1.The Equilibrium of the Consumer (Optimal Point) (Note: a to c are only demonstrations)

a.The budget equation is I = Px + Py

b.The optimal point occurs at that bundle of good x and good y where

MRS = MU_X/MU_Y = - P_X/P_Yand MUx/Px = MUy/Py

c.Sample Computation:

Given U = 2x 5yCalculate for the MRS

1.Get the first derivative of U with respect to x:MUx = dU/dx = 2 (5y) = 10y

2.Get the first derivative of U with respect to y:MUy = dU/dy = 5 (2x) = 10x

3.MRS = MUx/MUy = 10y/10xor y/x

Question to be answered:

d.Given U = x^1/3 y^2/3 Calculate for the MRS only (Hint: get the first partial derivatives of the utility function) Show your solution & simplify your answer.

Answer:

MUx = dU/dx =

MUy = dU/dy =

Answer:

MRS = MUx / MUy = ____________________ (10)

e.Sample Calculation on how to get the Equilibrium of the Consumer:

Given U = x^1/3 y^2/3 Budget = $30Px = $2Py = $5

Calculate for the Optimum Point

Determine the Budget Equation based on data given above:

30 = 2X +5Y

Steps:

1.Based on the Utility function U = x^1/3 y^2/3 , calculate the MRS OR MUx/MUy

MRS = y/3x

2.Equate MRS to the slope Px/Py:

MRS = Px/Py

y/3x = 2/5

3.Calculate y from the equation in MRS = Px/Py :y/3x = 2/5

Cross multiply :y/3x = 2/5,5y = 6x becomesy = 6/5 x

4.Substitute y = 6/5xinto the budget equation to find x

Budget equation 30 = 2x + 5y

2x = 30 -5 (6/5x)

2x = 30 - 6x

2x + 6x = 30

8x = 30

x = 30/8 substitute this x value into the budget equation to find y

30 = 2 (30/8) + 5y

30 = 30/4 + 5y

30 - 30/4 = 5y

90/4 = 5y

y= 9/2

Hence, equilibrium of the consumer is when x = 30/8 units and y = 9/2 units

Question to be answered: Show your computations(no.11-15)

Given: U = x ² y ³ Budget = $120Price of good x = $2Price of good y = $3

Calculate for the optimum point by getting the MRS, slope and units of x and y

Highlight or encircle your final answers:MRSxy or MUx/MUy,Slope,Value of x,Value of y

Show your solutions.

MUx = dU/dx =

MUy = dU/dy =

MRS = MUx/MUy =

MUx/MUy = Px/Py

Solution: