcS1GX3E2JlEjNDBGJ0YvRjJGNUZkb0ZkcEZkcEZkcC1GY3E2JUZlcS1GIzYmLUYsNiZRI0tLRidGL0YyRjVGL0YyRjVGXXJGW3EtRl9xNiZRIzIwRidGL0YyRjVGZG9GZHBGZHBGZHBGZHBGZHBGZHBGZHBGZHBGZHAtRmNxNiVGZXEtRiM2Ji1GLDYmUSJMRidGL0YyRjVGL0YyRjVGXXJGW3EtRl9xNiZRIjVGJ0YvRjJGNUZkb0ZRRmBwRmBwRlEtRiw2JlEjeDFGJ0YvRjJGNUZbcS1GLDYmUSVwbG90RidGL0YyRjUtRl1vNiYtRiM2QS1JJm1mcmFjR0YkNigtRiM2JkZocEYvRjJGNS1GIzYmRl1zRi9GMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZddS8lKWJldmVsbGVkR0Y+LUY5Ni9RKCZtaW51cztGJ0YvRjJGNUY8RmdvRkFGQ0ZFRkdGSS9GTFEsMC4yMjIyMjIyZW1GJy9GT0ZmdS1GYnQ2KC1GIzYoRmJxLUY5Ni1RMSZJbnZpc2libGVUaW1lcztGJy9GNlEnbm9ybWFsRidGPEZnb0ZBRkNGRUZHRklGS0ZncEZqcUYvRjJGNUZmdEZodEZbdUZedUZgdS1GOTYvUSIsRidGL0YyRjVGPEY/RkFGQ0ZFRkdGSUZLL0ZPUSwwLjMzMzMzMzNlbUYnRmpxLUY5Ni9RIj1GJ0YvRjJGNUY8RmdvRkFGQ0ZFRkdGSUZob0ZOLUZfcTYmUSIwRidGL0YyRjUtRjk2L1EjLi5GJ0YvRjJGNUY8RmdvRkFGQ0ZFRkdGSUZldUZncEZgckZhdkZkcC1GLDYmUSZjb2xvckYnRi9GMkY1RmZ2LUYsNiZRJmJsYWNrRidGL0YyRjVGYXYtRiw2JlEqdGhpY2tuZXNzRidGL0YyRjVGZnYtRl9xNiZRIjFGJ0YvRjJGNUZhdkZkcC1GLDYmUSdsYWJlbHNGJ0YvRjJGNUZmdi1GXW82KC1GIzYoRmpxRmF2RmFzRi9GMkY1Ri9GMkY1LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmF2RmRwLUYsNiZRJXZpZXdGJ0YvRjJGNUZmdi1GXW82KC1GIzYsRml2Rlx3RmByRmF2Rml2Rlx3LUZfcTYmUSQyMjBGJ0YvRjJGNUYvRjJGNUYvRjJGNUZieEZleEYvRjJGNUYvRjJGNUZkb0ZRLUYsNiZRI3gyRidGL0YyRjVGW3FGanMtRl1vNiYtRiM2QUZhdEZidS1GYnQ2KC1GIzYoRmNyRmRwRmpxRi9GMkY1RmZ0Rmh0Rlt1Rl51RmB1RmF2RmpxRmZ2Rml2Rlx3RmByRmF2RmRwRl93RmZ2RmJ3RmF2RmV3RmZ2LUZfcTYmUSIzRidGL0YyRjVGYXZGZHBGW3hGZnZGXnhGYXZGZHBGaHhGZnZGW3lGL0YyRjVGL0YyRjVGZG9GZHBGUUZkcEZRLUYsNiZRI3gzRidGL0YyRjVGW3EtRiw2JlEpdGV4dHBsb3RGJ0YvRjJGNS1GXW82Ji1GIzYqLUZdbzYoLUYjNistRl9xNiZRIzEwRidGL0YyRjVGYXYtRl9xNiZRIzg1RidGL0YyRjVGYXYtSSNtc0dGJDYjUTZOZXd+QnVkZ2V0fkNvbnN0cmFpbnRGJy1GLDYjUSFGJ0YvRjJGNUYvRjJGNUZieEZleEZhdi1GLDYmUSZhbGlnbkYnRi9GMkY1RmZ2LUZdbzYoLUYjNigtRiw2JlEmYWJvdmVGJ0YvRjJGNUZhdi1GLDYmUSZyaWdodEYnRi9GMkY1Ri9GMkY1Ri9GMkY1L0ZjeFEifGZyRicvRmZ4USJ8aHJGJ0YvRjJGNUYvRjJGNUZkb0ZRRmBwRlEtRiw2JlEjeDRGJ0YvRjJGNUZbcUZjei1GXW82Ji1GIzYqLUZdbzYoLUYjNistRl9xNiZRJDAuOEYnRi9GMkY1RmF2LUZfcTYmUSMzMEYnRi9GMkY1RmF2LUZlW2w2I1E2T2xkfkJ1ZGdldH5Db25zdHJhaW50RidGaFtsRi9GMkY1Ri9GMkY1RmJ4RmV4RmF2RltcbEZmdkZeXGxGL0YyRjVGL0YyRjVGZG9GZHBGUUZkcEZRRmBwLUYsNiZRKGRpc3BsYXlGJ0YvRjJGNS1GXW82Ji1GIzYqLUZdbzYoLUYjNixGZ3NGYXZGYnlGYXZGYHpGYXZGXF1sRi9GMkY1Ri9GMkY1RmhcbEZqXGxGYXZGZHAtRiM2KC1GLDYmUSVmb250RidGL0YyRjVGZnYtRl1vNigtRiM2Ki1GLDYmUSZUaW1lc0YnRi9GMkY1RmF2LUYsNiZRJlJvbWFuRidGL0YyRjVGYXYtRl9xNiZRIzE0RidGL0YyRjVGL0YyRjVGL0YyRjVGYnhGZXhGL0YyRjVGaFtsRi9GMkY1Ri9GMkY1RjhGL0YyLyUrZXhlY3V0YWJsZUdGPkY1%;We can see that the amount of labor remains the same L=160 but the budget constraint shifts out on the K-axis (x-axis). Since the price of capital decreases the consumer can afford a larger amount of capital We now note what happens when we increase the price of capital 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%;We can see that the amount of capital remains the same K=20 but the budget constraint shifts in on the L-axis (y-axis). Since the price of labor increases the consumer can afford a smaller amount of labor We now note what happens when we decrease the price of labor 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%;We can see that the amount of capital remains the same K=160 but the budget constraint shifts out on the L-axis (y-axis). Since the price of labor decreases the consumer can afford a larger amount of labor We can some further 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%;Income = use DocumentTools in
