Question

1. Suppose a consumer has the utility function over goods x and y u(x,y) = 3x{y} (a) Setup the utility maximization problem for this consumer using the general budget con- straint. (2 points) (b) Will the constraint be active/binding? Is the sufficient condition for interior solution satisfied? Prove your answers. (4 points)

Answer 1

Answer:

u(x , y) = 3 x^1/3 y^2/3

a) Consumer's objective is to maximize utility u(x , y) =3 x ^1/3 y ^2/3 subject to the budget constraint p _x x +p _y y = m ,

Where m = income to be sport on x and y , p _x and p _y are prices of good x and y respectively.

u(x , y) = 3 x ^1/3 y ^2/3 subject to p _x x+ p _y y =m

$\alpha$ = 3 x ^1/3 y ^2/3 +$\lambda$[M -p _x x - p _y y]

b)Mu _x = 3 *(1/3) x^-2/3 y ^2/3

= x^-2/3 y 2/3 and M V _y = 3*(2/3) x ^1/3 y^-1/3

$\Rightarrow$MRS =

2 -2/342/3 2.71/3y-1/3 22

d/ dx (MRS) =

=   

= -2 y/4 x ²

= -y/2 x ²<0

Since the performance are represent convex indifference curve

(i.e d/ dx MRS< 0 or diminishing MRS) , the constraint will be binding

and the sufficient condition for interior solution satisfied.

c) $\frac{\partial \alpha }{\partial x} = 3.\frac{1}{3}x^{-2/3}y^{2/3} -\lambda p_{x}=0$

  $\lambda = \frac{x^{-2/3}y^{2/3}}{p_{x}} \rightarrow (1)$

$\frac{\partial \alpha }{\partial y} = 3\frac{2}{3}x^{1/3}y^{-1/3} -\lambda p_{y}=0$

1-2137-1/3 - (2

$\frac{\partial\alpha }{\partial \lambda } = m-p_{x}x - p_{y}y = 0$

p _x x +p _y y = m $\rightarrow$ (3)

From (1) and (2) , $\lambda$ = $\frac{x^{-2/3}y^{2/3}}{p_{x}} = \frac{2x^{1/3}y^{-1/3}}{p_{y}}$

y = 2 p _x /p _y(X)  $\rightarrow$(4)

using (4) in (3) , p _xx +p _y[2(p _x/ p _y (x)] = M

3 p _x x =m

x= M/3 p _x$\rightarrow$(5)

y =2 $\frac{p_{x}}{p_{y}}.\frac{M}{3p_{x}}$

= 2 M /3 p

Hence, x^*(p _{x ,} p _{y , M)} = M/ 3 p _x

and y^*(p _x, p _y, M) = 2 M / 3 P _y

d) Since y = 2 m/3 p _y  

$\frac{\partial y}{\partial p_{x}}= 0$(goods x and y are unrelated , i.e they are neither substitudes nor complements

$\frac{\partial y}{\partial p_{x}} = \frac{2M}{3}\frac{\partial }{\partial p_{y}}\frac{1}{p_{y}} = \frac{2M}{3(-1)}.\frac{1}{p_{y^{2}}} = \frac{-2M}{3p_{y^{2}}}<0$

since $\frac{\partial y}{\partial p_{y}}< 0 ,$ y is an ordinary good and it is not a giffen good as $\frac{\partial y}{\partial p_{y}}>0$ for a giffen good.

Now$\frac{\partial }{\partial m}y = \frac{2}{3p_{y}}> 0$Since $\frac{\partial }{\partial m} > 0,$

good y is a normal good and not an inferrior good.

For an inferior good $\frac{\partial y}{\partial m} < 0,$