Question

(38pts) Suppose a consumer spends all of her income on only two goods, z and y. Her preferences over these two goods are represented by the utility function u(r,y) min(, 4y). The price of good y is given to be S8. Her income and price of z are represented by m and ps, respectively. (a) (10 pts) Find the demand for good z as a function of m and pa. (b) (5 pts) Is good z ordinary or Giffen good? Explain. (e) (5 pts) Is good z normal or inferior good? Explain. (d) (18 pts) Now suppose m $80 and pr S8. i. (8 pts) Draw her budget line and find the optimal consumption bundle. i. (10 pts) Suppose now she receives an extra coupon worth $24, which can only be spent on good y. Draw the new budget line and find the optimal consumption bundle.

Answer 1

a) Consumer's problem:

$\small max(min[x,4y])$

subject to $8y+xp_x=m$

The utility function can be represented by level curves which will have a kink at x = 4y Since the 2 goods are complements, they will always be consumed together. Hence, we will have L shaped level curves with kinks at x = 4y

8 6 4 2 2 4 6 8

Plugging the value x = 4y in the budget constraint, we get:

$\small 8y+p_x4y=m\rightarrow y=\frac{m}{8+4p_x}$

$\small \rightarrow x=4y=\frac{m}{2+p_x}$

b) As price of x falls, quantity demanded of x increases. This implies that x is an ordinary good. This can be proved algebraically as follows:

$\small \rightarrow x=\frac{m}{2+p_x}$

Say $\small p_x$ decreases to $\small p_x$ ''

$\small \rightarrow x=\frac{m}{2+p_x''}$

$\small \rightarrow 2+p_x>2+p_x''\rightarrow \frac{m}{2+p_x}< \frac{m}{2+p_x''}$

c) As income increases, demand for x increases. Hence, x is a normal good. This can be proved algebraically as follows:

$\small \rightarrow x=\frac{m}{2+p_x}$

Say $\small m$ increases to $\small m$ ''

$\small \rightarrow x=\frac{m''}{2+p_x}$

$\small \rightarrow m<m''\rightarrow \frac{m}{2+p_x}< \frac{m''}{2+p_x}$

d) i) Plugging the values given in the demand for x we found in the first part we get:

$\small x=\frac{80}{10}\rightarrow x=8$

$\small y=\frac{80}{40}\rightarrow y=2$

Plotting the budget line

5 5

ii) Since the consumer wants to maximize his utility, he would want to spend all his income in equilibrium. Hence, the extra $24 received for y would be spent in its entirety. Hence, the new budget constraint is:

$\small 8x+8y=104,{_}x \leq 10$

15 10 -5 15 0 5

Plugging the values of $\small m$ and $\small p_x$ in the equations obtained in the first part, we get

$\small x=10.4, {_} y=2.6$

But x cannot be greater than 10 so the optimal bundle is

$\small (x,y)=(10,3)$