Do(%Plot0=myplot(%Slider0,%Slider1,%Slider2));
end use;
Price Capital = use DocumentTools in
Do(%Plot0=myplot(%Slider0,%Slider1,%Slider2));
end use;
Price Labour = use DocumentTools in
Do(%Plot0=myplot(%Slider0,%Slider1,%Slider2));
end use;
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">2) Constrained Maximization Graphically</Font></Text-field>Our objective function (or Cobb Douglas production function) is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JlEocmVzdGFydEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiOkYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGVy8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GUzYmRlVGWEZlbi9GaG5RJWF1dG9GJ0ZSLUYsNiZRIkFGJ0YvRjJGNS1GOTYuUSomY29sb25lcTtGJ0YvRjxGPkZBRkNGRUZHRklGS0ZNRlAtSSNtbkdGJDYlUSMxMkYnRi9GPEY4RlItRiw2JlEiWUYnRi9GMkY1RmFvRl5vLUY5Ni5RJyZzZG90O0YnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSYwLjBlbUYnL0ZRRl9wLUklbXN1cEdGJDYlLUYsNiZRIktGJ0YvRjJGNS1GIzYmLUZlbzYlUSQwLjRGJ0YvRjxGL0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRltwLUZicDYlLUYsNiZRIkxGJ0YvRjJGNUZncEZccUY4RlJGam5GUi1GOTYuUSJ+RidGL0Y8Rj5GQUZDRkVGR0ZJRktGXnBGYHAtRiw2JlEncGxvdDNkRidGL0YyRjUtSShtZmVuY2VkR0YkNiUtRiM2WEZoby1GOTYuUSIsRidGL0Y8Rj4vRkJGNEZDRkVGR0ZJRktGXnAvRlFRLDAuMzMzMzMzM2VtRidGZHFGZHBGZHEtRjk2LlEiPUYnRi9GPEY+RkFGQ0ZFRkdGSUZLRk1GUEZkcS1GZW82JVEiMEYnRi9GPEZkcS1GOTYuUSMuLkYnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSwwLjIyMjIyMjJlbUYnRmBwRmRxLUZlbzYlUSMyMEYnRi9GPEZfckZkcUZhcUZkcUZlckZkcUZockZkcUZbc0ZkcS1GZW82JVEkMTIwRidGL0Y8Rl9yRmRxLUYsNiZRJWF4ZXNGJ0YvRjJGNUZkcUZlckZkcS1GLDYmUSZib3hlZEYnRi9GMkY1Rl9yRmRxLUYsNiZRJnN0eWxlRidGL0YyRjVGZXItRiw2JlEtcGF0Y2hjb250b3VyRidGL0YyRjVGX3JGZHEtRiw2JlEmY29sb3JGJ0YvRjJGNUZlci1GLDYmUSVncmV5RidGL0YyRjVGX3JGZHEtRiw2JlEpY29udG91cnNGJ0YvRjJGNUZlckZgc0ZfckZkcS1GLDYmUSdsYWJlbHNGJ0YvRjJGNUZlci1GW3I2Jy1GIzYqRmRwRl9yRmFxRl9yLUYsNiZRJ091dHB1dEYnRi9GMkY1Ri8vJStleGVjdXRhYmxlR0ZARjxGL0Y8LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmRxRi9GZXVGPEYvRjxGZHFGL0ZldUY8%;As previously our constraint is given 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(); # input placeholderIn order to use the above budget constraint in constrained optimization (Lagrange) we have to express the above equation in an
equal to zero form. We therefore start by subtracting 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 on both 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print(); # input placeholderThis is the expression for our budget constraint in an equal to zero form. We can now plot such a budget 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%;JSFHWe can now plot the objective function and the budget constraint 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%;which also can be plotted (the above plot simply rotated 90 degrees) 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%;Note that we have constrained the plot above to only display where the optimal allocation is.We can see that the optimal point is L=80 and K=10LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">3) Constrained Maximization with Lagrange</Font></Text-field>We can now solve for the optimal allocation by using Lagrange. We assume as previously that the objective function is given by: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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=we also assume, as previously, that our budget constraint is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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(); # input