1a. What is an indifference curve?
b. How can you relate it to an indifference map?
2. Distinguish between the law of diminishing returns and the law of return to scale.
3. In a closed economy, net investment expenditure is at the rate of ₦1000m. APC = MPC = 0.8 of disposable income. There is no government activity.
- a. Calculate the equilibrium level of national income
- b. What would be the effect of increasing the rate of investment by 50%?
- c. An income tax of ₦500M per annum was introduced and this is accompanied by government expenditure of ₦500 per annum. What is the new equilibrium level of national income, assuming that APS, MPS and the level of investment remain unchanged?
4a. Market survey shows that the quantity of commodity X and the price of commodity Y for the month of April 2022 were 90 unit and ₦120 respectively. This is different from May survey of 301 units and ₦180 respectively. Determine the cross elasticity of demand of commodity X and their relationship.
b. Define the concept of price elasticity of demand.
5a. What is scale of preference?
b. Discuss how consumers and producers maximize utility.
6. Compute the mean, variance, standard deviation and the range of the following observations.
9, 12, 8, 10, 16.
---
FULL ANSWERS
## Question 1a: Indifference Curve & Indifference Map
### What is an Indifference Curve?
An **indifference curve** is a graphical representation showing all possible combinations of two goods that give a consumer the **same level of satisfaction (utility)**, such that the consumer is indifferent between any combination on the curve.
**Key Properties of an Indifference Curve:**
1. **Slopes Downward from Left to Right (Negative Slope)** — To maintain the same level of satisfaction, if a consumer gets more of one good, they must give up some of the other.
2. **Convex to the Origin** — Reflects the **diminishing marginal rate of substitution (MRS)** — as a consumer gets more of one good, they are willing to give up less and less of the other.
3. **Cannot Intersect** — Two indifference curves can never cross because each curve represents a different level of satisfaction. Intersection would imply two different utility levels are equal, which is a contradiction.
4. **Higher Curves Represent Higher Satisfaction** — A curve farther from the origin represents a higher level of utility.
5. **Never Touches the Axes** — Assumes the consumer desires positive quantities of both goods.
---
**Diagram:**
```
Quantity
of Good Y
|
| IC₃ (highest utility)
| IC₂
| IC₁ (lowest utility)
|________________________
Quantity of Good X
```
---
### 1b: Relating an Indifference Curve to an Indifference Map
An **indifference map** is a **collection or family of indifference curves** drawn on the same graph, each representing a different level of utility or satisfaction for a consumer.
**Relationship:**
- Each **single indifference curve** shows one specific utility level.
- An **indifference map** is made up of **multiple indifference curves** — each at a different level of satisfaction.
- Moving to a **higher indifference curve** (farther from the origin) indicates **greater total utility**.
- Moving to a **lower indifference curve** (closer to the origin) indicates **lower total utility**.
- The indifference map thus represents the **complete preference ordering** of the consumer for all possible combinations of the two goods.
- Together, all curves on the map help analysts understand consumer behavior and predict choices when prices or income change.
**In summary:** An indifference curve is a **single line** of equal utility; an indifference map is the **entire set of such lines**, providing a complete picture of consumer preferences.
---
## Question 2: Law of Diminishing Returns vs. Law of Returns to Scale
### Law of Diminishing Returns (Short Run)
**Definition:** The **Law of Diminishing Returns** (also called the Law of Variable Proportions) states that as successive units of a **variable factor** (e.g., labor) are added to a **fixed factor** (e.g., land or capital), beyond a certain point, the **marginal product of the variable factor will begin to decline**, holding all other factors constant.
**Key Features:**
- Applies in the **short run** when at least one factor of production is fixed
- Concerns **one variable input** being increased while others remain constant
- Focuses on **marginal product** of the variable factor
- Results in three stages: increasing returns → diminishing returns → negative returns
**Example:**
Adding more workers to a fixed piece of farmland:
- 1st worker: produces 10 bags
- 2nd worker: produces 18 bags (marginal product = 8)
- 3rd worker: produces 24 bags (marginal product = 6)
- 4th worker: produces 28 bags (marginal product = 4)
- 5th worker: produces 28 bags (marginal product = 0)
Marginal product eventually falls — this is the law of diminishing returns.
