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

(b) In the case described in part (a), assuming that I does not change, what quantities of each good are consumed before and after the price change? How much of each change is a substitution effect? How much is an income effect?

(c) P_x = $2; I =$15. Graph the amount of Y you consume as a function of P_y , for values of P_y ranging from $0 to $10 (your ordinary demand curve for Y).

(d) With both prices equal to $1, show how consumption of each good varies as I changes from $0 to $100.

Answer 1

let the utility function be

$U=xy+x+y$

Before proceeding lets determine the slope of indifference curve:

by total differential

$dU=xdy+ydx+dx+dy$

Along any indifference curve, dU=0 and

$xdy+ydx+dx+dy=0\\so \ dy(x+1)=-dx(y+1)$

$\frac{dy}{dx}=-\frac{y+1}{x+1}$

$or\ MRS_{xy}=-\frac{y+1}{x+1}$

the slope of indifference curve is negative

Now again differentiate

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

Hence the indifference curve is strict convex

The prices are $p_x \ and\ p_y$ respectively and I is the income

Hence the budget Equation is

$p_xx+p_yy=I$

Now you can solve this in general

$Max:U=xy+x+y\\subject\ to: p_xx+p_yy=I$

$L=xy+x+y+\lambda[I-xp_x-yp_y]$

The first order conditions are

$\frac{\delta L}{\delta x}=y+1-\lambda\ p_x=0.......(1)$

$\frac{\delta L}{\delta y}=x+1-\lambda\ p_y=0.......(2)$

$\frac{\delta L}{\delta \lambda}=xp_x+yp_y=0.......(3)$

from (1) and (2)

$\frac{y+1}{x+1}=\frac{px}{py}$

$p_xx=yp_y+p_y-p_x$

substitute in (3)

$2yp_y+p_y-p_x=I$

$y=\frac{I+p_x-p_y}{2p_y}$

This is the demand function of good y

$p_yy=\frac{I+p_x-p_y}{2p_y}$

for good x it is

$p_xx=\frac{I+p_y-p_x}{2p_x}$

(a) The budget function is

$2x+y=15$

$max: U=xy+x+y\\subject\ to:2x +y=15$

Now substitute all values in the demand function for each good

$x=\frac{I+p_y-p_x}{2p_x}$

$x=\frac{15+1-2}{2\times2}=3.5$

and for y

$y=\frac{15+2-1}{2\times1}=8$

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

Now, price of y rises to $2 but individuals want to consume initial level of utility

the new budget equqtion is

$2x+2y=I'$

where I is unknown

The initial level of utility is

$U=3.5(8)+3.5(8)=39.5=40$

$now \ x^s=\frac{I'+2-2}{2\times2}=\frac{I'}{4} \ and \ y^s=\frac{I'+2-2}{2\times2}=\frac{I'}{4}$

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

$I^{'2}+8I'-640=0$

You can sridharachariyyan method

$I^{'}=\frac{-8+\sqrt{64+4\times640}}{2}$

$I^{'}=\frac{-8+\sqrt{64+4\times640}}{2}$

$so\ I'=21.61$

Hence income should be increased by $(21.61-15)=6.61$

New compensated demand for good x and y are $x^s=\frac{21.61}{4}=5.40=y^s$

(b) Before price change the demand for x is 3.5 and demand for y is 8 and after change in price demand for x is

$x=\frac{I+p_y-p_x}{2p_x}$

$x^{**}=\frac{15+2-2}{2\times2}=3.75$

and demand for y is

$y^{**}=\frac{15+2-2}{2\times2}=3.75$

Now the substitution effect is change in price of one good but the utility at previous level

hence substitution effect on good x is

$(x^*-x^s)=(3.5-5.4)=-1.9$

for y is

$(y^*-y^s)=(8-5.4)=2.6$

The income effect is when price of good y increases then there is the decrease in real income and the demand for both goods fall.

Income effect of good x is

$(x^*-x^s)=(5.4-3.75)=1.7$

and income effect for good y is

$(y^s-y^{**})=(5.4-3.75)=1.7$

Here good y is drawn in vertical axis and drawn in horizontal axis for convenience. From $E_1$ to $E^s$ , there is the substitution effect and from $E^s\ to\ E_2$ there is the income effect and A"B" is the compensated budget line

(c) Now, price of y is a variable then the demand for y is:

$y=\frac{I+p_x-p_y}{2p_y}$

$y=\frac{15+2-p_y}{p_y}$

If price of y is zero then demand for y is infinite . if price of y is $10 then

$y=\frac{15+2-10}{10}=0.7$

The graph is

This is the demand curve that is D and maximum willingness to pay is $17 and if price is $10, demand is 0.7

(d) The prices are $1 but income is the variable then, demand for x is

$x=\frac{I'+1-1}{2}=\frac{I}{2} \ and \ y=\frac{I'+1-1}{2}=\frac{I}{2}$

If the income is zero , the demand for x is zero and also y is zero but if income increases then demand for x and y are increasing.

Hence if income is $100 the demand for x is $x=\frac{100}{2}=50=y$