placeholderOur Lagrange is therefore given 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(); # input placeholderthe first order conditions are given byfoc-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print(); # input 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print(); # input 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print(); # input placeholderfoc-2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlElZGlmZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEjTExGJ0YvRjJGNS1JI21vR0YkNi5RIixGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJMRidGL0YyRjVGL0ZERi9GRC1GQTYuUSI9RidGL0ZERkYvRkpGSEZLRk1GT0ZRRlMvRlZRLDAuMjc3Nzc3OGVtRicvRllGXW8tSSNtbkdGJDYlUSIwRidGL0ZERi8vJStleGVjdXRhYmxlR0ZIRkQ=print(); # input 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print(); # input 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(); # input placeholderfoc-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print(); # input placeholderwhich is our initial budget constraintManipulation foc-1 and foc-2we now set A=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(); # input placeholderWe cross multiply the above equationLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbW5HRiQ2JVEmMC4xMjBGJy8lJXNpemVHUSMxNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RJyZzZG90O0YnRi9GMi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOy8lKXN0cmV0Y2h5R0Y7LyUqc3ltbWV0cmljR0Y7LyUobGFyZ2VvcEdGOy8lLm1vdmFibGVsaW1pdHNHRjsvJSdhY2NlbnRHRjsvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZKLUY2Ni5RIn5GJ0YvRjJGOUY8Rj5GQEZCRkRGRkZIRkstSSVtc3VwR0YkNiUtSSNtaUdGJDYmUSJMRidGLy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRictRiM2KC1GLDYlUSQwLjRGJ0YvRjJGL0ZXLyUrZXhlY3V0YWJsZUdGOy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0ZaLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y1LUZRNiVGUy1GIzYoLUYsNiVRJDAuNkYnRi9GMkYvRldGW29GXW9GWkZgby1GNjYuUSI9RidGL0YyRjlGPEY+RkBGQkZERkYvRklRLDAuMjc3Nzc3OGVtRicvRkxGXnAtRlE2JS1GVDYmUSJLRidGL0ZXRlpGZW9GYG9GNS1GLDYlUSYwLjk2MEYnRi9GMkY1Rk0tRlE2JUZicEZmbkZgb0YvRltvRjI=print(); # input placeholderWe now solve for 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print(); # input placeholderwe plug in this value into to the budget constraint represented by 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print(); # input placeholderwe solve for KLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRiw2JlEmc29sdmVGJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2JS1GIzYoLUYsNiZRIiVGJ0YvRjJGNS1GOTYuUSIsRidGL0Y8Rj4vRkJGNEZDRkVGR0ZJRksvRk5RJjAuMGVtRicvRlFRLDAuMzMzMzMzM2VtRidGK0YvLyUrZXhlY3V0YWJsZUdGQEY8Ri9GPEYvRl9vRjw=print(); # input placeholderwe plug in this value into to the budget constraint represented by 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print(); # input placeholderWe now solve for 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print(); # input placeholderThis means that the optimal quantities of K and L are given 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print(); # input placeholderThese two quantities if K and L will produce a maximum value of the objective function Y equal 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print(); # input placeholderWe now note that we could have derived the above by simply typing: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print(); # input placeholderWhich is the same result that we had previously.
