Question

In: Economics

A consumer has an income of $120 to buy two goods (X, Y). the price of...

A consumer has an income of $120 to buy two goods (X, Y). the price of X is $2 and the price of Y is $4. The consumer utility function is given by

UX,Y=X2/3*Y1/3

                      You are also told that his marginal utilities are

MUX=23YX1/3

                                                                                        MUY=13XY2/3

  1. Find the slope of the budget constrain. (1 point)
  2. Calculate the optimal quantity of good X and Y for this consumer (Point A). (2 points)

Assume that the price of good X increases to PX'=$8, and holding everything else constant:

  1. Graph all your findings by showing the change in price if good X along the points A and B. (2 points)

Hint: Do not compute and graph point C.

    

Research studies found the demand for good X as follow:

                                X=8-0.4*PX+0.2*PY+0.05*M  

where, PX is the price of good X, PY is the price of good Y, and M is consumer’s income.

  1. Based on this research finding and holding everything else constant, can the consumer afford to buy 12 units of good X. Explain. (2 points)

Solutions

Expert Solution

Income(M)= 120

Price of X= Px= 2

Price of Y= Py= 4

Budget line: XPx+YPy=M

4Y= 120-2X

Y= (120-2X)/ 4

Differentiate it wrt X:

dY/dX= -2/4= -1/2 Slope of budget constraint

U=X2/3Y1/3

MUX= (2/3)X-1/3 Y1/3

MUY= (1/3) X2/3Y-2/3

MRS= MUX/MUY= 2Y/X

For optimal bundle:

MRS= Slope of budget constraint

2Y/X= 1/2

X= 4Y Equation 1

Use this equation in Budget line:

2X+4Y=120

2X+X= 120

X= 120/3= 40

Put X=40 in equation 1:

X= 4Y

40/4= Y

Y= 10

Optimal bundle= (40,10)

If Px'= 8:

New budget line: 8X+4Y= 120

Slope of budget line= (-)2

For optimal bundle:

MRS= Slope of budget line

2Y/X= 2

Y=X Equation 2

Use this equation in new budget line:

8X+4Y=120

12X=120

X=10

Y=10

New optimal bundle= (10,10)

X=8-0.4*PX+0.2*PY+0.05*M

With PY=4 and M=120

X= 8-0.4PX+0.8+6

X= 14.8-0.4PX

For X=12:

12= 14.8-0.4PX

0.4PX= 2.8

PX= 7

A consumer is ready to pay price for each= 7 for Quantity of X=12. If PX is less than or equals to 7 then consumer can afford and if price is more than 7 then Consumer can not afford to buy 12 units of good X.

Rahul Sunny answered 2 years ago
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