---
### Law of Returns to Scale (Long Run)
**Definition:** The **Law of Returns to Scale** examines what happens to output when **all factors of production are increased proportionately** in the long run — when there are no fixed inputs.
**Three Possibilities:**
1. **Increasing Returns to Scale** — Output increases by a **greater proportion** than the increase in inputs.
- Example: Inputs doubled → Output more than doubles
- Caused by: specialization, economies of scale, improved technology
2. **Constant Returns to Scale** — Output increases by the **same proportion** as inputs.
- Example: Inputs doubled → Output exactly doubles
3. **Decreasing Returns to Scale** — Output increases by a **smaller proportion** than inputs.
- Example: Inputs doubled → Output less than doubles
- Caused by: managerial inefficiency, coordination problems, diseconomies of scale
---
### Key Differences
| Basis | Law of Diminishing Returns | Law of Returns to Scale |
|---|---|---|
| Time Period | Short run | Long run |
| Factors Changed | Only one variable factor | All factors changed proportionately |
| Fixed Inputs | At least one input is fixed | No fixed inputs |
| Focus | Marginal product of variable input | Overall scale of production |
| Result | Marginal product eventually declines | Output may increase more, less, or equally |
| Application | Farm labor on fixed land | Factory expansion |
---
## Question 3: National Income — Calculations
### Given Information:
- Net Investment (I) = ₦1,000m
- APC = MPC = 0.8
- Closed economy with no government activity
- Therefore: MPS = 1 − MPC = 1 − 0.8 = **0.2**
---
### 3a: Calculate the Equilibrium Level of National Income
**Method 1 — Using the Multiplier:**
$$\text{Multiplier (K)} = \frac{1}{1 - MPC} = \frac{1}{1 - 0.8} = \frac{1}{0.2} = 5$$
$$\text{Equilibrium National Income (Y)} = K \times I$$
$$Y = 5 \times 1000 = \textbf{₦5,000m}$$
**Method 2 — Using the Equilibrium Condition (Y = C + I):**
Since APC = MPC = 0.8 (no autonomous consumption implied):
$$C = 0.8Y$$
$$Y = C + I$$
$$Y = 0.8Y + 1000$$
$$Y - 0.8Y = 1000$$
$$0.2Y = 1000$$
$$Y = \frac{1000}{0.2} = \textbf{₦5,000m}$$
**∴ Equilibrium National Income = ₦5,000m**
---
### 3b: Effect of Increasing Investment by 50%
New Investment = 1,000 + (50% × 1,000) = 1,000 + 500 = **₦1,500m**
$$\text{New Equilibrium Income} = K \times \text{New Investment}$$
$$Y = 5 \times 1,500 = \textbf{₦7,500m}$$
**Change in National Income:**
$$\Delta Y = 7,500 - 5,000 = \textbf{₦2,500m increase}$$
**Alternatively using the multiplier effect:**
$$\Delta Y = K \times \Delta I = 5 \times 500 = \textbf{₦2,500m}$$
**∴ Increasing investment by 50% raises equilibrium national income from ₦5,000m to ₦7,500m — an increase of ₦2,500m.**
---
### 3c: New Equilibrium with Tax and Government Expenditure
**New Information:**
- Income Tax (T) = ₦500m per annum
- Government Expenditure (G) = ₦500m per annum
- MPS and MPC remain the same (MPC = 0.8, MPS = 0.2)
- Investment remains at ₦1,000m
**With tax introduced, disposable income changes:**
$$Y_d = Y - T$$
Consumption function becomes:
$$C = MPC \times (Y - T) = 0.8(Y - 500)$$
**Equilibrium condition:**
$$Y = C + I + G$$
$$Y = 0.8(Y - 500) + 1000 + 500$$
$$Y = 0.8Y - 400 + 1500$$
$$Y - 0.8Y = 1100$$
$$0.2Y = 1100$$
$$Y = \frac{1100}{0.2} = \textbf{₦5,500m}$$
**∴ The new equilibrium level of national income = ₦5,500m**
**Note:** The balanced budget multiplier applies here:
- Tax reduces income but Government spending injects equal amount back
- Net effect is a **₦500m increase** in national income (from ₦5,000m to ₦5,500m)
- This confirms the **balanced budget multiplier = 1**
---