Finally we save the maximized value of Y for use in the next sectionPkkjbWZHNiImSSIlR0YkNiMiIiI=print(); # input placeholderJSFH
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">4) Interpretation of the Lagrange Multiplier</Font></Text-field>We know that our optimal quantities are given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSMxMEYnRi9GPC1GOTYuUSI7RidGL0Y8Rj4vRkJGNEZDRkVGR0ZJRksvRk5RJjAuMGVtRidGUC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Zbby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYmUSJMRidGL0YyRjVGOC1GUzYlUSM4MEYnRi9GPEZWLyUrZXhlY3V0YWJsZUdGQEY8print(); # input placeholderWe also know that the Lagrange multiplier lambda can be written asLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRi8vJStleGVjdXRhYmxlR0ZARjw=print(); # input placeholderor as:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUrZXhlY3V0YWJsZUdGQEY8print(); # input placeholderIf we substitute in the optimal values of K and L in any of the above equations we 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print(); # input 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print(); # input placeholderWhat does the value of the Lagrange multiplier mean?
The value for the Lagrange multiplier is the increase in the objective function we would get if we allowed the constraint (in our case income I)
to increase by one unit. This means that we can interpret the Lagrange multiplier as the marginal utility of income.
How much utility increases when we increase income by one unit. We can easily test if this is true by doing some cheating and quickly solve the optimization with the Maximize function in Maple when we have
increased income by one unit from 800 to 801. If the above reasoning is correct then output should increase by 0.1739
As previously our objective function is given 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print(); # input placeholderJSFHour budget constraint is given by (since we have increased income by one unit)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print(); # input placeholderWe now maximize our objective function subject to the budget 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This means that the maximum value of the objective function is 174.12 when I=801We know since previously that the maximum value of the objective function when I=800 i given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEjbWZGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjZRJ25vcm1hbEYnprint(); # input placeholderWhich means that the difference between the two values are given 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print(); # input placeholderwhich is exactly the same value that we had for the Lagrange multiplier.
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">5) Constrained Maximization Through Substitution</Font></Text-field>We assume as previously that the objective function is given by: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print(); # input placeholderJSFHand that our budget constraint is given 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(); # input placeholderWe now rewrite the budget constraint instead of plug in the values into LagrangeThe above budget constraint can be written as:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbW5HRiQ2JVEkODAwRicvJSVzaXplR1EjMTRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSgmbWludXM7RidGL0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkotRiw2JVEjNDBGJ0YvRjItRjY2LlEnJnNkb3Q7RidGL0YyRjlGPEY+RkBGQkZERkYvRklRJjAuMGVtRicvRkxGVC1GNjYuUSJ+RidGL0YyRjlGPEY+RkBGQkZERkZGU0ZVLUkjbWlHRiQ2JlEiS0YnRi8vJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLUY2Ni5RIj1GJ0YvRjJGOUY8Rj5GQEZCRkRGRi9GSVEsMC4yNzc3Nzc4ZW1GJy9GTEZgby1GLDYlUSI1RidGL0YyRlBGVi1GWjYmUSJMRidGL0ZnbkZqbi8lK2V4ZWN1dGFibGVHRjtGMg==print(); # input placeholderif we solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiTEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== we get: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print(); # input placeholderWe now plug in this expression into our objective function which gives us: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print(); # input placeholderWe now maximize this expression by setting the partial derivative with respect to K equal to 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(); # input placeholderif we solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== we get: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print(); # input placeholderif we plug in this into our rewritten budget constraint we 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print(); # input placeholderThis means that the optimal quantities of K and L are given 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print(); # input placeholderwhich are the same quantities that we had previouslyIn order to be completely sure that we have solved for the maximization quantities of K and L and not the minimization quantities of K and L
we need to check the second derivative. If the second derivative is negative then we have maximization quantities of K and 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print(); # input 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print(); # input 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print(); # input placeholderSince this second derivative is negative we can conclude that we indeed have solved for the maximizing quantities of K and L
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">6) Optimality Conditions</Font></Text-field>###################### Slope Indifference Curve ###################LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEocmVzdGFydEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRi8vJStleGVjdXRhYmxlR0ZARjw=Our general production function is given 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print(); # input placeholderThe derivative with respect to time (chain rule) for the above function is given 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print(); # input placeholderNote that any movement along the same indifference curve (also called a isoquant or contour line) means that the change in Q is equal to zero.
Which means that dQ=0. The increase in Q due to an increase in K is completely offset by the decrease in Q due to an decrease in L.
