Cost & Revenue Analysis: Practical Applications (Profit, Elasticity, Optimization)

TAKEAWAYS:

• Cost functions (TC, TFC, TVC, AVC, AC, MC) are derived from production data and determine a firm’s break-even point and profit zones.
• Revenue functions (TR, AR, MR) interact with cost curves to find profit-maximizing output (where MR = MC) and price elasticity determines pricing power.
• Long-run cost curves (LAC, LMC) show economies/diseconomies of scale, while short-run curves (SAC, SAC) reveal shutdown rules (P < AVC).
• Monopoly vs. competitive firms use different optimization rules: monopolies set MR = MC but charge higher prices due to downward-sloping demand.
• Real-world applications include Daraz’s dynamic pricing (elasticity), Ncell’s network expansion (economies of scale), and bank loan interest calculations (cost of capital).
• Exam focus: Derive functions from given data, complete cost-revenue tables, and solve optimization problems (e.g., profit-maximizing output/price).

1. Cost Functions: Breaking Down the Numbers

Cost analysis starts with the total cost (TC) function, which combines fixed and variable costs. For a firm producing Q units, TC is often expressed as: $$TC=TFC+TVC$$ where:

• TFC (Total Fixed Cost): Costs that do not change with output (e.g., rent, salaries). Example: A Daraz warehouse lease of Rs 500,000/month.
• TVC (Total Variable Cost): Costs that vary with output (e.g., raw materials, labor). Example: Rs 20 per order for packaging and delivery.

Worked Example: Deriving Cost Curves

Given the TC function: $$TC=200+10Q+3Q^2$$ Let’s derive all cost curves for Q = 10 (a typical order volume for a small Daraz seller):

1. Total Fixed Cost (TFC): Constant term = Rs 200.
2. Total Variable Cost (TVC): Remaining terms = $10Q+3Q^2$. At Q = 10: $$TVC=10(10)+3(10{)}^2=100+300=Rs400$$
3. Average Cost (AC): $AC=\frac{TC}{Q}=\frac{200+10Q+3Q^2}{Q}$ At Q = 10: $$AC=\frac{200+100+300}{10}=Rs60$$
4. Marginal Cost (MC): Derivative of TC = $\frac{d(TC)}{dQ}=10+6Q$. At Q = 10: $$MC=10+6(10)=Rs70$$

graph LR
    TC["Total Cost (TC)"] --> TFC["Total Fixed Cost\n(Rs 200)"]
    TC --> TVC["Total Variable Cost\n(Rs 400 at Q=10)"]
    TC --> AC["Average Cost\n(Rs 60 at Q=10)"]
    TC --> MC["Marginal Cost\n(Rs 70 at Q=10)"]

Key Relationships

• AC = AVC + AFC (where AFC = TFC/Q).
• MC cuts AC at its minimum point (this is a golden rule for optimization).
• Shutdown Rule: A firm shuts down if P < AVC (cannot cover variable costs).

2. Revenue Functions: How Firms Price and Profit

Revenue comes in two forms:

1. Total Revenue (TR): $TR=P\times Q$.
2. Average Revenue (AR): $AR=\frac{TR}{Q}=P$ (price per unit).
3. Marginal Revenue (MR): Additional revenue from selling one more unit.
   • Perfect Competition: $MR=AR=P$ (horizontal demand).
   • Monopoly: $MR$ is less than $AR$ (downward-sloping demand).

Worked Example: Profit Maximization for a Monopoly

Given:

• Demand: $P=400-10Q$
• Cost: $TC=280+8Q$

Step 1: Find TR and MR $$TR=P\times Q=(400-10Q)Q=400Q-10Q^2$$ $$MR=\frac{d(TR)}{dQ}=400-20Q$$

Step 2: Find MC $$MC=\frac{d(TC)}{dQ}=8$$

Step 3: Set MR = MC for profit maximization $$400-20Q=8$$ $$20Q=392$$ $$Q=19.6\approx 20\text{ units}$$

Step 4: Find Price (P) $$P=400-10(20)=Rs200$$

Step 5: Calculate Profit $$TR=200\times 20=Rs4,000$$ $$TC=280+8(20)=Rs440$$ $$\text{Profit}=TR-TC=Rs3,560$$

3. Elasticity and Pricing Power

Price Elasticity of Demand (PED) measures how sensitive quantity demanded is to price changes: $$PED=\frac{\%\text{ change in Q}}{\%\text{ change in P}}$$

• PED > 1: Elastic (quantity changes more than price). Example: Daraz discounts on electronics (high PED).
• PED < 1: Inelastic (quantity changes less than price). Example: Ncell’s basic call packages (low PED).
• PED = 1: Unit elastic (revenue maximized).

Worked Example: Marginal Revenue and Elasticity

Given:

• $PED=1.5$
• $AR=Rs300$

Step 1: Relate MR to AR and PED $$MR=AR\times (1+\frac{1}{PED})$$ $$MR=300\times (1+\frac{1}{1.5})=300\times 1.667=Rs500$$

Real-World Tie-In: Ncell adjusts prices for different call packages based on elasticity. A Rs 100 increase in a Rs 500 package (inelastic) raises revenue more than a Rs 100 increase in a Rs 50 package (elastic).

4. Long-Run vs. Short-Run Costs

Visual: Long-Run Average Cost (LAC) Curve

Real-World Example: Nepal’s NTC built a new fiber-optic network (long-run decision). Initially, costs per kilometer were high (small scale), but as they expanded, the LAC decreased due to bulk purchasing of equipment and shared infrastructure.

