# Economic Applications

Business I · Mathematics II · https://tryals.app/learn/business-i/economic-applications

## Elasticity & Production

**Price elasticity of demand** measures responsiveness: $\varepsilon = \frac{\%\Delta Q}{\%\Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$. It is unitless. If $|\varepsilon| > 1$, demand is elastic and revenue falls if price rises. If $|\varepsilon| < 1$, it is inelastic and revenue rises. Revenue is maximised at $|\varepsilon| = 1$.

**Cobb-Douglas production** models output as $Q = AK^\alpha L^{1-\alpha}$. Here, $\alpha$ is capital's share of output. Since exponents sum to 1, it exhibits **constant returns to scale**: doubling inputs doubles output. Marginal products are $MP_K = \alpha Q/K$ and $MP_L = (1-\alpha) Q/L$, always proportional to average products.

| Tool | Formula | Key Insight |
| --- | --- | --- |
| Elasticity | $\varepsilon = \frac{dQ}{dP}\cdot\frac{P}{Q}$ | Measures responsiveness |
| Cobb-Douglas | $Q = AK^{\alpha}L^{1-\alpha}$ | Returns to scale sum to 1 |

## Input-Output & Envelope Theorem

**Input-output analysis** (Leontief) relates total output $\mathbf{x}$, inter-industry matrix $A$, and final demand $\mathbf{d}$ via $\mathbf{x} = A\mathbf{x} + \mathbf{d}$. Solving gives the **Leontief inverse**: $\mathbf{x} = (I - A)^{-1}\mathbf{d}$, which accounts for all supply chain ripples.

The **envelope theorem** states that in an optimised system, a parameter's effect on the optimised value $V^*$ equals the direct effect only: $\frac{dV^*}{d\theta} = \frac{\partial \mathcal{L}}{\partial \theta}\Big|_\text{opt}$. Indirect effects through variables vanish at the optimum.

| Tool | Formula | Answers |
| --- | --- | --- |
| Leontief inverse | $(I-A)^{-1}\mathbf{d}$ | Total output including ripples |
| Shadow price | $\lambda = \frac{\partial \mathcal{L}}{\partial \theta}$ | What the constraint is worth |

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. In $Q = AK^{0.4}L^{0.6}$, the exponent 0.4 is the marginal product of capital.

**Answer:** False

**Why:** False. The exponent is the output **elasticity** of capital, and under competitive factor pricing it equals capital's *share* of output. The marginal product is $\partial Q/\partial K = 0.4Q/K$, which changes with the input levels, the exponent does not. Exponents summing to 1 is what makes the function constant-returns.

Page: https://tryals.app/practice/business-i/economic-applications/in-q-ak-4l-6-the-exponent-0-4-is-the-marginal-product-of-capital

### 2. If $\varepsilon = -1.5$ and price rises by 10%, demand falls by approximately:

**Answer:** 15 (within ±2)

**Why:** $\%\Delta Q = \varepsilon \times \%\Delta P = -1.5 \times 10\% = -15\%$. Demand falls 15%, the elasticity is a direct multiplier from price changes to quantity changes.

Page: https://tryals.app/practice/business-i/economic-applications/if-1-5-and-price-rises-by-10-demand-falls-by-approximately

### 3. The envelope theorem says $dV^*/d\theta = \partial\mathcal{L}/\partial\theta$: only the direct effect counts. Why can the indirect effect be ignored?

A. Because the constraint prevents the choice variables from moving
B. Because indirect effects are typically small in economic applications
C. Because the choice variables do not in fact change when $\theta$ changes
D. Because the first-order conditions already hold at an optimum

**Answer:** D. Because the first-order conditions already hold at an optimum

**Why:** You are already at the top, so moving slightly sideways costs nothing to first order. This is why a shadow price can be read straight off the Lagrangian without re-solving the problem.

Page: https://tryals.app/practice/business-i/economic-applications/the-envelope-theorem-says-dv-d-mathcall-only-the-direct

### 4. Match each concept to its mathematical tool.

**Answer:**

- Elasticity → Derivative times price-quantity ratio
- Returns to scale → Sum of exponents in Cobb-Douglas
- Leontief inverse → $(I-A)^{-1}$: total output multiplier
- Envelope theorem → Direct effect only at the optimum

**Why:** Each economic concept maps to a precise mathematical operation, economics is applied mathematics.

Page: https://tryals.app/practice/business-i/economic-applications/match-each-concept-to-its-mathematical-tool

### 5. Elasticity is defined as a ratio of percentage changes rather than a plain derivative. What does that buy?

A. The ability to handle non-linear demand curves, which derivatives cannot
B. Greater accuracy than the derivative, which depends on the measurement scale
C. A guarantee that the value is constant along the demand curve
D. Unit-free comparability across goods and countries

**Answer:** D. Unit-free comparability across goods and countries

**Why:** Switching units rescales the derivative and leaves the elasticity untouched, so it travels between markets and datasets. Option 4 is a common error, elasticity varies along a linear demand curve, running from infinite at the intercept to zero at the axis, which is why the tax-incidence result depends on where you are.

Page: https://tryals.app/practice/business-i/economic-applications/elasticity-is-defined-as-a-ratio-of-percentage-changes-rather-than-a

### 6. Sort by elasticity type.

**Answer:**

- Elastic ($|\varepsilon| > 1$): Luxury goods (many substitutes), A specific brand of cereal, Restaurant meals
- Inelastic ($|\varepsilon| < 1$): Petrol in the short run, Insulin for diabetics, Electricity

**Why:** Goods with substitutes (luxuries, brands, restaurants) are elastic; necessities without substitutes (fuel, medicine, utilities) are inelastic.

Page: https://tryals.app/practice/business-i/economic-applications/sort-by-elasticity-type