This gives us: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print(); # input placeholderWe now solve for 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(); # input placeholderwe divide both sides by 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print(); # input placeholderWe note that 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 is equal to the Marginal Product of Labor (MPL) and 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 is equal to the Marginal Product
of Capital (MPK). MPL is the extra output the firm gets from one extra unit of labor and MPC is the extra output the firm gets from one extra
unit of capital. This gives us: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(); # input placeholderwe now note that:##############################################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 and 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 which means 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print(); # input placeholderWe now note that the slope of a function is defined as: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(); # input placeholderWe can therefore 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print(); # input placeholder##############################################This means that we can write our previous equation as:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlE5c2xvcGVfaW5kaWZmZXJlbmNlX2N1cnZlRicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSI9RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1GOTYuUSomdW1pbnVzMDtGJ0YvRjxGPkZBRkNGRUZHRklGSy9GTlEsMC4yMjIyMjIyZW1GJy9GUUZWLUkmbWZyYWNHRiQ2KC1GLDYmUSRNUENGJ0YvRjJGNS1GIzYoLUYsNiZRJE1QTEYnRi9GMkY1Ri9GMi8lK2V4ZWN1dGFibGVHRkAvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGNS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZ28vJSliZXZlbGxlZEdGQEYvRl1vRjw=print(); # input 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 is called the Marginal Rate of Substitution for labor and capital and answer the question:If labor is reduced by one unit, how much more capital is needed to produce the same output###################### Slope Budget Constraint ###################Again our budget constrain is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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print(); # input placeholderWe assume that we have L on the y-axis which means that the equation for L would be given 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print(); # input placeholder which can be written 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(); # input placeholderor 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print(); # input placeholderwhere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiZRJyYjOTIxO0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUklbXN1YkdGJDYlLUYvNiZRIlBGJ0YyL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnLUYjNiYtRi82JlEiTEYnRjJGQ0ZFRjJGQ0ZFLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkZDRkUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlQvJSliZXZlbGxlZEdGN0YyRjg= is the intercept and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JJW1zdWJHRiQ2JS1JI21pR0YkNiZRIlBGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYyNiZRIktGJ0Y1RjhGO0Y1RjhGOy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYjNiYtRi82JUYxLUYjNiYtRjI2JlEiTEYnRjVGOEY7RjVGOEY7RkNGNUY4RjsvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlQvJSliZXZlbGxlZEdRJmZhbHNlRidGNS9GPFEnbm9ybWFsRic= is the slope of the budget line and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== is the x-variable###################### Optimal Allocation ###################For an optimal allocation the slope of the indifference curve has to be equal to the slope of the budget 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(); # input placeholderwhich can be written as: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print(); # input placeholderor as: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print(); # input placeholderwe divide both sides by 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print(); # input placeholder###################### Example ###################We assume that Production is a function of capital K and labor 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print(); # input placeholderwe can plot such a production function as follows: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%;The Marginal Product of Capital (MPC) is given by: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print(); # input placeholderThe Marginal Product of Labor (MPL) is given by: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print(); # input placeholderWhich means that the Marginal rate of Substitution is given by: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(); # input placeholderwe know since previously that our budget constraint is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KS1GLDYmUSJMRicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSI9RidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GIzYpLUkmbWZyYWNHRiQ2KC1GIzYnLUYsNiZRJyZJb3RhO0YnRjQvRjhGRUZBRjQvJStleGVjdXRhYmxlR0Y5LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0ZBLUYjNictSSVtc3ViR0YkNiUtRiw2JlEiUEYnRjRGN0Y6LUYjNidGMUY0RlxvRl5vRkEvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y0RlxvRl5vRkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmNwLyUpYmV2ZWxsZWRHRkUtRj42LlEoJm1pbnVzO0YnRjRGQUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRlxxLUZaNigtRiM2KS1GZG82JUZmby1GIzYnLUYsNiZRIktGJ0Y0RjdGOkY0RlxvRl5vRkFGW3AtRj42LVExJkludmlzaWJsZVRpbWVzO0YnRkFGQ0ZGRkhGSkZMRk5GUC9GU1EmMC4wZW1GJy9GVkZdckZmcUY0RlxvRl5vRkFGYW9GXnBGYXBGZHBGZnBGNEZcb0Zeb0ZBRitGXG9GXm9GQUYrRkE=JSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2JlEnJklvdGE7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGL0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiVRJDgwMEYnRi9GNS1GOTYuUSI6RidGL0Y1RjxGPkZARkJGREZGRkhGSkZNLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRmVuLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUklbXN1YkdGJDYlLUYsNiZRIlBGJ0YvL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUYjNiUtRiw2JlEiS0YnRi9GZG9GZm9GL0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOC1GUDYlUSM0MEYnRi9GNUZTRlYtRl9vNiVGYW8tRiM2JS1GLDYmUSJMRidGL0Zkb0Zmb0YvRjVGXXBGOC1GUDYlUSI1RidGL0Y1RlMtRjk2LlEifkYnRi9GNUY8Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmFxRlYtRiM2J0ZncC1GOTYuUSI9RidGL0Y1RjxGPkZARkJGREZGRkhGSkZNLUYjNictSSZtZnJhY0dGJDYoLUYjNiVGK0YvRjUtRiM2JUZjcEYvRjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZyLyUpYmV2ZWxsZWRHRjQtRjk2LlEoJm1pbnVzO0YnRi9GNUY8Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl9zLUZbcjYoLUYjNidGXm8tRjk2LVExJkludmlzaWJsZVRpbWVzO0YnRjVGPEY+RkBGQkZERkZGSEZgcUZicUZqb0YvRjVGX3JGYXJGZHJGZ3JGaXJGL0Y1LUYsNiNRIUYnRjVGaHNGNQ==print(); # input placeholderWe can see from the above formula that the slope of the budget line is given by 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 . This gives us: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(); # input placeholderfor optimal we require 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 which means that we get: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(); # input placeholderWe now solve for 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print(); # input placeholderIf we substitute that into the budget constraint we get: 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print(); # input placeholderJSFHWe now solve for 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print(); # input placeholderIf we substitute that into the budget constraint we get: 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(); # input placeholderJSFHwhich are the same quantities that we had in our previous example.
