Question

Utility Maximization

Bernie loves notebooks and Beyonce CDs. The accompanying table shows the utility Bernie receives from each product.

Quantity of Notebooks	Utility from notebooks (utils)	Quantity of CD's	Utility from CDs (utils)
0	0	0	0
2	70	1	80
4	130	2	150
6	180	3	210
8	220	4	260
10	250	5	300

The price of a notebook is $5, the price of a CD is $10, and Bernie has $50 of income to spend.

(a) Which consumption bundles of notebooks and CDs can Bernie consume if he spends all his income?

(b) Illustrate Bernie’s budget line with a diagram, putting notebooks on the horizontal axis and CDs on the vertical axis.

(c) Calculate the marginal utility of each notebook and the marginal utility of each CD.

(d) Then calculate the marginal utility per dollar spent on notebooks and the marginal utility per dollar spent on CDs.

(e) Find the bundle Bernie will choose given his budget constraint using the optimal consumption rule.

Answer 1

a) Which consumption bundles of notebooks and CDs can Bernie consume if he spends all his income?

Budget line has an equation of M = xPx + yPy which becomes 50 = 5Notebooks + 10CDs

From the given bundles, we see that following bundles are income exhausting (using all his income)

• 0 notebooks and 5 CDs = (0*5 + 5*10 = 50)
• 2 notebooks and 4 CDs = (2*5 + 4*10 = 50)
• 4 notebooks and 3 CDs = (4*5 + 3*10 = 50)
• 6 notebooks and 2 CDs = (6*5 + 2*10 = 50)
• 8 notebooks and 1 CDs = (8*5 + 1*10 = 50)
• 10 notebooks and 0 CDs = (10*5 + 0*10 = 50)

(b) Illustrate Bernie’s budget line with a diagram, putting notebooks on the horizontal axis and CDs on the vertical axis.  

Budget line is shown below

(c) Calculate the marginal utility of each notebook and the marginal utility of each CD.

MU or marginal utility is the additional utility when one more unit is consumed. It is calculated by subtracting two sucessive utilities for both CDs and notebooks

(d) Then calculate the marginal utility per dollar spent on notebooks and the marginal utility per dollar spent on CDs.

We use the formula MU/price for both commodities

(e) Find the bundle Bernie will choose given his budget constraint using the optimal consumption rule.

Optimal bundle should satisfy 50 = 5Notebooks + 10CDs as well as (MU/price)^Notebook = (MU/price)^CDs

Such a bundle is 4 notebooks and 3 CDs because the income is exhausted and MU/price is 6 for both commodities

Quantity of Notebooks	Utility from notebooks (utils)	Marginal utility	Marginal utility per dollar	Quantity of CD's	Utility from CDs (utils)	Marginal utility	Marginal utility per dollar
0	0			0	0
2	70	35	7	1	80	80	8
4	130	30	6	2	150	70	7
6	180	25	5	3	210	60	6
8	220	20	4	4	260	50	5
10	250	15	3	5	300	40	4