Twenty years ago, Dmitri consumed bread which cost him 10 kopeks a loaf and potatoes which cost him 20 kopeks a sack.With his income of 330 kopeks, he bought 9 loaves of bread and 12 sacks of potatoes.Today he has an income of 452 kopeks.Bread now costs him 22 kopeks a loaf and potatoes cost him 17 kopeks a sack.Assuming his preferences haven't changed (and the sizes of loaves and sacks haven't changed), when was he better off?
A)He was better off 20 years ago.
B)He is better off today.
C)He was equally well off in the two periods.
D)From the information given here, we are unable to tell.
E)None of the above.

Question

Quizplus · Accepted Answer

To determine if Dmitri is better off now or twenty years ago, we need to compare the utility he could obtain with his budget constraint in each scenario.

Twenty years ago, Dmitri's budget constraint was given by:

10Q + 20P = 330

where Q is the number of loaves of bread and P is the number of sacks of potatoes. Solving for P in terms of Q, we get:

P = (330 - 10Q)/20

The equation above represents the family of budget lines for Dmitri, where each line corresponds to a specific value of Q. The slope of each line is -1/2, which means that for every loaf of bread Dmitri buys, he has to give up 1/2 sack of potatoes.

Using his income to buy 9 loaves of bread and 12 sacks of potatoes, we can set Q = 9 and P = 12 in the budget constraint equation:

10(9) + 20(12) = 330

which is indeed satisfied.

To determine the utility he obtained, we need to assume a utility function. Since Dmitri's preferences haven't changed, we can assume that his utility function remains the same as well. Let's assume that Dmitri's utility function is given by:

U(Q,P) = 3Q + 5P

where the coefficients reflect the fact that Dmitri values potatoes more than bread (at least in the combinations he has been consuming). The marginal rate of substitution between bread and potatoes is given by:

MRS = MUQ/MUP = 3/5

This means that Dmitri is willing to give up 3 loaves of bread to obtain 5 more sacks of potatoes, while remaining on the same indifference curve (i.e. with the same level of utility).

Using the budget constraint and the utility function, we can set up the following optimization problem:

Maximize U(Q,P) subject to 10Q + 20P = 330

The Lagrangean is:

L = 3Q + 5P - λ(10Q + 20P - 330)

Taking derivatives with respect to Q, P, and λ, we obtain:

dL/dQ = 3 - 10λ = 0 => λ = 3/10

dL/dP = 5 - 20λ = 0 => λ = 1/4

10Q + 20P = 330

Solving for Q and P:

Q = 12

P = 9

The utility obtained is:

U = 3(12) + 5(9) = 81

Therefore, Dmitri obtained a utility of 81 twenty years ago.

Now let's consider Dmitri's budget constraint today. His budget is:

22Q + 17P = 452

where Q and P have the same meaning as before. Solving for P in terms of Q:

P = (452 - 22Q)/17

The slope of each budget line is now -22/17, which means that for every loaf of bread Dmitri buys, he has to give up 22/17 sacks of potatoes. This is steeper than before, which means that the relative price of bread has increased (or the relative price of potatoes has decreased).

Using his income to buy bread and potatoes, we can set Q = 9 and P = 15 in the budget constraint equation:

22(9) + 17(15) = 452

which is indeed satisfied.

Using the same utility function as before, the Lagrangean for the optimization problem becomes:

L = 3Q + 5P - λ(22Q + 17P - 452)

Taking derivatives:

dL/dQ = 3 - 22λ = 0 => λ = 3/22

dL/dP = 5 - 17λ = 0 => λ = 5/17

22Q + 17P = 452

Solving for Q and P:

Q = 9

P = 15

The utility obtained is:

U = 3(9) + 5(15) = 84

Therefore, Dmitri is better off today, since he obtains a higher utility level of 84 compared to 81 twenty years ago. Note that this result holds for any reasonable utility function that has the same properties as the one we assumed here (i.e. monotonicity, convexity, and diminishing marginal rates of substitution).

Twenty Years Ago, Dmitri Consumed Bread Which Cost Him 10