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.
Solution
(a)
Maximize:
"U(x,y)=xy+x+y"
Subject to "I = P_x x + P_y y"
Lagrangian for the problem:
"L(x,y,\\lambda) = xy+x+y+\u03bb(I-P_xx-P_yy)"
First-order conditions(derivatives):
"\u2202L\/\u2202x=y+1-\u03bb^* P_x=0" (1)
"\u2202L\/\u2202y=x+1-\u03bb^* P_y=0" (2)
"\u2202L\/\u2202\u03bb=I-P_x x-P_y y=0" (3)
Using the first two conditions(1 and 2) we have
"(y+1)\/P_x=(x+1)\/P_y"
So,
"y^*=((P_x\/P_y)(x+1))-1\n\u200b" (4)
Substitute this into I function
"I=P_xx+P_y(((P_x\/P_y)(x+1))-1)"
"I=P_xx+P_xx+P_x-P_y"
"I=2P_xx+P_x-P_y" (5)
solve for x using (5)
"2P_xx=I-P_x+P_y"
"x^*=(I-P_x+P_y)\/2P_x" (6)
Substitute x* into (4)
"y^*=((P_x\/P_y)((I-P_x+P_y)\/2P_x+1))-1\n\u200b"
"y^*=(I-P_y+P_x)\/2P_y"
Demand function of Both Goods are:
"x^*=(I-P_x+P_y)\/2P_x"
"y^*=(I-P_y+P_x)\/2P_y"
I= 15, Px=2 and Py=1
Before Increase
"x^*=(15-2+1)\/2*2=3.5"
"y^*=(15-1+2)\/2*1=8"
Initial Utility
"U(3.5,8)=3.5(8)+3.5+8=39.5\\to40"
To make same utility after Py increased to $2
"x=(I+2-2)\/2*2=I\/4"
"y=(I+2-2)\/2*2=I\/4"
So,
"U(2,2)=(I\/4)^2+I\/4+I\/4=40"
"=(I^2+8I)\/16=40"
"I^2+8I-640"
Solving for I
"I=-8\u00b1 ( \u221a(64+(4*640)))\/2=(8\u00b151.22)\/2=\u00b121.61"
Income is non-negative so it's $21.61
Therefore I should be increased by: "21.61-15=" $6.61
(b)
After an increase of Py to $2
"x^*=(15-2+2)\/2*2=3.75"
"y^*=(15-2+2)\/2*2=3.75"
consumption of X and Y is 3.75 each
Substitution effect:
"Y= 8-3.75= 4.25"
"X=3.5-3.75=-0.25"
So more of X will be consumed and amount of Y consumed
(c)
Px = $2; I =$15
"Y=(I-P_y+P_x)\/2P_y =(15-P_y+2)\/2*2"
"Y=(17-P_y)\/4"
Graph of Y as a function of Py
(d)
"P_x=P_y=" $1
"x^*=(I-1+1)\/2(1)=I\/2"
"y^*=(I-1+1)\/2(1)=I\/2"
Therefore, when Income is at 0 the, demand for both x and y is 0, and as income increases from 0 to 100, their demand tends to increase also.
at 100 demands are:
"X=Y=100\/2=50"
So, demand for X, and Y become 50 each
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