John makes $200 a week at his summer job and spends his entire weekly income on new sweaters and designer jeans since these are the only two items that provide utility to him. Furthermore, John insists that for every sweater he buys, he must also buy a pair of jeans (without jeans the new sweater is worthless ). Therefore, he buys the same number of sweaters and jeans in any given week.
(i) If jeans cost $20 and sweaters cost $30, how many will John buy of each?
(ii) If John's income rises to $300 per week how many sweaters and jeans will he buy?
(iii) Use the results in parts (i) and (ii) to construct John's income expansion path and Engel curve. Are John's preferences homothetic?