Linear Demand, Market Failures & Policy Responses – Core Concepts & Applications

1. Linear Demand Function

A linear demand function has the form

$$P=a-bQ$$

where $P$ is price, $Q$ quantity demanded, $a$ is the intercept (price when $Q=0$) and $b>0$ is the slope coefficient.

1.1 Graphical Representation

The curve is a straight line descending from the price intercept $a$ to the quantity intercept $a/b$.

1.2 Elasticity Along a Linear Demand

Elasticity $\epsilon =\frac{dQ}{dP}\frac{P}{Q}=-\frac{1}{b}\frac{P}{Q}$.

• At the upper left (high $P$, low $Q$) elasticity > 1 (elastic).
• At the midpoint ($P=a/2,Q=a/(2b)$) elasticity = –1 (unit‑elastic).
• At the lower right elasticity < 1 (inelastic).

Worked Example – Wheat Market

The demand schedule is linear: $P=500-5Q$.
At price 400 Rs, $Q=20$ (midpoint). Elasticity

$$\epsilon =-\frac{1}{5}\frac{400}{20}=-4$$

Thus demand is elastic at the equilibrium price (400 Rs).

2. Market Failures

When the invisible hand does not lead to a Pareto‑efficient allocation, a market failure occurs. The four canonical types are:

2.1 Externalities

A negative externality imposes an uncompensated cost on third parties. Graphically, the social marginal cost (SMC) lies above private marginal cost (PMC).

Dead‑weight loss (DWL) is the triangular area between SMC and PMC from $Q_{opt}$ to $Q^{\ast }$.

Real‑World Example – eSewa Transaction Fees

eSewa charges a 1 % fee on digital payments. The fee raises the private cost of using the platform, creating a negative externality for the broader digital‑economy (reduced adoption). The Nepalese government’s “Digital Payment Incentive” subsidizes 0.5 % of the fee, moving the effective cost closer to the social optimum and expanding e‑commerce (e.g., Daraz orders).

2.2 Public Goods

Public goods are non‑rival (one person’s consumption does not diminish another’s) and non‑excludable (cannot prevent non‑payers from using). The market fails because firms cannot charge each user.

Example: Kathmandu’s street lighting. Private firms cannot exclude households that do not pay, so the government funds the service through taxes.

2.3 Information Asymmetry

When sellers know more about product quality than buyers, market outcomes can be inefficient.

Example – Tea Export Certification

Nepalese tea growers must obtain a “organic” certification to access premium export markets. Without reliable certification, foreign buyers may distrust the product, reducing demand. The government’s Tea Board provides third‑party testing, reducing asymmetry and raising export revenues.

2.4 Monopoly Power

A single firm with price‑setting ability produces where MR = MC, not where P = MC, creating a dead‑weight loss.

Example – Ncell’s 4G Spectrum

Ncell holds a dominant share of 4G spectrum, allowing it to set data prices above marginal cost. The Telecom Authority of Nepal imposes a price cap to limit the welfare loss.

3. Policy Instruments & Their Graphical Effects

Worked Example – Tax on Negative Externality

Suppose a factory emits pollution costing society Rs 2 per unit. Private marginal cost: $PMC=10+Q$.

• Without tax: equilibrium where $D=PMC$.
• With tax (t = 2): effective marginal cost becomes $PMC+t=12+Q$, aligning with SMC.

The tax eliminates the triangular DWL shown between the two quantities.

4. Comparative Summary of Market Failures

5. Real‑World Applications

Ilam tea farms – certification reduces information asymmetry for export (Image: Pravinchapagain, CC BY-SA 4.0, via Wikimedia Commons)

5.1 eSewa & Transaction Tax

eSewa charges a 1 % fee (private cost). The government’s 0.5 % subsidy lowers the effective cost, moving the market outcome closer to the socially optimal level of digital payments, increasing transaction volume by ~15 % (2023 data).

5.2 Daraz Order Queue

Daraz’s algorithm matches buyers and sellers at a market‑clearing price each hour. The linear demand for “smartphones” can be approximated by $P=1200-0.2Q$. When a flash sale reduces price to Rs 800, the quantity demanded jumps from 2,000 to 2,000 + (1200‑800)/0.2 = 4,000 units, illustrating elasticity at different price points.

5.3 Tea Export Certification

The Tea Board’s certification adds a fixed cost of Rs 500 per ton but raises export price by Rs 1,200 per ton. Net profit per ton rises from Rs 300 to Rs 1,000, demonstrating how reducing information asymmetry expands market size.

5.4 Kathmandu Traffic Congestion

Each additional car imposes an external cost of about Rs 150 in time loss to other drivers. A congestion charge of Rs 200 per vehicle (Pigouvian tax) is projected to cut peak‑hour traffic by 12 %, moving the system toward the socially optimal flow.

6. Worked Numerical Problem – Profit Maximisation with Linear Demand

Given:

$$P=50-0.5Q,C=350+10Q$$

a) TR‑TC Approach

• Total Revenue (TR): $TR=P\times Q=(50-0.5Q)Q=50Q-0.5Q^2$
• Total Cost (TC): $TC=350+10Q$

Set $TR=TC$ for break‑even, but profit maximisation requires $MR=MC$.

b) MR‑MC Approach

• Marginal Revenue (MR): derivative of TR → $MR=50-Q$
• Marginal Cost (MC): derivative of C → $MC=10$

Set $MR=MC$:

$$50-Q=10\Rightarrow Q^{\ast }=40$$

• Price: $P^{\ast }=50-0.5(40)=30$
• TR: $30\times 40=1,200$
• TC: $350+10(40)=750$
• Profit: $\pi =TR-TC=1,200-750=450$

Thus the profit‑maximising output is 40 units, price Rs 30, and maximum profit Rs 450.

7. Determinants of Demand (Quick Recap)

1. Income – normal vs. inferior goods.
2. Prices of Related Goods – substitutes and complements.
3. Tastes & Preferences – shifts due to advertising, trends.
4. Expectations – future price or income expectations.
5. Number of Buyers – population growth, demographic changes.

8. In the Real World

• Google Ads Auction uses a linear demand approximation to set the price per click; higher bids shift the demand curve rightward, increasing ad impressions.
• NTC’s Rural Broadband Expansion faces a positive externality: each new connection raises overall digital literacy, justifying a government subsidy that aligns private marginal cost with social marginal benefit.
• NEPSE (Nepal Stock Exchange) exhibits information asymmetry; mandatory quarterly disclosures aim to reduce the “lemons” problem for investors, improving market efficiency.

9. Exam Tip

• Graph‑first strategy: For any question on externalities, public goods or monopoly, sketch the relevant curves (D, S, MC, MR, SMC, etc.) before writing algebra. Mark equilibrium, socially optimal points, and dead‑weight loss clearly.
• Elasticity shortcut: On a linear demand, remember that elasticity = –1 at the midpoint; use the ratio $\frac{P}{Q}$ multiplied by $-\frac{1}{b}$ for other points.
• Formula checklist:
  • Demand: $P=a-bQ$ → $Q=\frac{a-P}{b}$
  • TR = $P\times Q$ → $TR=aQ-b{Q}^2$
  • MR = $a-2bQ$ (or simply differentiate TR)
  • MC = $dC/dQ$
  • DWL (tax) = $\frac{1}{2}(Q_{notax}-Q_{tax})(Tax)$
• Mark allocation: Allocate 2 marks for a correct diagram, 1 mark for each correctly labeled axis, 2 marks for identifying the welfare loss, and the remaining marks for explanation. Write concise bullet points for determinants or policy tools to secure easy marks.