## Question 4a: Cross Elasticity of Demand
### Given Information:
- **April 2022:** Quantity of X = 90 units; Price of Y = ₦120
- **May 2022:** Quantity of X = 301 units; Price of Y = ₦180
### Formula for Cross Elasticity of Demand:
$$E_{xy} = \frac{\% \text{ Change in Quantity Demanded of X}}{\% \text{ Change in Price of Y}}$$
$$E_{xy} = \frac{\Delta Q_x / Q_{x1}}{\Delta P_y / P_{y1}}$$
**Step 1: Calculate changes:**
$$\Delta Q_x = 301 - 90 = 211 \text{ units}$$
$$\Delta P_y = 180 - 120 = ₦60$$
**Step 2: Calculate percentage changes:**
$$\% \Delta Q_x = \frac{211}{90} \times 100 = 234.4\%$$
$$\% \Delta P_y = \frac{60}{120} \times 100 = 50\%$$
**Step 3: Calculate Cross Elasticity:**
$$E_{xy} = \frac{234.4\%}{50\%} = \textbf{4.69}$$
### Interpretation and Relationship:
Since **E_xy = +4.69 (positive and greater than zero)**:
- Commodities X and Y are **SUBSTITUTES**
- When the price of Y **rose** from ₦120 to ₦180, consumers shifted to buying more of X (quantity rose from 90 to 301 units)
- The **positive cross elasticity** confirms a substitute relationship
- The high value (4.69) indicates they are **close substitutes** — consumers readily switch from Y to X when Y's price rises
**General Rule:**
| Cross Elasticity Value | Relationship |
|---|---|
| Positive (+) | Substitutes |
| Negative (−) | Complements |
| Zero (0) | Unrelated goods |
---
### 4b: Price Elasticity of Demand
**Definition:**
**Price Elasticity of Demand (PED)** is a measure of the **degree of responsiveness** of the quantity demanded of a good to a **change in its own price**, ceteris paribus (all other things being equal).
$$PED = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}}$$
**Types:**
1. **Perfectly Elastic (PED = ∞)** — Any price rise causes demand to fall to zero
2. **Elastic (PED > 1)** — % change in quantity > % change in price (luxury goods)
3. **Unit Elastic (PED = 1)** — % change in quantity = % change in price
4. **Inelastic (PED < 1)** — % change in quantity < % change in price (necessities)
5. **Perfectly Inelastic (PED = 0)** — Quantity demanded does not change regardless of price
**Factors Affecting PED:**
- Availability of substitutes
- Necessity vs. luxury nature of the good
- Proportion of income spent on the good
- Time period considered
- Breadth of market definition
---
## Question 5a: Scale of Preference
### Definition
A **scale of preference** is an **orderly arrangement of an individual's wants or needs** in order of their **urgency, priority, or importance** — from the most pressing to the least pressing. It reflects the fact that human wants are unlimited but resources to satisfy them are limited, making it necessary to rank wants and satisfy the most important ones first.
**Key Points:**
- It is based on the **economic problem of scarcity** — not all wants can be satisfied simultaneously
- It guides rational decision-making by helping individuals prioritize
- It is **subjective** — each individual has their own scale based on values, income, and circumstances
- It forms the basis for the concept of **opportunity cost** — choosing to satisfy one want means forgoing another
**Example:**
| Priority | Want |
|---|---|
| 1st | Food |
| 2nd | School fees |
| 3rd | Clothing |
| 4th | Rent |
| 5th | Entertainment |
---
### 5b: How Consumers and Producers Maximize Utility
#### How Consumers Maximize Utility
**Utility** is the satisfaction a consumer derives from consuming a good or service.
Consumers maximize utility by following the **Law of Equi-Marginal Utility**, which states:
> A consumer maximizes total utility when the **marginal utility per unit of money spent** is **equal across all goods purchased**.
