3. Given: Total Utility = 40x - x 2 + 20y - y 2 Budget or Income, I= 50 Px=6 Py=

Question

3. Given:
Total Utility = 40x - x2 + 20y - y2
Budget or Income, I= 50
Px=6
Py=2

a.) What is the Marginal Utility of x (MUx)?


b.) What is the Marginal Utility of y (MUy)?


c.) What is the slope of the indifference curve for this individual?


d.) What is the slope of the budget constraint for this individual?


c.) What bundle of x and y would maximize utility subject to the constraint?(*Note: you can consume fractions of units, like eating only half of an apple. If you get a fraction/decimal as an answer it does not necessarily mean you did something wrong)

Explanation / Answer

Utility function: U = 40X - X2 + 20Y - Y2

Budget Line: I = X. PX + Y. PY, Or

50 = 6X + 2Y

Or,

25 = 3X + Y (Dividing both sides by 2)

(1) MUX = dU / dX = 40 - 2X

(2) MUY = dU / dY = 20 - 2Y

(3) Slope of indifference curve = Marginal rate of Substitution

= MUX / MUY = (40 - 2X) / (20 - 2Y)

Or, MRS = (20 - X) / (10 - Y) [Dividing numerator & denominator by 2]

(4) Slope of Budget Constraint = - PX / PY

= - 6 / 2 = - 3

(5) Utility will be maximized at the point where slope of indifference curve = slope of budget line

Or, MRS = - (PX / PY)

(20 - X) / (10 - Y) = - 3

20 - X = 3Y - 30

Or, X + 3Y = 50 ...... (1)

And

3X + Y = 25 ...... (2) [Budget Line]

(1) x 3 gives:

3X + 9Y = 150 .... (3)

(3) - (2) Gives:

8Y = 125

Y = 125 / 8 = 15.63

X = 50 - 3Y [From (1)]

= 50 - (3 x 15.63) = 3.13

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