Theory of Consumer Behavior

Given/Concept: The budget set relates to what a consumer can afford given her income and market prices.

Definition: The budget set of a consumer is the collection of all bundles of goods that the consumer can buy with her given income at the prevailing market prices.

Explanation: If a consumer has income $M$ and the prices of two goods are $p_1$ and $p_2$, then the budget set consists of all bundles $(x_1, x_2)$ such that:
$$p_1 x_1 + p_2 x_2 \leq M, \quad x_1 \geq 0, \quad x_2 \geq 0$$

In other words, it includes all affordable combinations — those that cost exactly $M$ (on the budget line) as well as those that cost less than $M$ (inside the budget line). The budget set is also called the opportunity set of the consumer.

Definition: The budget line represents all bundles of two goods which cost the consumer her entire income at the prevailing market prices.

Mathematical Expression: If the consumer's income is $M$ and prices of good 1 and good 2 are $p_1$ and $p_2$ respectively, the budget line is:
$$p_1 x_1 + p_2 x_2 = M$$

Key Points:
- All bundles lying on the budget line exhaust the consumer's entire income.
- The budget line is the boundary of the budget set.
- It can be rewritten as: $x_2 = \dfrac{M}{p_2} - \dfrac{p_1}{p_2} x_1$, which is a straight line with slope $-\dfrac{p_1}{p_2}$ and vertical intercept $\dfrac{M}{p_2}$.

Concept: The slope of the budget line reflects the trade-off between the two goods.

Explanation:

The equation of the budget line is:
$$p_1 x_1 + p_2 x_2 = M$$

Rearranging:
$$x_2 = \frac{M}{p_2} - \frac{p_1}{p_2} x_1$$

The slope of the budget line is $-\dfrac{p_1}{p_2}$, which is negative (since both p_1 > 0 and p_2 > 0).

Intuitive Reason: The consumer has a fixed income $M$. If she wants to consume more of good 1, she must spend more on good 1, and since her income is fixed, she must reduce her expenditure on good 2, thereby consuming less of good 2. Thus, as $x_1$ increases, $x_2$ must decrease — making the budget line downward (negatively) sloping.

Conclusion: The budget line is downward sloping because of the constraint of a fixed income — consuming more of one good necessarily means consuming less of the other good.

Given:
- Price of good 1: $p_1 = \text{Rs } 4$
- Price of good 2: $p_2 = \text{Rs } 5$
- Consumer's income: $M = \text{Rs } 20$

Let $x_1$ = quantity of good 1 and $x_2$ = quantity of good 2.

(i) Equation of the Budget Line:
$$p_1 x_1 + p_2 x_2 = M$$
$$4x_1 + 5x_2 = 20$$

(ii) Maximum consumption of good 1:

If the consumer spends her entire income on good 1, then $x_2 = 0$:
$$4x_1 = 20 \implies x_1 = \frac{20}{4} = 5 \text{ units}$$

The consumer can consume 5 units of good 1.

(iii) Maximum consumption of good 2:

If the consumer spends her entire income on good 2, then $x_1 = 0$:
$$5x_2 = 20 \implies x_2 = \frac{20}{5} = 4 \text{ units}$$

The consumer can consume 4 units of good 2.

(iv) Slope of the budget line:
$$\text{Slope} = -\frac{p_1}{p_2} = -\frac{4}{5} = -0.8$$

The slope of the budget line is $\mathbf{-\dfrac{4}{5}}$ (or $-0.8$).

Given (from Q.4):
- $p_1 = \text{Rs } 4$, $p_2 = \text{Rs } 5$
- New income: $M' = \text{Rs } 40$

New Budget Line:
$$4x_1 + 5x_2 = 40$$

Effect on the Budget Line:

- New horizontal intercept (max $x_1$): $\dfrac{40}{4} = 10$ units (earlier it was 5 units)
- New vertical intercept (max $x_2$): $\dfrac{40}{5} = 8$ units (earlier it was 4 units)
- Slope remains unchanged: $-\dfrac{p_1}{p_2} = -\dfrac{4}{5}$

Conclusion: When income doubles from Rs 20 to Rs 40 with prices unchanged, the budget line shifts outward (to the right) in a parallel manner. The slope remains the same ($-4/5$), but the consumer can now afford more of both goods. The budget set expands.