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">7) Ways to Solve it Directly in Maple</Font></Text-field>1)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print(); # input placeholder2)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(); # input placeholder3)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(); # input placeholder4)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print(); # input placeholder5)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZmbi1JI21pR0YkNiZRKHJlc3RhcnRGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIjpGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRlcvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRlM2JkZVRlhGZW4vRmhuUSVhdXRvRictRiw2JlEidkYnRi9GMkY1LUY5Ni5RKiZjb2xvbmVxO0YnRi9GPEY+RkFGQ0ZFRkdGSUZLRk1GUC1JKG1mZW5jZWRHRiQ2Jy1GIzYqLUYsNiZRIktGJ0YvRjJGNS1GOTYuUSIsRidGL0Y8Rj4vRkJGNEZDRkVGR0ZJRksvRk5RJjAuMGVtRicvRlFRLDAuMzMzMzMzM2VtRictRiw2JlEiTEYnRi9GMkY1RlxwLUYsNiZRImFGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdGQEY8Ri9GPC8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGOEZSLUYsNiZRImZGJ0YvRjJGNUZhby1JI21uR0YkNiVRIzEyRidGL0Y8LUY5Ni5RJyZzZG90O0YnRi9GPEY+RkFGQ0ZFRkdGSUZLRmBwL0ZRRmFwLUklbXN1cEdGJDYlRmlvLUYjNiQtRmZxNiVRJDAuNEYnRi9GPEY8LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZpcS1GXnI2JUZkcEZgckZlckY4RlItRiw2JlEiZ0YnRi9GMkY1RmFvLUZmcTYlUSQ4MDBGJ0YvRjwtRjk2LlEoJm1pbnVzO0YnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSwwLjIyMjIyMjJlbUYnL0ZRRmRzLUZmcTYlUSM0MEYnRi9GPEZpcS1GOTYuUSJ+RidGL0Y8Rj5GQUZDRkVGR0ZJRktGYHBGXHJGaW9GYHMtRmZxNiVRIjVGJ0YvRjxGaXFGaXNGZHBGOEZSLUYsNiZRI0xMRidGL0YyRjVGYW9GYnEtRjk2LlEiK0YnRi9GPEY+RkFGQ0ZFRkdGSUZLRmNzRmVzRmdwRmlxRmpyRjhGUi1GLDYmUSJXRidGL0YyRjVGYW9GaXMtRmVvNictRiM2Ly1GLDYmUSVkaWZmRidGL0YyRjUtRmVvNiUtRiM2KEZfdEZccEZpb0YvRmpwRjxGL0Y8RlxwRmlzRlx1LUZlbzYlLUYjNihGX3RGXHBGZHBGL0ZqcEY8Ri9GPEZccEZpc0ZcdS1GZW82JS1GIzYoRl90RlxwRmdwRi9GanBGPEYvRjxGL0ZqcEY8Ri9GPEZccUZfcUY4RlJGam5GUkZqbi1GLDYmUSZzb2x2ZUYnRi9GMkY1LUZlbzYlLUYjNipGZXRGXHBGaXMtRmVvNictRiM2Jy1GLDYmUSRzZXFGJ0YvRjJGNS1GZW82JS1GIzYtRl5vLUZlbzYnLUYjNiYtRiw2JlEiaUYnRi9GMkY1Ri9GanBGPEYvRjwvRl1xUSJbRicvRmBxUSJdRidGXHBGYXctRjk2LlEiPUYnRi9GPEY+RkFGQ0ZFRkdGSUZLRk1GUC1GZnE2JVEiMUYnRi9GPC1GOTYuUSMuLkYnRi9GPEY+RkFGQ0ZFRkdGSUZLRmNzRlxyLUZmcTYlUSIzRidGL0Y8Ri9GanBGPEYvRjxGL0ZqcEY8Ri9GPEZccUZfcUZpc0YvRmpwRjxGL0Y8RmlzLUY5Ni5RIjtGJ0YvRjxGPkZfcEZDRkVGR0ZJRktGYHBGUEYvRmpwRjw=print(); # input placeholder6)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print(); # input placeholder7)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print(); # input placeholder8)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print(); # input placeholder
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">8) Constrained Minimization with Lagrange </Font></Text-field>Our objective function is given 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%;our constraint is given by: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%;Our constrained minimization can be illustrated as follows: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%;which also can be plotted (the above plot simply rotated 90 degrees) 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%;The optimal amount of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiS0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vRjZRJ25vcm1hbEYn is therefore given by: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print(); # input placeholder
<Text-field style="Heading 1" layout="Heading 1"><Font underline="true" size="22">9) Further Examples </Font></Text-field>Example-1############## Outline #########Our budget constraint is given 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print(); # input placeholderwe assume thatLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2JlEnJiM5MjE7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGL0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiVRJDEwOEYnRi9GNS1GOTYuUSI7RidGL0Y1RjwvRj9RJXRydWVGJ0ZARkJGREZGRkgvRktRJjAuMGVtRidGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Zpbi8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1JJW1zdWJHRiQ2JS1GLDYmUSJQRidGLy9GM0ZXL0Y2USdpdGFsaWNGJy1GIzYoLUYsNiZRIktGJ0YvRmhvRmlvRi9GaG8vJStleGVjdXRhYmxlR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRmlvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOC1GUDYlUSIzRidGL0Y1RlNGWi1GY282JUZlby1GIzYoLUYsNiZRIkxGJ0YvRmhvRmlvRi9GaG9GYHBGYnBGaW9GZXBGOC1GUDYlUSI0RidGL0Y1RlNGL0ZgcEY1print(); # input