Question

Michael is a particular man. He likes to drink just milk and sugar to get himself going in the morning rather than coffee. Furthermore, the ratios must be 1:3. That is, he must have 1 cup of milk (M) for every three cubes of sugar (S). The utility that he receives must contain both milk and sugar.

(a) What is Michael’s utility function? What is the type of preference that he has for these two goods?

(b) What is the optimality condition here? Also write down the optimality condition using words.

(c) Draw three indifference curves, making sure to clearly label all points.

(d) use the optimal ratio equation from part (b) along with the budget constraint to solve for this consumer’s demand functions, M∗ and S∗

(e) Suppose I = 50,PM = $2,PS = $0.50. Write down the budget con- straint.

(f) Using the numbers from part (e), how many cups of milk and cubes of sugar will Michael consume?

(g) What would Michael’s utility be if he spent all of his income on sugar cubes?

Answer 1

(a) Michael's utility function is given by
U (M, S) = min(M, 3S)
For every 1 cup of milk, he requires 3 cubes of sugar.

The two goods are complements.

(b) The optimality condition in this case is given by
M^* = 2S^*

Any more cube of sugar with respect to 1 cup of milk would not give him any additional utility. Similarly, any more cup of milk with respect to 3 cubes of sugar would provide him with additional utility. Therefore, the utility is maximized exactly at the ratio of 1:3.

(c)

(d) The budget constraint is PmM + PsS = I,
where Ps is the price 1 cube of sugar and Pm is the price of 1 cup of milk.
Using the optimality condition, i.e. M^* = 2S^*, we get
Pm(2S) + PsS = I
S ( 2Pm + Ps) = I
=> S^* = I / (2Pm + Ps)

M^* = 2I / (2Pm + Ps)