Given (from Q.4):
- $p_1 = \text{Rs } 4$, $M = \text{Rs } 20$
- New price of good 2: $p_2' = 5 - 1 = \text{Rs } 4$

New Budget Line:
$$4x_1 + 4x_2 = 20$$

Effect on the Budget Line:

- Horizontal intercept (max $x_1$): $\dfrac{20}{4} = 5$ units — unchanged
- New vertical intercept (max $x_2$): $\dfrac{20}{4} = 5$ units (earlier it was 4 units) — increases
- New slope: $-\dfrac{p_1}{p_2'} = -\dfrac{4}{4} = -1$ (earlier it was $-4/5$)

Conclusion: When the price of good 2 falls, the budget line rotates outward around the horizontal intercept (the point on the $x_1$-axis stays fixed at 5 units, but the point on the $x_2$-axis moves upward from 4 to 5 units). The budget line becomes less steep (slope changes from $-4/5$ to $-1$ in absolute terms it becomes steeper — from $0.8$ to $1$). The consumer can now afford more of good 2.

Given (from Q.4):
- Original: $p_1 = 4$, $p_2 = 5$, $M = 20$
- New: $p_1' = 8$, $p_2' = 10$, $M' = 40$

New Budget Line:
$$8x_1 + 10x_2 = 40$$

Dividing both sides by 2:
$$4x_1 + 5x_2 = 20$$

This is identical to the original budget line.

Verification:
- New horizontal intercept: $\dfrac{40}{8} = 5$ units (same as before)
- New vertical intercept: $\dfrac{40}{10} = 4$ units (same as before)
- New slope: $-\dfrac{8}{10} = -\dfrac{4}{5}$ (same as before)

Conclusion: When both prices and income double simultaneously, the budget set remains unchanged. This is because the purchasing power of the consumer does not change — the ratio of income to prices stays the same. The budget line is exactly the same as before.

Given:
- Quantity of good 1: $x_1 = 6$ units
- Quantity of good 2: $x_2 = 8$ units
- Price of good 1: $p_1 = \text{Rs } 6$
- Price of good 2: $p_2 = \text{Rs } 8$
- The consumer spends her entire income on this bundle.

Formula:
$$M = p_1 x_1 + p_2 x_2$$

Calculation:
$$M = 6 \times 6 + 8 \times 8 = 36 + 64 = 100$$

The consumer's income is Rs 100.

Given:
- Price of good 1: $p_1 = \text{Rs } 10$
- Price of good 2: $p_2 = \text{Rs } 10$
- Income: $M = \text{Rs } 40$
- Both goods available only in integer units.

Budget Constraint: $10x_1 + 10x_2 \leq 40$, i.e., $x_1 + x_2 \leq 4$, where $x_1, x_2 \in \{0, 1, 2, 3, 4, ...\}$

(i) All bundles available to the consumer (i.e., $x_1 + x_2 \leq 4$, non-negative integers):

$$\{(0,0),(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(2,0),(2,1),(2,2),(3,0),(3,1),(4,0)\}$$

There are 15 bundles in total.

(ii) Bundles that cost exactly Rs 40 (i.e., $x_1 + x_2 = 4$):

$$\{(0,4),(1,3),(2,2),(3,1),(4,0)\}$$

These 5 bundles lie exactly on the budget line and exhaust the consumer's entire income of Rs 40.

Definition: A consumer's preferences are said to be monotonic if, between any two bundles, the consumer prefers the bundle which has more of at least one good and no less of the other good.

In other words: If bundle $A = (x_1, x_2)$ and bundle $B = (y_1, y_2)$, then the consumer prefers $A$ over $B$ if:
- $x_1 \geq y_1$ and $x_2 \geq y_2$, with at least one strict inequality.

Implication: Monotonic preferences mean that the consumer always prefers more to less — having more of a good (without having less of any other good) always makes the consumer better off.

Example: If bundle $(5, 6)$ is compared with $(4, 6)$, a consumer with monotonic preferences will prefer $(5, 6)$ because it has more of good 1 and the same amount of good 2.

Key implication for indifference curves: Monotonicity implies that indifference curves are downward sloping.

Given:
- Bundle A = $(10, 8)$
- Bundle B = $(8, 6)$
- The consumer has monotonic preferences.

Analysis:

Comparing the two bundles:
- Good 1: Bundle A has 10 units > 8 units in Bundle B.
- Good 2: Bundle A has 8 units > 6 units in Bundle B.

Bundle A has strictly more of both goods compared to Bundle B.

Conclusion: According to the definition of monotonic preferences, the consumer strictly prefers the bundle with more of both goods. Therefore, the consumer will strictly prefer bundle $(10, 8)$ over bundle $(8, 6)$.

No, a consumer with monotonic preferences cannot be indifferent between $(10, 8)$ and $(8, 6)$. She will always prefer $(10, 8)$ over $(8, 6)$.

