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?

(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.

Answer 1

Solution

(a)

Maximize:

$U(x,y)=xy+x+y$

Subject to $I = P_x x + P_y y$

Lagrangian for the problem:

$L(x,y,\lambda) = xy+x+y+λ(I-P_xx-P_yy)$

First-order conditions(derivatives):

$∂L/∂x=y+1-λ^* P_x=0$ (1)

$∂L/∂y=x+1-λ^* P_y=0$ (2)

$∂L/∂λ=I-P_x x-P_y y=0$ (3)

Using the first two conditions(1 and 2) we have

$(y+1)/P_x=(x+1)/P_y$

So,

$y^*=((P_x/P_y)(x+1))-1 ​$ (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)

$y^*=((P_x/P_y)((I-P_x+P_y)/2P_x+1))-1 ​$

$y^*=(I-P_y+P_x)/2P_y$

Demand function of Both Goods are:

$x^*=(I-P_x+P_y)/2P_x$

$y^*=(I-P_y+P_x)/2P_y$

I= 15, P_x=2 and P_y=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 P_y 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

$I=-8± ( √(64+(4*640)))/2=(8±51.22)/2=±21.61$

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 P_y 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