Question

1. Assume that Roberts’s utility from consuming good X and good Y is given by the following function:

U = 5X^0.2Y^0.8

Where X is the quantity of good X while Y is the quantity of good Y.

Assume the price of X (P_X) is £5, the price of Y (P_Y) is £8 and he has a budget of £100 to spend on the two goods.

1. Using the Lagrangian multiplier, calculate the quantities of good X and good Y Robert should purchase to maximise his utility.
2. How much utility does Robert receive at his optimal consumption bundle?
3. Calculate l at the optimal consumption bundle and provide an explanation of its value.
4. Using your previous answers, sketch the budget constraint and indifference curve at the optimum consumption point. Clearly explain and label (i) the quantities of X and Y (ii) the intercepts of the budget constraint (iii) the slope of the budget constraint and (iv) the level of utility.

Answer 1

Given: Utility function: U (X, Y) = 5x^0.2y^0.8…………………(1)

      Budget line for P_X = 5 and P_Y =8 as prices of X & Y with income m =100
      = (X)*5 + (Y)*8 = 100……….(2)

      MRS (X, Y) = d(UX)/d(UY) = 5*(0.2X ^-0.8 Y^0.8)/ (0.8X ^0.2 Y^-0.8)

                         = 5* (0.2Y^0.8/ X^0.8) / (0.8X^0.2/Y^0.2)

                         = 5*0.2/0.8*Y/X

                         = 5 * 2/8 * Y/X

                         = 5/4*Y/X……………………(3)

And, Slope of budget line = P_X / P_{Y =} 5/8 _{………………………..}(4)

Solving (3) and (4), 5/4*Y/X = 5/8

or, Y/X = 1/2

or, X = 2Y ……………………….(5)

now putting (5) in (2), we get

(2Y)*5 + Y*8 = 100

18Y = 100

Or, Y = 50/9

And X = 2*Y = 100/9

And U(X, Y) = 5(100/9)^0.2(50/9)^0.8