Question

Do each of a-d, both geometrically (you need not be precise) and using calculus. There are only two goods; x is the quantity of one good and y of the other. Your income is I and u(x,y) = xy + x + y.

(a) Px = $2; Py = $1; I = $15. Suppose Py rises to $2. By how much must I increase in order that you be as well off as before?

Answer 1

Assume the utility function to be:

$U=xy+x+y$

Determining the slope indifference curve:

$dU=xdy+ydx+dx+dy$

Along any indifference curve, $dU=0$

$xdy+ydx+dx+dy=0\\dy(x+1)+dx(y+1)=0\\dy(x+1)=-dx(y+1)\\\frac{dy}{dx}=-\frac{y+1}{x+1}\\MRS_{xy}=-\frac{y+1}{x+1}$

The slope of indifference curve is negative

Differentiating it again

$\frac{d(MRS_{xy})}{dx}=-[\frac{(x+1)\frac{dy}{dx}-(y+1)}{(x+1)^2}]\\\frac{d(MRS_{xy})}{dx}=-[-\frac{(x+1)\frac{y+1}{x+1}-(y+1)}{(x+1)^2}]>0$

Therefore, the indifference curve is strict convex

The budget equation is:

$P_xx+P_yy=I$

Solving this in general:

max: $U=xy+x+y$

subject to: $P_xx+P_yy=I$

The Lagrangian expression is:

$L=xy+x+y+\lambda[I-xP_x-yP_y]$

The first order conditions are:

$\frac{\partial L}{\partial x}=y+1-\lambda P_x=0-----(1)\\\frac{\partial L}{\partial y}=x+1-\lambda P_y=0-----(2) \\\frac{\partial L}{\partial \lambda}=I=xP_x+yP_y-----(3)$

From (1) and (2):

$\frac{y+1}{x+1}=\frac{P_x}{P_y}\\P_xx+P_x=yP_y+P_y\\P_xx=yP_y+P_y-P_x$

Substitute this in (3):

$yP_y+P_y=P_x+yP_y=I\\2yP_y+P_y-P_x=I\\y=\frac{I+P_x-P_y}{2P_y}$

This is the demand function of good y

$P_yy=\frac{I+P_x-P_y}{2}\\P_xx=yP_y+P_y-P_x\\P_xx=\frac{I+P_x-P_y}{2}+P_y-P_x\\P_xx=\frac{I+P_x-P_y+2P_y-2P_x}{2}\\P_xx=\frac{I+P_y-P_x}{2}$

$x=\frac{I+P_y-P_x}{2P_x}$

This is the demand function for good x

The prices are $2 and $1 for good x and good y respectively and the income is $15

Thus, the budget function is:

$2x+y=15$

max: $U=xy+x+y$

subject to: $2x+y=15$

$x=\frac{I+P_y-P_x}{2P_x}=\frac{15+1-2}{2*2}=3.5\\y=\frac{I+P_x-P_y}{2P_y}=\frac{15+2-1}{2*1}=8$

The demand function for x is 3.5 and y is 8 units

Now, price of y rises to $2 but individual wants to consume initial level of utility

The new budget equation is:

$2x+2y=I'$

Where $I'$ is unknown

The initial level of utility is:

$U=3.5*8+3.5+8=39.5\approx40$

Now, $x^s=\frac{I'+2-2}{2*2}=\frac{I'}{4}$

and

$y^s=\frac{I'+2-2}{2*2}=\frac{I'}{4}$

$U=\frac{I'^2}{16}+\frac{I'}{4}+\frac{I'}{4}=40$

Then:

$\frac{I'^2+8I'}{16}=40\\I'^2+8I'=640\\I'^2+8I'-640=0$

You can solve this by the quadratic formula:

$I'=\frac{-8\pm\sqrt{64+(4*640)}}{2}=\frac{-8\pm51.22}{2}$

Income should not be negative, so $I'=21.61$

Therefore, income should be increased by:

$21.61-15=6.61$