placeholderwe assume that the production function is given 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(); # input placeholder############## Solution #########Our objective function is given by: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print(); # input placeholderJSFHour budget constraint is given 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print(); # input placeholderOur Lagrange is therefore given 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(); # input placeholderthe first order conditions are given byfoc-1LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlElZGlmZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNigtRiw2JlEjTExGJ0YvRjJGNS1JI21vR0YkNi5RIixGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJLRidGL0YyRjVGLy8lK2V4ZWN1dGFibGVHRkhGREYvRkQtRkE2LlEiPUYnRi9GREZGL0ZKRkhGS0ZNRk9GUUZTL0ZWUSwwLjI3Nzc3NzhlbUYnL0ZZRl9vLUkjbW5HRiQ2JVEiMEYnRi9GREYvRmhuRkQ=print(); # input 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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1GLDYmUSZldmFsZkYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNiktRiw2JlEoaXNvbGF0ZUYnRi9GMkY1LUZWNiUtRiM2KC1GLDYmUSIlRidGL0YyRjUtRjk2LlEiLEYnRi9GPEY+L0ZCRjRGQ0ZFRkdGSUZLL0ZOUSYwLjBlbUYnL0ZRUSwwLjMzMzMzMzNlbUYnLUYsNiZRJyYjOTU1O0YnRi8vRjNGQEY8Ri8vJStleGVjdXRhYmxlR0ZARjxGL0Y8Rl5vLUkjbW5HRiQ2JVEiM0YnRi9GPEYvRmpvRjxGL0Y8Ri9Gam9GPA==print(); # input placeholderfoc-2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlElZGlmZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEjTExGJ0YvRjJGNS1JI21vR0YkNi5RIixGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJMRidGL0YyRjVGL0ZERi9GRC1GQTYuUSI9RidGL0ZERkYvRkpGSEZLRk1GT0ZRRlMvRlZRLDAuMjc3Nzc3OGVtRicvRllGXW8tSSNtbkdGJDYlUSIwRidGL0ZERi8vJStleGVjdXRhYmxlR0ZIRkQ=print(); # input 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print(); # input 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(); # input placeholderfoc-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print(); # input placeholderwhich is our initial budget constraintManipulation foc-1 and foc-2we now set A=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(); # input placeholderWe cross multiply the above 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(); # input placeholderWe now solve for 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print(); # input placeholderwe plug in this value into to the budget constraint represented by 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print(); # input placeholderwe solve for 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print(); # input placeholderwe plug in this value into to the budget constraint represented by 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print(); # input placeholderWe now solve for 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print(); # input placeholderThis means that the optimal quantities of K and L are given 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print(); # input placeholderThese two quantities if K and L will produce a maximum value of the objective function Y equal 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print(); # input placeholderWe can confirm that K=16 and L=15 indeed are our optimal quantities (quantities that maximize Y) by using the
simple Maximize function in 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print(); # input placeholderWhich is the same result that we had previously.Example-2############## Outline #########Our budget constraint is given 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print(); # input placeholderwe assume 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(); # input placeholderwe assume that the production function is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JlEiWUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRjk2LlEifkYnRi9GPEY+RkFGQ0ZFRkdGSUZLL0ZOUSYwLjBlbUYnL0ZRRlYtSSVtc3VwR0YkNiUtRiw2JlEiS0YnRi9GMkY1LUYjNigtSSNtbkdGJDYlUSQwLjRGJ0YvRjxGL0YyLyUrZXhlY3V0YWJsZUdGQC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0Y1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZSLUY5Ni5RJyZzZG90O0YnRi9GPEY+RkFGQ0ZFRkdGSUZLRlVGVy1GWTYlLUYsNiZRIkxGJ0YvRjJGNS1GIzYoLUZbbzYlUSQwLjZGJ0YvRjxGL0YyRl5vRmBvRjVGY29GL0Zeb0Y8print(); # input