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

(a) The budget function is;

$2x+y=15$

Max: $U=xy+x+y$

Substituting all values in the demand function for each good;

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

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

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

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

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

Now, price of y rises to $\$2$ but individual wants to consume initial level of utility.

The new budget equation is;

$2x+2y=I'$ where $I'$ is unknown.

The initial level of utility is;

$U^*=3.5\times 8+3.5+8=39.5\approx40$

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

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

Solving for I';

$I'=\frac{(-8\pm\sqrt{(64+4\times640)}}{2}$ ]

$I'=21.61$

$\therefore$ Income should be increased by;

$\$(21.61-15)$$=$ $\$6.61$

b)After price change, demand for x and y are;

$x^*=\frac{I-p_y-p_x}{2P_x}$

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

$y^*=\frac{I-p_x-p_y}{2P_y}$

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

In the above diagram, from E₁ TO E ^s there is the substitution effect and from E ^s to E_2, there is income effect and A"B" is the compensated budget line.

c)When $p_y$ 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 ;

d)The prices are $\$1$ but income is the variable then, demand for x and y is;

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

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

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

Hence if income is $\$100$ ,then demand for x becomes;

$x=\frac{100}{2}=50=y$