Advanced Managerial Economics

A starting point is the demand curve, which shows the relationship between price and quantity demanded, holding other factors constant (income, tastes, substitutes, etc.). For most normal goods, the demand curve slopes downward: as price rises, quantity demanded falls.

To build a model you need two ingredients: 1. A functional form: how you relate price to quantity (linear, log-linear, constant elasticity, etc.). 2. Parameter estimates: the numbers that make the model fit your data.

For MBA coursework, the most common “first model” is the linear demand model, written in plain language as: - Quantity demanded changes at a constant rate as price changes.

So you can treat demand as: “quantity equals a starting level minus a slope times price,” where the slope tells you how sensitive quantity is to price.

A concrete example of “model building” Suppose a gym sells monthly memberships and tracks sales across a few price points. You might observe:

• At price 40, quantity demanded is 120 memberships/month - At price 50, quantity demanded is 100 memberships/month

From just two points, you can infer a linear relationship between price and quantity. The slope is the change in quantity divided by the change in price.

Here’s the key learning: the slope is not just math - it’s sensitivity. A steep negative slope means quantity reacts strongly when price changes.

Step-by-step: turning data into a linear demand equation 1. Pick units and keep them consistent. If price is in dollars per month and quantity is memberships per month, keep those exact units. 2. Choose a linear form. Write quantity as Q = a + bP, where b will be negative for a typical demand curve. 3. Use your data to solve for a and b. Two data points are enough for two unknowns. 4. Use the model for predictions. Plug in a new price to forecast quantity. 5. Check plausibility. If the model predicts negative quantity at some price, that’s a sign the linear form isn’t valid outside the observed range.

Practical takeaway prompt Ask yourself: “If my model slope is -2, what does that mean in human terms?” It means quantity changes by about 2 units for every 1 unit increase in price, within the range where the model fits.

Price Elasticity and Revenue: The “Revenue Switch” Logic

A price elasticity of demand measures how responsive quantity demanded is to a change in price. The most used version in managerial decisions is price elasticity:

• Elastic demand: quantity responds a lot to price (elasticity magnitude greater than 1). - Inelastic demand: quantity responds a little to price (elasticity magnitude less than 1). - Unit elastic: elasticity magnitude equals 1.

Elasticity is usually reported as a magnitude (a positive number), even though the true relationship between price and quantity is negative. So you can think: “Elasticity of 2” means a 1% price change causes about a 2% change in quantity, in the opposite direction.

Why elasticity matters for revenue Revenue is R = P × Q. When price changes, revenue depends on which effect dominates: - Price effect: higher price tends to raise revenue. - Quantity effect: lower quantity tends to reduce revenue.

Elasticity tells you which effect wins.

A simple rule of thumb (the “revenue switch” logic): - If demand is elastic (elasticity > 1), raising price tends to reduce revenue because quantity drops a lot. - If demand is inelastic (elasticity < 1), raising price tends to increase revenue because quantity drops a little. - If demand is unit elastic (elasticity = 1), revenue stays approximately constant for small price changes.

Elasticity with a linear demand curve With a linear demand curve, elasticity is not constant across prices. That’s a common exam trap and a common business reality: sensitivity often changes as you move along the curve.

Ask yourself: when price is high, quantity is low - does that crowd out buyers quickly or slowly? The elasticity calculation will reflect that.

Policy connection: who pays under a per-unit tax A per-unit tax is a fixed amount charged per unit sold (for example, $1 tax on each membership sold). The key insight is that the burden of the tax depends on relative elasticities: - The side with more inelastic demand/supply absorbs more of the tax. - The side with more elastic response can avoid the tax by reducing quantity more easily.

In practical terms: if consumers don’t change buying much when price rises (inelastic demand), they bear more of the tax cost through higher effective prices.

Section takeaway prompt Before doing any calculations, ask: “If I raise the price slightly, do customers cut back a little or a lot?” Elasticity turns that intuition into a decision tool.

Worked Example: Build a Demand Model, Compute Elasticity, Predict Revenue and Tax Impact

Let’s walk through a full, numbers-first example you can reuse on homework problems.

Scenario: A streaming service sets a monthly price and observes demand in a limited range. You estimate that demand is linear. You have two data points: - When price is 10, quantity demanded is 5000 subscribers/month - When price is 15, quantity demanded is 4000 subscribers/month

You want to: 1. Build the demand model. 2. Compute price elasticity at price 12. 3. Predict what happens to revenue if price increases from 12 to 13. 4. Use elasticity intuition to reason about who would bear a per-unit tax of $1 (assume supply is perfectly elastic for simplicity, so the tax shifts the effective price on consumers).

Step 1: Build the linear demand model Assume Q = a + bP.