5. Practical Applications: Case Studies

Case 1: Daraz’s Dynamic Pricing (Elasticity)

Daraz uses price elasticity to adjust prices for the same product across regions. For example:

• In Kathmandu, demand for smartphones is inelastic (PED < 1) due to high income and necessity. Daraz charges a premium.
• In Pokhara, demand is elastic (PED > 1). Discounts are offered to boost sales.

Worked Example: If Daraz raises the price of a Rs 20,000 smartphone by 10% in Kathmandu:

• Quantity demanded falls by 5% (PED = 0.5).
• Revenue increases because demand is inelastic.

Case 2: Ncell’s Network Expansion (Economies of Scale)

Ncell’s long-run cost curve shows economies of scale:

• Small-scale: High cost per subscriber (limited coverage).
• Large-scale: Lower cost per subscriber (shared towers, bulk deals with Huawei).

Case 3: Bank Loan Interest (Cost of Capital)

Banks like NMB calculate loan interest using cost of funds:

• Fixed Costs: Branch rent, salaries.
• Variable Costs: Per-loan processing fees, risk assessment.
• Profit Maximization: Banks set interest rates where MR = MC of lending.

Worked Example: For a Rs 1,000,000 loan:

• TFC: Rs 500,000/year (branch costs).
• TVC: Rs 20,000 per loan (processing).
• Optimal Interest Rate: Banks charge 12% to cover costs and maximize profit.

6. Government Intervention and Market Efficiency

Governments intervene to correct market failures:

1. Price Ceilings/Floors: Example: NTC caps internet prices to prevent monopolistic exploitation.
2. Subsidies: Example: Government subsidies for solar energy reduce costs, shifting the MC curve downward.
3. Taxes: Example: Excise tax on cigarettes shifts the demand curve left, reducing consumption.

In the Real World

1. eSewa and Khalti (Transaction Costs)

   • These apps reduce transaction costs (TVC) for digital payments by leveraging economies of scale. Their LAC curve is downward-sloping because processing millions of transactions per day lowers the cost per transaction.

2. Daraz’s Warehouse Network (Fixed vs. Variable Costs)

   • Daraz’s TFC includes renting large warehouses in Kathmandu and Pokhara. Their TVC includes last-mile delivery costs, which vary with the number of orders. During sales (e.g., Dashain), Daraz optimizes Q to balance AC and MC.

3. Nepal Electricity Authority (NEA) – Cost of Generation

   • NEA’s MC curve for hydroelectricity is steep at low output (dams need minimum water flow) but flatter at higher output (economies of scale). During monsoon, NEA increases generation to take advantage of lower MC.

4. Pathao’s Surge Pricing (Elasticity)

   • Pathao uses dynamic pricing based on PED. During peak hours (e.g., 7–9 PM), demand is inelastic (PED < 1), so prices rise. At night, demand is **elastic** (PED > 1), so prices drop to attract riders.

5. Nepal Rastra Bank (NRB) – Interest Rate Decisions

   • NRB adjusts repo rates (cost of borrowing for banks) based on MC of money supply. If inflation is high, NRB raises rates to shift the demand for loans left, reducing aggregate demand.

Exam Tip

1. Derive Functions First: Always start by deriving TC, TVC, AVC, AC, MC from given data. Example: If $TC=6000+400Q-20Q^2+Q^3$, then:

   • $TVC=400Q-20Q^2+Q^3$
   • $MC=\frac{d(TVC)}{dQ}=400-40Q+3Q^2$

2. Profit Maximization: For any firm (monopoly, competitive), the rule is MR = MC. For perfect competition, P = MR = MC.

3. Complete Tables Carefully:

   • Calculate TR = P × Q.
   • Calculate TC from the given function.
   • Profit = TR – TC.
   • Common Mistake: Forgetting to calculate AFC = TFC/Q and AVC = TVC/Q.

4. Elasticity Shortcuts:

   • If PED > 1, a price increase reduces TR.
   • If PED < 1, a price increase increases TR.
   • MR = AR × (1 + 1/PED) is your friend.

5. Long-Run vs. Short-Run:

   • Short-run: Firms can only adjust Q (not plant size).
   • Long-run: Firms adjust all inputs, leading to LAC curves with economies/diseconomies of scale.

6. Graphical Questions:

   • Always label equilibrium points (e.g., where MR = MC).
   • Show shutdown point (P = min AVC).
   • For monopoly, show deadweight loss (area between demand and MC).

Final Worked Example: Comprehensive Problem

Given:

• Demand: $P=400-0.4Q$
• Cost: $TC=280+8Q$

Questions:

1. Find profit-maximizing Q and P.
2. Calculate TR and Profit.
3. If the government imposes a Rs 20/unit tax, how does this affect Q, P, and Profit?

Solutions:

1. Profit Maximization:

   • $TR=400Q-0.4Q^2$
   • $MR=400-0.8Q$
   • $MC=8$
   • Set $MR=MC$: $$400-0.8Q=8$$ $$Q=490\text{ units}$$
   • $P=400-0.4(490)=Rs204$

2. TR and Profit:

   • $TR=204\times 490=Rs100,960$
   • $TC=280+8(490)=Rs4,180$
   • Profit = Rs 96,780

3. Tax Impact:

   • New $MC=8+20=Rs28$
   • Set $MR=MC$: $$400-0.8Q=28$$ $$Q=465\text{ units}$$
   • New $P=400-0.4(465)=Rs214$
   • Consumer pays Rs 10 more, but Q falls (tax burden shared).

Summary Table: Key Formulas

Visual Summary: Cost and Revenue Curves