Chapter 2: Consumer Choice, Demand, and Revealed Preference. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Splitting a price rise into substitution and income

How much of the fall in food demand comes from the price change itself, and how much from lost purchasing power?

Compensation shifts the new budget line out until the consumer can afford the old bundle (Slutsky) or reach the old utility (Hicksian). The move from A to S is substitution; the move from S to C is the income effect.

$Equation, written in LaTeX: x(p,m)=\frac{m}{2p}, y(p,m)=\frac{m}{2}$

Equation, written in LaTeX: -15=-7.5-(30)(0.25)

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x is food, y is a composite good with price one, p is the food price and m = 120 is income. Utility is sqrt(xy). Point A is the old choice, C the new one and S the choice after compensation.

Predict first. As the price rise gets larger, does the gap between the Slutsky and Hicksian substitution effects widen or close?

Your prediction It widens It closes

New food price

Compensation rule

Figure: Splitting a price rise into substitution and income. Old, new and compensated budget lines for a food price of 3.0. Food moves from 30 at A to 25.00 on the compensated line and 20.00 at C. — New food price: 3, Compensation rule: Slutsky (old bundle affordable)
Constructed example: the chapter's hypothetical consumer (income 120, price 2 rising to 3); prices 2.5, 4 and 5 are added for comparison.

Food before (A): 30.00
Food after (C): 20.00
Compensated income: 150.00
Compensated food (S): 25.00
Substitution effect: -5.00
Income effect: -5.00
Total change: -10.00

Ordinary demand falls from 120 / (2 x 2) = 30 to 120 / (2 x 3.0) = 20.00. Slutsky income = 120 + (3.0 - 2) x 30 = 150.00, so compensated food is 150.00 / (2 x 3.0) = 25.00. Substitution = 25.00 - 30 = -5.00 and income = 20.00 - 25.00 = -5.00; together -5.00 + (-5.00) = -10.00.

Worked steps

x before = 120 / (2 x 2) = 30.00
x after = 120 / (2 x 3.0) = 20.00
Slutsky income = 120 + (3.0 - 2) x 30 = 150.00
Compensated x = 150.00 / (2 x 3.0) = 25.00
Substitution = 25.00 - 30 = -5.00
Income effect = 20.00 - 25.00 = -5.00

Use the idea

When a price rise hits a household, separate the part of the response a cash transfer would undo from the part that reflects changed relative prices.

Where the conclusion applies

Cobb-Douglas utility sqrt(xy), a fixed income and a single price change. With other preferences the split differs, and the two compensation rules answer different questions.

Check your understanding: If the food price rises from 2 to 4, what are the Slutsky substitution and income effects?

Compensated income is 120 + (4 - 2) x 30 = 180, so x = 180 / 8 = 22.5 and substitution = 22.5 - 30 = -7.5. Ordinary x = 120 / 8 = 15, so the income effect is 15 - 22.5 = -7.5.

Chapter 2 source: section "Slutsky equation".

Demonstration 2 of 4

Compensating vs equivalent variation

How much money is a fare increase worth to the rider, measured at new prices or at old ones?

The compensating variation is the extra income that restores the old utility at the new price; the equivalent variation is the income that, taken at old prices, leaves the consumer as badly off as the price rise does. Both are vertical gaps between expenditure curves and income.

$Equation, written in LaTeX: h_x(p,\bar u)=\frac{\bar u}{\sqrt p}, h_y(p,\bar u)=\bar u\sqrt p, e(p,\bar u)=2\bar u\sqrt p$

$Equation, written in LaTeX: e(2,50)=100\sqrt2\approx141.42$

$Equation, written in LaTeX: e(1,35.36)\approx70.71$

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p is the transit price, income is 100 and the composite good has price one. Utility is sqrt(xy), so the expenditure function e(p, u) = 2u sqrt(p) is the least spending that reaches utility u. u0 = 50 before the rise and u1 after it.

Predict first. Is CV always larger than EV for a price rise here?

Your prediction Yes, at every price shown No, EV is sometimes larger

New transit price

Figure: Compensating vs equivalent variation. Expenditure functions at the old and new utility against the transit price. At p = 2.0 the gap above income 100 is the CV, 41.42; at p = 1 the gap below 100 is the EV, 29.29. — New transit price: 2
Constructed example: the chapter's hypothetical transit rider (income 100, price 1 rising to 2); prices 1.5, 3 and 4 are added for comparison.

New utility u1: 35.36
e(p1, 50): 141.42
Compensating variation: 41.42
e(1, u1): 70.71
Equivalent variation: 29.29
Hicksian bundle at p1: (35.36, 70.71)

At p = 2.0 ordinary demand is 100 / (2 x 2.0) = 25.00 rides, so u1 = sqrt(25.00 x 50) = 35.36. CV = 2 x 50 x sqrt(2.0) - 100 = 141.42 - 100 = 41.42. EV = 100 - 2 x 35.3553 x sqrt(1) = 100 - 70.71 = 29.29. CV is the larger of the two for this price rise.

