Boost 3:38 PM Back Homework 2.doc PLEASE STAPLE YOUR HOMEWORK!! l· ( 10 total po

Question

Boost 3:38 PM Back Homework 2.doc PLEASE STAPLE YOUR HOMEWORK!! l· ( 10 total points) Suppose a consumer's utility function is given by U(X,Y)-X Also, the consumer has S32 to spend, and the price of X, Px S1, and the price of Y, Py $A a) (2 points) How much X and Y should the consumer purchase in order to maximize her utility? b) (2 points) How much total utility does the consumer receive? c) (2 points) Now suppose Px increases to S4. What is the new bundle of X and Y that the consumer will demand? d) (2 points) How much additional money would the consumer need in order to have the same utility level after the price change as before the price change? (Note: this amount of additional money is called the Compensating Variation.) Dashboard Calendar To Do Notifications Inbox

Explanation / Answer

1.U(x,y) = x1/2y1/2. Price of x (Px) = $1, Price of y (Py) = $4 and income (M) = $32

a) First we will prite the budget constraint of the consumer. The budget sonstraint is given by:

=>(Price of x * quantity of x ) + (price of y*Quantity of y) = Income

We will denote quantity of x = Qx and quantity of y = Qy .

Therefore budget constraint becomes:

=>(Px * x) + (Py * y) = M

Putting the values of Px and Py we get:

=> x + 4y = 32

Now in equlibrium, MUx/MUy = Px/Py where MUx is the marginal utlity of X and MUy is the marginal utility of Y. Or in other words, Marginal rate of substitution or MRS = price ratio

Marginal utlity = extra utlity a consumer gets by consuming one more unit of a good.

Therefore in order to find marginal utility of a good, we have to differentiate the utlity function with respect to that good.

That is: MUx = d(U(x,y))/dx and MUy = d(U(x,y))/dy .

=> MUx = d(U(x,y))/dx

=> MUx = d(x1/2y1/2) / dx   

(Rules of differentation: d(axn)/dx = an*xn-1 where a and n are constants. Also any other variable present in the equation will be treated as a constant. Therefore while calculating MUx, y will be treated as a constant and vice versa).

Therefore MUx = 1/2*(y1/2/x1/2)

Similarly MUy= d(U(x,y))/dy

=> MUy = d(x1/2y1/2) / dy  

=> MUy = 1/2*(x1/2/y1/2)

Therefore MUx/MUy = (1/2*(y1/2/x1/2))/(1/2*(x1/2/y1/2))

=> MUx/MUy = y/x

Price ratio Px/Py = 1/4

Therefore y/x = 1/4

=> y = x/4

Putting this term in budget constraint, we get:

=> x + 4*x/4 = 32

=> x + x = 32

=> 2x = 32

=> x = 16

Therefore y = x/ 4 = 16/4 = 4

Hence in equilibrium the consumer should purchase 16 units of good x and 4 units of good y to maximise utlity.

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b) In order to find total utlity, we have to plug the equlibrium values of x and y we got in the earlier question in the utlity function.

=> x = 16 and y = 4

Therefore, U(x,y) = x1/2y1/2

=> U(16,4) = 161/241/2

=>U(16,4) = 4*2 = 8

Therefore the consumer's utlity is 8.

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c. Now the Px has increased to 4. Hence the new price ratio,

=> Px/Py = 4/4 = 1/1

Also the new budget constraint : Px*x + Py*y = M =

=> 4x + 4y = 32

Therefore the MUx/MUy = Px/Py relation becomes:

=> y/x = 1/1

=> y = x

Putting the value of this y in the new budget constraint, we get:

=> 4x + 4x = 32

=> 8x = 32

Therefore x = 4

Hence y = x = 4

Hence after the price increase the consumer will demand 4 units of good x and 4 units of good y in equlibrium.

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d) In order to calculate the additional money the consumer will need, in order to have the same utlity after the price change as it was before the price change, we first need to calculate the demand functions of x and y.

So we know that MUx / MUy = Px/Py

and MUx / MUy = y/x

Also the budget constraint = Px*x + Py*y = M

=>Therefore, y/x = Px/Py

=> y = x *(Px/Py)

Putting this value of y in the budget equation, we get:

Px*x + Py * x *(Px/Py) = M

The Py term cancels out, and we get:

=>Px*x + Px * x = M

=> 2Px*x = M

=> x = M/2Px -------(1)

This is the demand function of x

And since y = x *(Px/Py)

Therefore putting the value of x from equation 1 we get:

=> y = M/2Px * Px/Py

=> y = M/2Py ----------> (2)

This is the demand function of y.

Now we have x and y in terms of M and Px and Py. Therefore we can plug these values into the utlity function to get the utlity function in terms of M, Px and Py :

=> U(x,y) = x1/2y1/2

=> U(M/2Px, M/2Py) = (M/2Px)1/2(M/2Py)1/2

We know that this should equal 8 as we have calculated before (in order for consumer to get same utlity as before).

=>  U(M/2Px, M/2Py) = (M/2Px)1/2(M/2Py)1/2 = 8

We also know that Px = Py = 4

Therefore: (M/(2*4))1/2(M/(2*4))1/2 = 8

=> (M/8)1/2(M/8)1/2 = 8

=> M/8 = 8 (as x1/2*x1/2 = x1/2 + 1/2 = x1 = x)

=> M = 8*8 = 64

Therefore the new income which the consumer will need in order to remain at original utlity = 64.

But the additional income = new income - old income

=> 64 - 32

=> 32

Therefore the additional income or the Compensating variation = $32.  

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