Use the two points:

• 5000 = a + b(10) - 4000 = a + b(15)

Subtract the first equation from the second: - 4000 - 5000 = b(15 - 10) - -1000 = 5b - b = -200

Now plug back into 5000 = a + (-200)(10): - 5000 = a - 2000 - a = 7000

So the demand model is: - Q = 7000 - 200P

Final result (demand model): Q = 7000 - 200P

Step 2: Compute elasticity at price P = 12 First find quantity at P = 12: - Q = 7000 - 200(12) = 7000 - 2400 = 4600

Now compute elasticity for a linear demand model using the definition in percentage terms. For small changes, elasticity can be approximated by: - Elasticity ≈ (dQ/dP) × (P/Q)

Here, dQ/dP is the slope of the demand curve: - dQ/dP = -200

So: - Elasticity ≈ (-200) × (12 / 4600) - = -200 × 0.0026087... - ≈ -0.5217

Price elasticity magnitude is about: - |E| ≈ 0.52

Interpretation: demand at price 12 is inelastic (since 0.52 < 1). That means a price increase should raise revenue (at least for a small move).

Final result (elasticity at P = 12): |E| ≈ 0.52 (inelastic)

Step 3: Predict revenue change from P = 12 to P = 13 Compute revenue at each price.

At P = 12: - Q = 4600 - R = P × Q = 12 × 4600 = 55,200

At P = 13: - Q = 7000 - 200(13) = 7000 - 2600 = 4400 - R = 13 × 4400 = 57,200

Revenue increases from 55,200 to 57,200, a rise of 2,000.

Final result (revenue impact): Revenue increases to 57,200 (up by 2,000)

This matches the elasticity conclusion: with inelastic demand, price hikes tend to increase revenue.

Step 4: Reason about the impact of a per-unit tax of $1 Assume supply is perfectly elastic (so the supplier doesn’t change much; the tax effectively raises the consumer’s price by $1). Then the consumer faces an increase from 12 to 13.

We already computed the quantity response from 12 to 13: - Quantity falls from 4600 to 4400

So the immediate demand effect is: - Subscribers drop by 200 (a 4.35% decline, since 200/4600 ≈ 4.35%)

Who bears the burden? - Because demand is inelastic at this range (elasticity magnitude about 0.52), consumers cannot easily reduce quantity much. - Therefore, consumers bear more of the tax through higher effective prices.

If you were computing exact tax incidence with both elasticities, you’d need supply elasticity too. But the direction is clear from the demand elasticity alone: inelastic demand means the tax bites consumers more (in the sense that effective price rises and quantity doesn’t fall enough to offset it).

Final result (policy direction): With inelastic demand, most of a $1 per-unit tax shows up as higher consumer price, and quantity falls modestly (4600 → 4400).

Section takeaway prompt When you get an elasticity below 1 in magnitude, force yourself to say out loud: “Revenue should move in the same direction as price.” Then check it with the numbers.

Check Your Understanding: Practice Demand Models and Elasticity Decisions

1. Two-point demand model You observe: at P = 20, Q = 300; at P = 30, Q = 250. (a) Build the linear demand model Q = a + bP. (b) Predict Q when P = 25. Hint: find b from the slope (change in Q divided by change in P), then solve for a using one point.

2. Elastic vs inelastic and revenue A product has price elasticity magnitude of 1.3 at the current price. If price increases slightly, what happens to revenue? Hint: elastic demand means the quantity drop dominates the price gain.

3. Compute elasticity from a linear slope Suppose demand is Q = 800 - 50P. Find price elasticity magnitude at P = 8. Hint: use |E| ≈ |dQ/dP| × (P/Q). Here dQ/dP = -50.

4. Revenue prediction using the model Demand is Q = 1200 - 100P. Compute revenue at P = 6 and P = 7. Does revenue increase or decrease? Hint: compute Q first, then multiply by price. Don’t rely on intuition - verify.

5. Tax intuition For a per-unit tax, consumers face a higher price. Demand at the relevant price is inelastic (elasticity magnitude 0.6). Compared with elastic demand (elasticity magnitude 2.0), which group experiences a larger effective price increase and why? Hint: inelastic demand means quantity doesn’t fall much, so prices must move more to deliver the tax effect.

Answer Key 1. b = -5, a = 400, so Q = 400 - 5P; at P = 25, Q = 275. 2. Revenue decreases when price rises (elastic demand). 3. Q = 800 - 50(8) = 400; |E| ≈ 50 × (8/400) = 1.0. 4. At P = 6: Q = 600, R = 3,600. At P = 7: Q = 500, R = 3,500, so revenue decreases. 5. The group on the inelastic side faces the larger effective price increase; with inelastic demand, quantity doesn’t adjust enough to “soften” the price impact.

The unifying takeaway is simple: once you can describe demand with a usable model and translate sensitivity into elasticity, you stop treating price changes as guesses. You treat them as predicted outcomes - revenue first, policy impact next.

End of chapter one. 4 more chapters in the full book.