Given: Consumer has monotonic preferences.

Bundles:
- Bundle A = $(10, 10)$
- Bundle B = $(10, 9)$
- Bundle C = $(9, 9)$

Comparison:

A vs B: $(10, 10)$ vs $(10, 9)$
- Good 1: equal (10 = 10)
- Good 2: 10 > 9
- Bundle A has more of good 2 and no less of good 1 → A is preferred to B.

B vs C: $(10, 9)$ vs $(9, 9)$
- Good 1: 10 > 9
- Good 2: equal (9 = 9)
- Bundle B has more of good 1 and no less of good 2 → B is preferred to C.

Conclusion (by transitivity):
$$A \succ B \succ C$$
$$(10,10) \succ (10,9) \succ (9,9)$$

The consumer's preference ranking is: $(10,10)$ is most preferred, followed by $(10,9)$, and $(9,9)$ is least preferred.

Given:
- Bundle A = $(5, 6)$
- Bundle B = $(6, 6)$
- The friend is indifferent between A and B.

Check for Monotonicity:

Comparing the two bundles:
- Good 1: Bundle B has 6 units > 5 units in Bundle A.
- Good 2: Both bundles have 6 units (equal).

Bundle B has strictly more of good 1 and the same amount of good 2 compared to Bundle A.

According to monotonic preferences, if one bundle has more of at least one good and no less of the other, the consumer should strictly prefer that bundle. Therefore, the friend should strictly prefer $(6, 6)$ over $(5, 6)$.

However, the friend is indifferent between them, which violates the condition of monotonic preferences.

Conclusion: No, the preferences of the friend are NOT monotonic.

Given:
$$d_1(p) = \begin{cases} 20 - p & \text{if } p \leq 20 \\ 0 & \text{if } p > 20 \end{cases}$$
$$d_2(p) = \begin{cases} 30 - 2p & \text{if } p \leq 15 \\ 0 & \text{if } p > 15 \end{cases}$$

Concept: Market demand = Sum of individual demands at each price.

$$D(p) = d_1(p) + d_2(p)$$

Case 1: When p > 20
$$D(p) = 0 + 0 = 0$$

Case 2: When 15 < p \leq 20
$$D(p) = (20 - p) + 0 = 20 - p$$

Case 3: When $p \leq 15$
$$D(p) = (20 - p) + (30 - 2p) = 50 - 3p$$

Market Demand Function:
$$D(p) = \begin{cases} 50 - 3p & \text{if } p \leq 15 \\ 20 - p & \text{if } 15 < p \leq 20 \\ 0 & \text{if } p > 20 \end{cases}$$

Given:
- Number of consumers = 20
- Each consumer has identical demand:
$$d(p) = \begin{cases} 10 - 3p & \text{if } p \leq \dfrac{10}{3} \\ 0 & \text{if } p > \dfrac{10}{3} \end{cases}$$

Concept: Market demand = Number of consumers $\times$ Individual demand (when consumers are identical).

$$D(p) = 20 \times d(p)$$

Case 1: When $p \leq \dfrac{10}{3}$
$$D(p) = 20 \times (10 - 3p) = 200 - 60p$$

Case 2: When p > \dfrac{10}{3}
$$D(p) = 20 \times 0 = 0$$

Market Demand Function:
$$D(p) = \begin{cases} 200 - 60p & \text{if } p \leq \dfrac{10}{3} \\ 0 & \text{if } p > \dfrac{10}{3} \end{cases}$$

Concept: Market demand at each price = $d_1 + d_2$

Calculation:

| Price ($p$) | $d_1$ | $d_2$ | Market Demand $D = d_1 + d_2$ |
|---|---|---|---|
| 1 | 9 | 24 | $9 + 24 = \mathbf{33}$ |
| 2 | 8 | 20 | $8 + 20 = \mathbf{28}$ |
| 3 | 7 | 18 | $7 + 18 = \mathbf{25}$ |
| 4 | 6 | 16 | $6 + 16 = \mathbf{22}$ |
| 5 | 5 | 14 | $5 + 14 = \mathbf{19}$ |
| 6 | 4 | 12 | $4 + 12 = \mathbf{16}$ |

The market demand at each price level is shown in the last column. As price increases, market demand decreases, which is consistent with the Law of Demand.

Definition: A normal good is a good whose demand increases when the consumer's income increases, and decreases when the consumer's income decreases, keeping the price of the good and other factors constant.

Mathematical Expression: For a normal good, the income effect is positive:
\frac{\Delta q}{\Delta M} > 0
where $q$ is the quantity demanded and $M$ is the income.

