Question

1. Suppose a consumer has the utility function over goods x and y u(x, y) = 3x}}} (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:

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) = 3x^1/3 y^2/3 subject to p_x x+p_y y =m

$\alpha$ = 3x^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 y2/3and MV_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) =

2ray-y20 (2.)2

=   

0-y:2 (2.c)

= -2y/4x²

= -y/2x²<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_xx +p_yy = 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 = 2p_x/p_y(X)  $\rightarrow$(4)

using (4) in (3) , p_xx +p_y[2(p_x/p_y(x)] = M

3p_xx =m

x= M/3p_x$\rightarrow$(5)

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

= 2M/3p

Hence, x^*(p_{x ,} p_{y, M)} = M/3p_x

and y^*(p_x, p_y, M) = 2M/3P_y

d) Since y = 2m/3p_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 $\frac{\partial y}{\partial m} < 0,$

e) Min p_xx +p_yy

s.t v^- = 3x^1/3 y^2/3

$\alpha$ = p_x x =p_yy +$\lambda$[v^--3x^1/3y^2/3]

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

$\lambda$ = p_x/x^-2/3 y^-1/3 equation (2)

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

v-=3x^1/x y^-2/3 equation (3)

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

p_{x. x.2 =} p_y.y

y= 2p_x/p_y(x) equation(4)

E(p_x ,p_y,v_-) = [p_x(2.p_x/p_y)^-2/3+p_y(2.p_x/p_y]^1/3v^-/3