Question

1. ?(?, ?) = ? ?^? ?^?

a) Write down the utility maximization problem.

b) Find the MRS if ? is 24, ? is 16, ? is 3, ? is 2 and ? is 1. How would you interpret this value?

Answer 1

Q1)

a) The utility maximization problem can be written as,

max u($\alpha$m^rn^s)

subject to : p_mm + p_nn = I , where p_m is the price of good m

  p_n is the price of good n

I is the income of the consumer.

b) Marginal rate of substitution = marginal utility from good m / marginal utility from good n

Marginal utility of good m can be calculated by differentiating the utility function with respect to m keeping n constant, So, marginal utility of good m = $\alpha$ X r X m^r-1 X n^s

Marginal utility of good n can be calculated by differentiating the utility function with respect to n keeping m constant. So, marginal utility of good n = $\alpha$ X m^r X s X n^s-1

So, marginal rate of substitution = ($\alpha$ X r X m^r-1 X n^s) / ( $\alpha$ X m^r X s X n^s-1)

= (r X m^r-1 X n^s) /  (m^r X s X n^s-1)

= [ 3 X (24)^3-1 X 16²] / [24³ X 2 X (16)^2-1]

= [ 3 X 24 X 24 X 16 X 16] / [24 X 24 X 24 X 2 X 16]

= 1 is the answer.

Marginal rate of substitution is the slope of the indifference curve. It measures the rate at which the consumer has to give up some units of one good in order to obtain the other good. In this case, since the marginal rate of substitution is 1, it means that the consumer has to give up 1 unit of good n in order to obtain one more unit of good m.