$$\frac{MU_x}{P_x} = \frac{MU_y}{P_y} = \frac{MU_z}{P_z} = \lambda \text{ (constant)}$$
Where:
- MU = Marginal Utility of the good
- P = Price of the good
- λ = Marginal utility of money
**Process:**
1. The consumer ranks goods by their marginal utility per naira spent
2. They allocate income first to the good with the highest MU/P ratio
3. They continue allocating until MU/P is equal across all goods
4. At this point, no reallocation of spending can increase total utility
5. The consumer is said to be in **consumer equilibrium**
**Using Indifference Curve Analysis:**
Consumer equilibrium occurs where the **budget line is tangent to the highest attainable indifference curve**, i.e.:
$$MRS_{xy} = \frac{P_x}{P_y}$$
This means the consumer's subjective valuation ratio equals the market price ratio.
---
#### How Producers Maximize Utility (Profit)
Producers maximize **profit** (the difference between total revenue and total cost) rather than utility per se.
**Condition for Profit Maximization:**
$$\text{Profit} (\pi) = \text{Total Revenue (TR)} - \text{Total Cost (TC)}$$
Profit is maximized where:
$$MC = MR$$
*(Marginal Cost = Marginal Revenue)*
**AND** the MC curve must be **rising** (cutting the MR curve from below).
**Explanation:**
- If **MR > MC**: Producing one more unit adds more revenue than cost → **increase output**
- If **MR < MC**: Producing one more unit adds more cost than revenue → **reduce output**
- If **MR = MC**: No further gain from changing output → **profit is maximized**
**Using Isoquant-Isocost Analysis:**
Producers also maximize output for a given budget (or minimize cost for a given output) where the **isocost line is tangent to the highest attainable isoquant:**
$$MRTS = \frac{w}{r}$$
Where w = wage rate and r = rental rate of capital.
This means the producer uses the **least-cost combination** of labor and capital.
---
## Question 6: Statistical Computations
### Given Data: 9, 12, 8, 10, 16
**n = 5**
---
### Step 1: Calculate the Mean (x̄)
$$\bar{x} = \frac{\sum x}{n} = \frac{9 + 12 + 8 + 10 + 16}{5} = \frac{55}{5} = \textbf{11}$$
---
### Step 2: Calculate the Variance
First, find the deviation of each value from the mean and square it:
| x | (x − x̄) | (x − x̄)² |
|---|---|---|
| 9 | 9 − 11 = −2 | 4 |
| 12 | 12 − 11 = +1 | 1 |
| 8 | 8 − 11 = −3 | 9 |
| 10 | 10 − 11 = −1 | 1 |
| 16 | 16 − 11 = +5 | 25 |
| **Total** | | **40** |
$$\text{Variance} (\sigma^2) = \frac{\sum(x - \bar{x})^2}{n} = \frac{40}{5} = \textbf{8}$$
---
### Step 3: Calculate Standard Deviation
$$\sigma = \sqrt{\text{Variance}} = \sqrt{8} = 2\sqrt{2} = \textbf{2.83 (approx.)}$$
---
### Step 4: Calculate the Range
$$\text{Range} = \text{Highest Value} - \text{Lowest Value}$$
$$\text{Range} = 16 - 8 = \textbf{8}$$
---
### Summary of Results
| Measure | Value |
|---|---|
| Mean | **11** |
| Variance | **8** |
| Standard Deviation | **2.83** |
| Range | **8** |
---
### Interpretation:
- The **mean of 11** is the average value of the dataset
- The **variance of 8** measures how spread out the values are from the mean
- The **standard deviation of 2.83** tells us most values fall within approximately 2.83 units of the mean (i.e., between 8.17 and 13.83)
- The **range of 8** shows the spread between the smallest (8) and largest (16) values
---