Answer to Question #273402 in Microeconomics for Stephany

Question #273402

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.


1
Expert's answer
2021-12-01T10:44:52-0500

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"





Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS