Question

A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below.

Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively.

The consumer has an income of $74 to spend.

Product X				Product Y				Product Z
Quantity	Utility	Marginal Utility per $		Quantity	Utility	Marginal Utility per $		Quantity	Utility	Marginal Utility per $
1	42	_____		1	14	_____		1	32	_____
2	82	_____		2	26	_____		2	60	_____
3	118	_____		3	36	_____		3	84	_____
4	148	_____		4	44	_____		4	100	_____
5	170	_____		5	50	_____		5	110	_____
6	182	_____		6	54	_____		6	116	_____
7	182	_____		7	56.4	_____		7	120	_____

(a)          Complete the table by computing the marginal utility per dollar for successive units of X, Y, and Z to one or two decimal places.

(b)          How many units of X, Y, and Z will the consumer buy when maximizing utility and spending all income?

(c)           Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z?

Answer 1

A.

Product X	Product Y	Product Z
Quantity	Utility	Marginal Utility per $	Quantity	Utility	Marginal Utility per $	Quantity	Utility	Marginal Utility per $
1	42	4.2	1	14	7	1	32	4
2	82	4	2	26	6	2	60	3.5
3	118	3.6	3	36	5	3	84	3
4	148	3	4	44	4	4	100	2
5	170	2.2	5	50	3	5	110	1.25
6	182	1.2	6	54	2	6	116	0.75
7	182	0	7	56.4	1.2	7	120	0.5

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B.

The income of $74 will be spent, on the basis of products producing highest marginal utility per dollar spent. On the basis of it:

First purchase will be of: Product Y ($72 left)

Second purchase will be of: Product Y ($70 left)

Third purchase will be of: Product Y    ($68 left)

Fourth purchase will be of: Product X    ($58 left)

Fifth purchase will be of: Product Z ($50 left)

Sixth purchase will be of: Product X ($40 left)

Seventh purchase will be of: Product Y ($38 left)

Eight purchase will be of: Product X ($28 left)

Ninth purchase will be of: Product Z    ($20  left)

Tenth purchase will be of: Product X    ($10 left)

Eleventh purchase will be of: Product Y ($8 left)

Twelfth purchase will be of: Product Z ($0 left)

Now total products purchased are:

4 units of product X

5 units of Product Y

3 units of product Z

Total utility achived = 148+50+84

Total utility achieved = 282 ( it is the maximum utility to be achieved with $74)

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C.

In this case, total utility = 82+44+32 = 158

Total income spent = 10*2 + 2*4+8*1 = $36

The first reason is that total income is not used, so utility is not maximized. Second, total utility achieve is 158 that is less that utility maximization level of utility. So, utility is not maximized with this level of consumption bundle.