Chapter 10, Problem 6P

To determine

Concept introduction:

Budget line:

This is defined as the combination of all goods that a consumer can purchase exhausting all his income. Formula for the budget line is:

Here,

• is the quantity of good X.
• is the quantity of good Y.
• is the total income.
• is the price of good X
• is the price of good Y.

Marginal Utility:

It is defined as the change in the total utility due to a change in an additional unit of a good. It may be diminishing, increasing, or constant.

Or

Here,

• is the marginal utility.
• is the total utility.
• X is the quantity of any good.

Maximizing utility Principle:

Maximizing utility in the case of two goods state that the equilibrium level of consumption of the two goods for a consumer is achieved when the Marginal Utility per dollar of the two goods are equal, provided the marginal utility of money is constant. This means that the following conditions must be fulfilled:

Here,

• is the marginal utility of good X.
• is the marginal utility of good Y.
• is the price of good X.
• is the price of good Y.

Substitution effects:

In this case, the demand of a good increases if the price of its substitute goods increase and the demand of a good decreases if the price of its substitute goods decrease. Take an example of tea and coffee, if the prices of tea increase, then the demand for coffee will increase.

Income effects:

It states that the demand for normal goods and the income are directly related which means that when income increases, the demand for normal goods also increases and vice versa.

Explanation of Solution

a. Mr. Cal’s budget line and optimal bundle Case I.

The total money with Cal is the cost of a cell phone is the cost of a pair of sun glasses is Mr. Cal’s budget line is given as:

Here,

• is the cell phone.
• is the sun glasses.
• is the total income.
• is the price of good .
• is the price of good

From the above formula, the consumption bundle is calculated as follows:

Consumption bundle	Quantity of cell phones	Quantity of sun glasses
A	0	4
B	1	2
C	2	0

Table(1)

Quantity of cell phones	Total Utility	Marginal Utility	Marginal Utility per dollar
0	0	-	-
1	400	400	4
2	700	300	3

Table(2)

Quantity of sun glasses	Total Utility (utils)	Marginal Utility (utils)	Marginal Utility per dollar
0	0	-	-
2	600	300	6
4	700	50	1

Table(3)

By the optimal consumption rule, Mr. Cal should consume bundles where,

Here,

• is the marginal utility of good
• is the marginal utility of good
• is the price of good
• is the price of good

From Table 2 and 3, the above condition is satisfied at (1, 2). Hence, the optimal consumption bundle is

Fig 1

• In the above figure, the x-axis represents the quantity of sun glasses and cell phones from right to left and from left to right respectively.

• The Y-axis represents marginal utility per dollar.

• As shown in Fig 1, the marginal per dollar of cell phone and sun glasses are equal at the consumption bundle

Conclusion:

The bundles that lie on the budget line are and the optimal bundle is that is one pair of sun glasses and 2 cell phones.

To determine

Concept introduction:

Budget line:

This is defined as the combination of all goods that a consumer can purchase exhausting all his income. Formula for the budget line is:

Here,

• is the quantity of good X.
• is the quantity of good Y.
• is the total income.
• is the price of good X
• is the price of good Y.

Marginal Utility:

It is defined as the change in the total utility due to a change in an additional unit of a good. It may be diminishing, increasing, or constant.

Or

Here,

• is the marginal utility.
• is the total utility.
• X is the quantity of any good.

Maximizing utility Principle:

Maximizing utility in the case of two goods state that the equilibrium level of consumption of the two goods for a consumer is achieved when the Marginal Utility per dollar of the two goods are equal, provided the marginal utility of money is constant. This means that the following conditions must be fulfilled:

Here,

• is the marginal utility of good X.
• is the marginal utility of good Y.
• is the price of good X.
• is the price of good Y.

Substitution effects:

In this case, the demand of a good increases if the price of its substitute goods increase and the demand of a good decreases if the price of its substitute goods decrease. Take an example of tea and coffee, if the prices of tea increase, then the demand for coffee will increase.

Income effects:

It states that the demand for normal goods and the income are directly related which means that when income increases, the demand for normal goods also increases and vice versa.

Explanation of Solution

b. Mr. Cal’s budget line and optimal bundle Case II.

The total money with Cal is the cost of a cell phone is the cost of a pair of sun glasses is Mr. Cal’s budget line is given as follows:

Using the above formula, the consumption bundles of Mr. Cal are:

Interpretation:	Consumption bundle	Quantity of cell phones	Quantity of sun glasses
A	0	4
B	1	3
C	2	2
D	3	1
E	4	0

Table(4)

Quantity of cell phones	Total Utility (utils)	Marginal Utility (utils)	Marginal Utility per dollar
0	0	-	-
1	400	400	8
2	700	300	6
3	900	200	4
4	1000	100	2

Table(5)

Quantity of sun glasses	Total Utility (utils)	Marginal Utility (utils)	Marginal Utility per dollar
0	0	-	-
1	325	325	6.5
2	600	275	5.5
3	825	225	4.5
4	700	125	-2.5

Table(6)

Fig 2

• In the above figure, the x-axis represents the quantity of sun glasses and cell phones from right to left and from left to right respectively.
• The Y-axis represents the marginal utility per dollar.
• The optimum bundle is at the intersection level, which is (2,2).

Conclusion:

Thus, the optimum bundle is (2,2).

To determine

Concept introduction:

Budget line:

This is defined as the combination of all goods that a consumer can purchase exhausting all his income. Formula for the budget line is:

Here,

• is the quantity of good X.
• is the quantity of good Y.
• is the total income.
• is the price of good X
• is the price of good Y.

Marginal Utility:

It is defined as the change in the total utility due to a change in an additional unit of a good. It may be diminishing, increasing, or constant.

Or

Here,

• is the marginal utility.
• is the total utility.
• X is the quantity of any good.

Maximizing utility Principle:

Maximizing utility in the case of two goods state that the equilibrium level of consumption of the two goods for a consumer is achieved when the Marginal Utility per dollar of the two goods are equal, provided the marginal utility of money is constant. This means that the following conditions must be fulfilled:

Here,

• is the marginal utility of good X.
• is the marginal utility of good Y.
• is the price of good X.
• is the price of good Y.

Substitution effects:

In this case, the demand of a good increases if the price of its substitute goods increase and the demand of a good decreases if the price of its substitute goods decrease. Take an example of tea and coffee, if the prices of tea increase, then the demand for coffee will increase.

Income effects:

It states that the demand for normal goods and the income are directly related which means that when income increases, the demand for normal goods also increases and vice versa.

Explanation of Solution

c. Income and substitution effects of a price change.

• Due to a change in the price of a cell phone from $100 to $50, the consumption changed from 1 to 2. It is due to the total price effect which has two parts one is substitution effect and other is income effect.
• Due to the substitution effect, Mr. Cal has an additional cell phone by sacrificing a less number of sunglasses. This has happened due to the law of demand.
• Due to the income effect, Mr. Cal has more money with him to buy more cell phones and less sun glasses.

Conclusion:

The consumption of the cell phone has increased due to a fall in its prices, due to the substitution and income effect.