7 Douglas preferences are represented but the utility functi

7. Douglas preferences are represented but the utility function U(ry) y. So his marginal utility will be U, 2ry and U, - 3r\'y\'. He faces the following prices (P.PI) (a) The slope of Douglas\'s indifference curve at the point (s, p) is (b) If Douglas\'s budget line is tangent to his indifference curve at (zi), then (e) When he is consuming the optimal bundle given his income, what fraction of his budget is spent on good r? What fraction of his income is spent on consuming good y? (a) If we didnt define the exponents in Douglas\'s utility function, so his utility fiunction is represented as U(z v) ,what fraction of his income would be spent on good and good y?

Solution

a) The slope is ratio of marginal utility of the two goods as follow:

Slope = MRS = MUx/MUy = 2xy³/3x²y² = 2y / 3x

b) When the budget line is tangent to indiffernece curve then it is the optimal consumption bundle and at this bundle the following condtion is met -

MUx/MUy = Px/Py

and MUx/MUy = 2y/3x

then

2y/3x = Px/Py

Multiplying bith side with x/y

2xy/3xy = xPx/yPy

2/3 = xPx/yPy

2/3 is answer.

c) From part b it is clear that 2/3 of income is spent on good X and 1/3 is spent on good y.

d) Then a/bth portion will be spent on X and 1/bth portion will be spent on Y.