Utility maximization

Which best expresses the law of diminishing marginal utility?

• The more consumption of a product, the smaller is the total and marginal utility

  from the consumption.

• The less consumption of a product, the greater is the total and marginal utility of

  the consumption

• The more consumption of a product, the smaller is the marginal utility from

  consuming an additional unit.

• The more consumption of a produc t, the smaller is the total and marginal utility

  from the consumption.

Answer explanation

Note that your total utility increases when you consume more, but at a lower rate => Thus marginal utility gets smaller as you consume more

A consumer with a fixed income will maximize utility when each good is purchased

in amounts such that the:

• total utility is the same for each good

• marginal utility of each good is maximized.

• marginal utility per dollar spent is the same for all good

• marginal utility per dollar spent is maximized for each good.

Answer explanation

When MUx/Px = MUy/Py, the consumer reaches the maximum utility

A utility-maximizing consumer buys only CDs and books. After all of his income is spent, he finds that the last CD he bought gave him 60 units of satisfaction, and the last book he purchased gave him 30 units of satisfaction. CDs cost $12 each and books cost $5 each. Which statement is true?

• he should buy fewer CDs and fewer books

• he should buy more CDs and more books

• he should buy more CDs and fewer books

• he should buy more books and fewer CDs

Answer explanation

MU(cd)/P(cd) = 60/12 = 5

MU(book)/P(book) = 30/5 = 6

MU per dollar for book is larger than MU per dollar for cd => The consumer can increase Total utility by buying more book

The price of product X is $4 and the price of product Y is $2. The income of the consumer is $20.

If the consumer can only buy product X, how much will the consumer buy and what will be the total utility?

• 4X and 20

• 4X and 104

• 5X and 16

• 5X and 120

The price of product X is $4 and the price of product Y is $2. The income of the consumer is $20.

If the consumer buys both product X and product Y, how much will the consumer buy of each to maximize utility?

• 4X and 2Y

• 3X and 4Y

• 4X and 3Y

• 5X and 3Y

Answer explanation

MUx/Px = MUy/Py

<=> MUx/4 = MUy/2

Look at the table, at

x = 3; MUx = 24 => MUx/4 = 6

y = 4; MUy= 12 => MUy/2 = 6

Check income:

3*4 + 4* 2= 20

=> so 3x and 4y is the optimal bundle

The price of product X is $4 and the price of product Y is $2. The income of the consumer is $20.

When the consumer purchases the utility-maximizing combination of product X and product Y, total utility will be:

• 72

• 3X and 4Y

• 136

• 156

Answer explanation

Recall: Sum of marginal utility = total utility

Total utility at 3x:

32 + 28 + 24 = 84

Total utility at 4y:

24 + 20 + 16 + 12 = 72

Total Utility at (3x,4y) = 84 + 72 = 156

Julie has preferences for food F and clothing C described by a utility function U(F, C) = FC. Her marginal utilities are MU(F) = C and MU(C) = F. Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing.

Using algebra, find the optimal choice of food and clothing.

Answer explanation

MU(F)/MU(C) = P(F)/P(C)

<=> C/F = 1/2

<=> F = 2C (eq.1)

Buget line: 1F + 2C = 12 (eq.2)

Replace (eq.1) in (eq.2)

2C + 2C = 12 <=> C = 3

=> F = 2C = 2*3 = 6

Optimal basket (6F,3C)

Julie has preferences for food F and clothing C described by a utility function U(F, C) = FC. Her marginal utilities are MU(F) = C and MU(C) = F. Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing.

What is the marginal rate of substitution of food for clothing at her optimal basket?

Answer explanation

MRS = MU(F)/MU(C) ;

MU(F) = C

MU(C) = F

At optimal basket (6F,3C):

MRS = 3/6 = 1/2

Julie has preferences for food F and clothing C described by a utility function U(F, C) = FC. Her marginal utilities are MU(F) = C and MU(C) = F. Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing.

Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget (instead of the optimal basket). Would her marginal utility per dollar spent on food be greater than or less than her marginal utility per dollar spent on clothing? What should be her rational behaviour at this basket (to increase her total utility)?

Answer explanation

At basket A = (4F,4C)

MU(f)/P(f) = 4/1 = 4

MU(c)/P(c) = 4/2 = 2

MU per dollar of food > MU per dollar of clothing

=> Julie can buy more food and less clothing to increase Total Utility