Explanation: When a consumer's income rises, she can afford more of the good, and she actually chooses to buy more of it. Most goods that we consume in daily life are normal goods.

Examples: Clothing, branded food items, electronics, furniture — demand for all these goods rises as income increases.

Key Point: For a normal good, the income effect and the substitution effect both work in the same direction (both lead to increased demand when price falls), making the demand curve downward sloping.

Definition: An inferior good is a good whose demand decreases when the consumer's income increases, and increases when the consumer's income decreases, keeping the price of the good and other factors constant.

Mathematical Expression: For an inferior good, the income effect is negative:
\frac{\Delta q}{\Delta M} < 0
where $q$ is the quantity demanded and $M$ is the income.

Explanation: As a consumer's income rises, she shifts away from inferior goods and substitutes them with better-quality (normal) goods. Conversely, when income falls, she consumes more of inferior goods as they are cheaper alternatives.

Examples:
1. Coarse grains (like jowar, bajra) — as income rises, people shift to wheat/rice.
2. Low-quality cooking oil — replaced by better oils as income increases.
3. Public bus transport — people shift to private vehicles as income rises.
4. Second-hand goods — demand falls as income increases.

Note: Whether a good is inferior or normal depends on the consumer's income level and preferences — a good can be normal at low income levels and inferior at higher income levels.

Definition: Two goods are called substitutes if an increase in the price of one good leads to an increase in the demand for the other good, and vice versa, keeping other factors constant.

Explanation: Substitute goods are those that can be used in place of each other to satisfy the same need or want. When the price of one good rises, consumers switch to the other (cheaper) good, increasing its demand.

Mathematical Condition: For substitute goods:
\frac{\Delta d_x}{\Delta p_y} > 0
where $d_x$ is the demand for good $X$ and $p_y$ is the price of good $Y$.

Examples:
1. Tea and Coffee — If the price of tea rises, consumers may switch to coffee, increasing the demand for coffee.
2. Pepsi and Coca-Cola — If the price of Pepsi rises, consumers may shift to Coca-Cola.
3. Butter and Margarine — These can be used in place of each other.

Key Point: Substitutes compete with each other in the market. They have a positive cross-price elasticity of demand.

Definition: Two goods are called complements (or complementary goods) if an increase in the price of one good leads to a decrease in the demand for the other good, and vice versa, keeping other factors constant.

Explanation: Complementary goods are those that are used together to satisfy a want. They are consumed jointly, so a rise in the price of one reduces its demand, which in turn reduces the demand for the other as well.

Mathematical Condition: For complementary goods:
\frac{\Delta d_x}{\Delta p_y} < 0
where $d_x$ is the demand for good $X$ and $p_y$ is the price of good $Y$.

Examples:
1. Car and Petrol — If the price of petrol rises sharply, demand for cars may fall.
2. Bread and Butter — They are consumed together; if the price of bread rises, demand for butter may fall.
3. Pen and Ink — Used together; a rise in the price of ink reduces demand for pens.

Key Point: Complements have a negative cross-price elasticity of demand.

Definition: The price elasticity of demand for a good is defined as the percentage change in the quantity demanded of the good divided by the percentage change in its price.

Formula:
$$e_D = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}} = \frac{\Delta q / q}{\Delta p / p} = \frac{\Delta q}{\Delta p} \times \frac{p}{q}$$

where $\Delta q$ = change in quantity demanded, $\Delta p$ = change in price, $p$ = original price, $q$ = original quantity.

Key Properties:
1. Pure number: Elasticity is dimensionless (has no units) because it is a ratio of two percentages.
2. Generally negative: Since demand and price move in opposite directions (Law of Demand), $e_D$ is typically negative.

Types based on value:

| Value of $|e_D|$ | Type | Meaning |
|---|---|---|
| |e_D| > 1 | Elastic | Demand changes proportionately more than price |
| |e_D| < 1 | Inelastic | Demand changes proportionately less than price |
| $|e_D| = 1$ | Unitary elastic | Demand changes proportionately equal to price |
| $|e_D| = 0$ | Perfectly inelastic | Demand does not change with price |
| $|e_D| = \infty$ | Perfectly elastic | Demand is infinitely responsive to price |

Relationship with Expenditure:
- If e_D < -1 (elastic): Price and expenditure move in opposite directions.
- If e_D > -1 (inelastic): Price and expenditure move in the same direction.
- If $e_D = -1$ (unitary): Expenditure remains unchanged.