placeholder############## Solution #########Our objective function is given by: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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=our budget constraint is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY/LUkjbWlHRiQ2JlEnJiM5MjE7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGL0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiVRJDIwMEYnRi9GNS1GOTYuUSI6RidGL0Y1RjxGPkZARkJGREZGRkhGSkZNLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRmVuLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUklbXN1YkdGJDYlLUYsNiZRIlBGJ0YvL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLUYjNiQtRiw2JlEiS0YnRi9GZG9GZm9GNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjgtRlA2JVEjNTBGJ0YvRjVGU0ZWLUZfbzYlRmFvLUYjNiQtRiw2JlEiTEYnRi9GZG9GZm9GNUZdcEY4LUZQNiVRIzIwRidGL0Y1RlMtRjk2LlEifkYnRi9GNUY8Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmFxRlYtRiw2JlEiZ0YnRi9GZG9GZm9GOEYrLUY5Ni5RKCZtaW51cztGJ0YvRjVGPEY+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZqcS1GX282JUZhby1GIzYmRmpvLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0Y1Rl1wLUY5Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y1RjxGPkZARkJGREZGRkhGYHFGYnFGam9GZnEtRl9vNiVGYW8tRiM2JkZncEZgckZickY1Rl1wRmVyRmdwLUY5Ni5RIjtGJ0YvRjVGPC9GP0Zlb0ZARkJGREZGRkhGYHFGTUY1print(); # input placeholderOur Lagrange is therefore given 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(); # input placeholderthe first order conditions are given byfoc-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print(); # input 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print(); # input 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print(); # input placeholderfoc-2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlElZGlmZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEjTExGJ0YvRjJGNS1JI21vR0YkNi5RIixGJ0YvL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJMRidGL0YyRjVGL0ZERi9GRC1GQTYuUSI9RidGL0ZERkYvRkpGSEZLRk1GT0ZRRlMvRlZRLDAuMjc3Nzc3OGVtRicvRllGXW8tSSNtbkdGJDYlUSIwRidGL0ZERi8vJStleGVjdXRhYmxlR0ZIRkQ=print(); # input 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print(); # input 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(); # input placeholderfoc-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print(); # input placeholderwhich is our initial budget constraintManipulation foc-1 and foc-2we now set A=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(); # input placeholderWe cross multiply the above 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print(); # input placeholderWe now solve for 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print(); # input placeholderwe plug in this value into to the budget constraint represented by CLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlElc3Vic0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNi4tRiw2JlEiTEYnRi9GMkY1LUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctSSNtbkdGJDYlUSUzLjc1RidGL0ZELUZBNi5RJyZzZG90O0YnRi9GREZGRklGS0ZNRk9GUUZTL0ZWUSYwLjBlbUYnL0ZZRlxvLUZBNi5RIn5GJ0YvRkRGRkZJRktGTUZPRlFGU0Zbb0Zdby1GLDYmUSJLRidGL0YyRjUtRkE2LlEiLEYnRi9GREZGL0ZKRjRGS0ZNRk9GUUZTRltvL0ZZUSwwLjMzMzMzMzNlbUYnRl5vLUYsNiZRIkNGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdGSEZERi9GREYvRl1wRkQ=print(); # input placeholderwe solve for 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print(); # input placeholderwe plug in this value into to the budget constraint represented by 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(); # input placeholderWe now solve for 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print(); # input placeholderThis means that the optimal quantities of K and L are given 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print(); # input placeholderThese two quantities if K and L will produce a maximum value of the objective function Y equal 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print(); # input placeholderWe can confirm that K=1.6 and L=6 indeed are our optimal quantities (quantities that maximize Y) by using the
simple Maximize function in 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print(); # input placeholderWhich is the same result that we had previously.