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?
(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) Px = $2; I =$15. Graph the amount of Y you consume as a function of Py , for values of Py 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.
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}"
"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"
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