Question

In: Economics

Meghan’s utility function is given byU( x, y) = x yShe has income I =240 and...

Meghan’s utility function is given byU( x, y) = x yShe has income I =240 and faces prices Px = $ 4and Py = $ 4 Find Meghan’s optimal basket given these prices and her income. If the price of x decreases to $ 2 and Meghan’s income is unchanged, what must the price of y rise to in order for her to be exactly as well off as before the change in Px? (Hint: You need to calculate p y such that, with the new prices, Meghan reaches exactly the same indifference curve as before).

Expert Solution

U(x,y)=xy

Px=$4 , Py=$4 , I=240

Budget equation

I=Px*X+Py*Y

240=4X+4Y

60=X+Y

Optmality is achieved when slope of indifference curve mathces with slope of budget line

MRS=Marginal rate of substitution is the slope of indiffernce curve and calculated as below

MRS=MUx/MUy where MUx=dU/dx=y amd MUy=dU/dy=x

SLope of budget line above is Px/Py

therefoe we have,

MRS=Px/Py

y/x=Px/Py=1

y=x...Equation 1)

Using equation 1) into budget equation we get

60=x+y=2x

x=y=30

Hence the optimal budget is (x,y)=(30,30) & Utility from this budget is U(X,Y)=30*30=900

Now if Price of X decreases to $2 then we need increased Y such that we get same utility

Again using MRS we get

y/x=2/(Py+k)

y=2x/(Py+k)

Using this into new budget equation

240=2X+(Py+k)Y

240=2X+2X

X=60 and Y=120/(Py+k)

Now originally Py=$4 therefore using the same value we get to obtain utility of 900 levels

900=XY=60(120/(4+k))

1/8=1/(4+k)

k=4
hence New Py'=Py+k=4+4=8

we need Px=$2 and Py'=$8 to maintain same utility levels

Rahul Sunny answered 2 years ago

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