Question

a) U = xy

b) U = (xy)^1/3

c) U = min(x,y/2)

d) U = 2x + 3y

e) U = x^2 y^2 + xy

2. All homogeneous utility functions are homothetic. Are any of the above

functions homothetic but not homogeneous? Show your work.

Answer #1

Utility function is homogeneous if for some degree k , U(tx,ty)
= t^{k} U(x,y) .

And it is homothetic , if it is a monotonic transformation of homogeeneous functions.

(a) U(x,y) = xy

U(tx,ty) = (tx) (ty)

= t^{2} xy

**It is homogeneous of degree 2**.

(b) U(x,y) = (xy)^{1/3}

U(tx,ty) = (tx)^{1/3}(ty)^{1/3}

= t^{2/3} (xy)^{1/3}

**It is homogeneous of degree 2/3.**

(c) U(x,y) = min(x,y/2)

**It is homogeneous of degree 1.**

(d) U(x,y) = 2x + 3y

U(tx,ty) = 2(tx) + 3(ty)

= t(2x + 3y)

**It is homogeneous of degree 1.**

(e) U(x,y) = x^{2}y^{2} + xy

Now, multiply t with x and y :

U(tx, ty) = (tx)^{2} (ty)^{2} + (tx)(ty)

= t^{4} (x^{2} y^{2}) + t^{2}
xy

= t^{2} (t^{2}x^{2} y^{2} +
xy)

**It is not homogeneous but it is homothetic.**

Hence, (e) is homothetic but not homogeneous.

b) The consumers utility function is given by U(X,Y) = MIN(2X,
3Y), and the given bundle is X = 3 and Y = 1.
i) MRS = __________________________________________________
ii) Draw your graph in this space:

f_X,Y(x,y)=xy 0<=x<=1, 0<=y<=2
f_X(x)=2x 0<=x<=1, f_Y(y)=y/2
0<=y<=2 choose the all correct things.
a. E[X]=1/2
b. E[XY]=8/9
c. COV[X,Y]=1
d. correlation coefficiet =1

PROVIDE EXPLANATION
- Kinko’s utility function is U(x, y) = min{2x, 6y}, where x is
whips and y is leather jackets. If the price of
whips were $20 and the price of leather jackets were $20, Kinko
would demand
a. 3 times as many whips as leather jackets.
b. 4 times as many leather jackets as whips.
c. 2 times as many leather jackets as whips.
d. 5 times as many whips as leather jackets.
e. only leather jackets.
-...

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cos(2x) / (x−1)^4 .
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Let f(x, y) = 2x^3y^2 + 3xy^3 4x^3 y. Find
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(c) fxx
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