5. Answer all parts of this question. Each part is an independent question. i) Consider a...

  

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5. Answer all parts of this question. Each part is an independent question. i) Consider a Cobb-Douglas production function ii) Q=al K Write an expression for the natural log of the output. By log differentiating the expression you just derived, prove that the elasticity of output with respect to labour, and the elasticity of output with respect to capital happen to be the exponents of the production function. Find the degree of homogeneity of the production function iii) y=x1 x2 and show that Euler's theorem holds here. An advertising company conducts a special campaign to promote sales of a certain product. They estimate that the benefits of the campaign will result in extra sales and, when the campaign is over, the extra sales, occurring in a continuous manner, will obey a curve of the form S(t)=4000e 0.31 where S is the amount of extra sales and 7 is the time in days after the campaign is over. How many extra sales are obtained 10 days after the close of the advertising campaign? iv) Consider a function f(x1, x2) = (0.5x1 + 0.5x22) with x1, x2>0. v) If this function happens to be a production function with x and x2 as factor inputs, what is the degree of homogeneity of this production function? Describe the returns to scale for this function based on your earlier answer in this part. Prove that the function f(x1, x2) = x + x2 with x1, x2>0 is quasi-convex. After you have shown the proof, next comment on whether the function is also strictly convex. 5. Answer all parts of this question. Each part is an independent question. i) Consider a Cobb-Douglas production function ii) Q=al K Write an expression for the natural log of the output. By log differentiating the expression you just derived, prove that the elasticity of output with respect to labour, and the elasticity of output with respect to capital happen to be the exponents of the production function. Find the degree of homogeneity of the production function iii) y=x1 x2 and show that Euler's theorem holds here. An advertising company conducts a special campaign to promote sales of a certain product. They estimate that the benefits of the campaign will result in extra sales and, when the campaign is over, the extra sales, occurring in a continuous manner, will obey a curve of the form S(t)=4000e 0.31 where S is the amount of extra sales and 7 is the time in days after the campaign is over. How many extra sales are obtained 10 days after the close of the advertising campaign? iv) Consider a function f(x1, x2) = (0.5x1 + 0.5x22) with x1, x2>0. v) If this function happens to be a production function with x and x2 as factor inputs, what is the degree of homogeneity of this production function? Describe the returns to scale for this function based on your earlier answer in this part. Prove that the function f(x1, x2) = x + x2 with x1, x2>0 is quasi-convex. After you have shown the proof, next comment on whether the function is also strictly convex.