Suppose that utility is a function of two goods: U(X,Y) = XY + 4X a. Find the ge

Question

Suppose that utility is a function of two goods: U(X,Y) = XY + 4X a. Find the generalized demand curve for X           

b. Find the generalized demand curve for Y              

c. If M = $100, Px = $1 and Py = $2. Find the optimal bundle of X and Y.             

d. Finally, if M = $100, Px = $1 and Py = $30, find the optimal bundle of X and Y.     

. An individual has a utility function given by: U = 2XY Income is $120, Px = $4 and Py = $1.

Next, suppose that the price of X falls to Px = $3.  

A) Calculate the total effect from this price reduction.             

B) Determine the substitution effect from this price reduction.          

C) Determine the income effect from this price reduction.   

D) Show the total effect, substitution effect, and income effect on a graph below.  

F. Calculate the gain in consumer welfare using the equivalent variation measure when price falls.                 

G. Calculate the gain in consumer welfare using the compensating variation measure when price falls

Explanation / Answer

U = XY + 4X

Budget constraint is:

I = X. Px + Y. Py

(a) In optimal condition, MRS = Px / Py Where

MRS = MUx / MUy

MUx = dU / DX = Y + 4

MUy = dU / dY = X

So, (Y + 4) / X = Px / Py

Y + 4 = X. Px / Py

Y = (X. Px / Py) - 4

And X = (Y + 4). Py / Px

Substituting in budget constraint:

I = X. Px + Y. Py

I = X. Px + [(X. Px / Py) - 4] Py

= X. Px + [X. Px - 4Py]

I = 2X. Px - 4Py

X = (I + 4Py) / 2Px [Demand curve for X]

(b) Again,

I = X. Px + Y. Py

= [Px. (Y + 4). Py / Px] + Y.Py

I = Py. (Y + 4) + Y.Py

I = 2Y. Py + 4Y

I = 2Y (Py + 2)

Y = I / 2(Py + 2) [Demand curve for Y]

(c)

In this case, Budget line is

100 = X + 2Y

So, MRS = (Y + 4) / X = Px / Py = 1/2

2Y + 8 = X

Y = (X - 8) / 2 = 0.5X - 4

Substituting in budget line:

100 = X + 2Y

100 = 2Y + 8 + 2Y = 4Y + 8

4Y = 92

Y = 23

X = 2Y + 8 = 46 + 8 = 54

(d) Here budget line is

100 = X + 30Y

Equating MRS with price ratio,

(Y + 4) / X = Px / Py = 1/30

30Y + 120 = X

So, Y = (X - 120) / 30

Substituting in budget line 100 = X + 30Y:

100 = 30Y + 120 + 30Y = 120 + 60Y

60Y = - 20, Y = - (1/3)

X = 30Y + 120 = - 10 + 120 = 110

Please cross check your data. Is Px = 1 & Py = 30 as you've written? Quantity cannot be negative but here Y = - (1/3) based on data provided by you. Py = 30 seems too high to me.

NOTE: Out of total 10 sb-questions, I've answered 1st 4 questions.

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