Given:
- Original price: $p = \text{Rs } 4$
- New price: $p + \Delta p = \text{Rs } 5$
- Original demand: $q = 25$ units
- New demand: $q + \Delta q = 20$ units

Step 1: Calculate changes.
$$\Delta p = 5 - 4 = \text{Rs } 1$$
$$\Delta q = 20 - 25 = -5 \text{ units}$$

Step 2: Apply the formula.
$$e_D = \frac{\Delta q}{\Delta p} \times \frac{p}{q}$$

$$e_D = \frac{-5}{1} \times \frac{4}{25}$$

$$e_D = -5 \times \frac{4}{25} = \frac{-20}{25} = -0.8$$

The price elasticity of demand is $\mathbf{-0.8}$.

Interpretation: Since |e_D| = 0.8 < 1, the demand is price inelastic — a 1% increase in price leads to only a 0.8% decrease in quantity demanded.

Given:
- Demand function: $D(p) = 10 - 3p$
- Price: $p = \dfrac{5}{3}$

Step 1: Find quantity demanded at $p = \dfrac{5}{3}$.
$$q = D\left(\frac{5}{3}\right) = 10 - 3 \times \frac{5}{3} = 10 - 5 = 5 \text{ units}$$

Step 2: Find $\dfrac{\Delta q}{\Delta p}$ (the slope of the demand function).
$$\frac{dD}{dp} = -3$$

So $\dfrac{\Delta q}{\Delta p} = -3$.

Step 3: Apply the elasticity formula.
$$e_D = \frac{\Delta q}{\Delta p} \times \frac{p}{q} = (-3) \times \frac{\dfrac{5}{3}}{5}$$

$$e_D = (-3) \times \frac{5}{3 \times 5} = (-3) \times \frac{1}{3} = -1$$

The price elasticity of demand at price $\dfrac{5}{3}$ is $\mathbf{-1}$ (unitary elastic).

Interpretation: At this price, a 1% change in price leads to exactly a 1% change in quantity demanded in the opposite direction. Total expenditure remains unchanged at this point.

Given:
- Price elasticity of demand: $e_D = -0.2$
- Percentage increase in price: $\dfrac{\Delta p}{p} \times 100 = 5\%$

Formula:
$$e_D = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}$$

Calculation:
$$-0.2 = \frac{\text{Percentage change in quantity demanded}}{5}$$

$$\text{Percentage change in quantity demanded} = -0.2 \times 5 = -1\%$$

The demand for the good will go down by $\mathbf{1\%}$.

Interpretation: Since |e_D| = 0.2 < 1, the demand is highly inelastic. A 5% rise in price causes only a 1% fall in demand.

Given:
- Price elasticity of demand: $e_D = -0.2$
- Percentage increase in price = $10\%$

Step 1: Find the percentage change in quantity demanded.
$$e_D = \frac{\% \Delta q}{\% \Delta p}$$
$$-0.2 = \frac{\% \Delta q}{10}$$
$$\% \Delta q = -0.2 \times 10 = -2\%$$

So demand falls by $2\%$.

Step 2: Analyze the effect on expenditure.

Expenditure $E = p \times q$

Using the relationship: $\% \Delta E \approx \% \Delta p + \% \Delta q = 10\% + (-2\%) = +8\%$

Alternatively, using the formula:
Since e_D = -0.2 > -1, the demand is inelastic. When demand is inelastic, price and total expenditure move in the same direction.

A $10\%$ increase in price with only a $2\%$ decrease in quantity means the price effect dominates.

Conclusion: The expenditure on the good will increase by approximately $8\%$.

This is consistent with the rule: when e_D > -1 (inelastic demand), an increase in price leads to an increase in total expenditure.

Given:
- Percentage change in price: $\% \Delta p = -4\%$ (decrease)
- Percentage change in expenditure: $\% \Delta E = +2\%$ (increase)

Step 1: Find the percentage change in quantity demanded.

Since $E = p \times q$:
$$\% \Delta E \approx \% \Delta p + \% \Delta q$$
$$+2 = -4 + \% \Delta q$$
$$\% \Delta q = 2 + 4 = +6\%$$

So quantity demanded increased by $6\%$.

Step 2: Calculate the price elasticity of demand.
$$e_D = \frac{\% \Delta q}{\% \Delta p} = \frac{+6\%}{-4\%} = -1.5$$

Step 3: Interpret the result.

Since $e_D = -1.5$ and |e_D| = 1.5 > 1, the demand is price elastic.

Conclusion: The price elasticity of demand is $\mathbf{-1.5}$. The demand is elastic (|e_D| > 1). This is consistent with the observation that when price decreased, expenditure increased — which happens when demand is elastic (price and expenditure move in opposite directions when e_D < -1).