Worked steps

x1 = 100 / (2 x 2.0) = 25.00
u1 = sqrt(25.00 x 50) = 35.3553
e(p1, 50) = 2 x 50 x sqrt(2.0) = 141.42
CV = 141.42 - 100 = 41.42
e(1, u1) = 2 x 35.3553 = 70.71
EV = 100 - 70.71 = 29.29

Use the idea

When valuing a price change, say which counterfactual you mean: payment to accept the change (CV) or payment to avoid it (EV).

Where the conclusion applies

Utility sqrt(xy), income fixed at 100 and no other price changes. These are partial-equilibrium measures for one household, not a verdict on social welfare.

Check your understanding: If the price rises from 1 to 4, what are CV and EV?

CV = 2 x 50 x sqrt(4) - 100 = 100. u1 = sqrt(12.5 x 50) = 25, so EV = 100 - 2 x 25 = 50.

Chapter 2 source: section "Hicksian-Marshallian demand duality".

Demonstration 3 of 4

When is an inferior staple Giffen?

How large must the budget share and how negative the income elasticity be for demand to rise with price?

Substitution always pushes demand down when price rises. For an inferior good the loss of purchasing power pushes it up, by the budget share times the income elasticity. The good is Giffen only when that push is larger.

$Equation, written in LaTeX: \varepsilon^{M}=-0.20-(0.70)(-0.60)=0.22$

$Equation, written in LaTeX: \varepsilon^{M}=-0.20-(0.70)(-0.10)=-0.13$

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eps_M is the ordinary own-price elasticity, -0.20 the compensated elasticity, s the staple's budget share and eta its income elasticity. The Slutsky equation in elasticities is eps_M = eps_H - s eta.

Predict first. At budget share 0.5, how negative must income elasticity be for Giffen behavior?

Your prediction Below -0.4 Below -0.2 Below -0.6

Budget share of the staple

Income elasticity

Figure: When is an inferior staple Giffen? Bars for the substitution term -0.20, the income term 0.42 and the ordinary elasticity 0.22 at budget share 0.70 and income elasticity -0.60. — Budget share of the staple: 0.7, Income elasticity: -0.6
Constructed example: the chapter's hypothetical calibration (-0.20, share 0.70, eta -0.60 and -0.10); shares 0.3 and 0.5 and eta of -0.3 and -0.9 are added for comparison.

Substitution term: -0.20
Income term -s x eta: 0.42
Ordinary elasticity: 0.22
Change in demand for a 10% price rise: 2.2%
Giffen?: yes
Threshold eta for Giffen: below -0.29

eps_M = -0.20 - (0.70)(-0.60) = -0.20 + 0.42 = 0.22. The income term outweighs substitution, so demand rises with price: Giffen behavior. At this budget share, Giffen behavior needs income elasticity below -0.20 / 0.70 = -0.29.

Worked steps

Income term = -(0.70) x (-0.60) = 0.42
eps_M = -0.20 + 0.42 = 0.22
A 10% price rise changes demand by 10 x 0.22 = 2.2%
Threshold: -0.20 / 0.70 = -0.29

Use the idea

Before reading a positive price response as Giffen behavior, check that the good takes a large share of the budget and is strongly inferior.

Where the conclusion applies

A local, hypothetical calibration with a fixed compensated elasticity of -0.20. It is not an estimate from any study; Giffen behavior is local to a price and income range.

Check your understanding: With share 0.5 and eta = -0.3, is demand Giffen?

eps_M = -0.20 - (0.5)(-0.3) = -0.05, still negative, so not Giffen; the threshold is eta below -0.4.

Chapter 2 source: section "Giffen paradox".

Demonstration 4 of 4

Add trips until marginal benefit falls below marginal cost

How many evening trips should run, and does a sunk depot cost change the answer?

Each trip is judged on its own forward benefit and cost. Trips run while the net increment is positive. A cost already paid that no choice can recover changes no increment, so it cannot change the count.

$Equation, written in LaTeX: (600,\ 320,\ 60,\ -180,\ -400)$

$Equation, written in LaTeX: (800,\ 520,\ 260,\ 20,\ -200)$

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Marginal benefits of the five successive trips are 900, 700, 520, 360 and 220; marginal costs are 300, 380, 460, 540 and 620, less any cost cut per trip. The vectors are the net increments.

Predict first. With a cost cut of 100, how many trips run?

Your prediction 3 4 5

Cost reduction per trip

Count the depot cost

Figure: Add trips until marginal benefit falls below marginal cost. Paired bars of marginal benefit and marginal cost for five trips with a cost cut of 0. The first 3 trips are shaded as run. — Cost reduction per trip: 0, Count the depot cost: No
Constructed example: the chapter's hypothetical transit planner (cuts of 0 and 200 and the sunk depot); cuts of 100 and 300 are added for comparison.

Marginal costs: 300, 380, 460, 540, 620
Net increments: 600, 320, 60, -180, -400
Trips run: 3
Net gain from trips run: 980
Depot cost: left out (sunk)

Net increments are MB - MC = 600, 320, 60, -180, -400. Trip 3 passes: 520 - 460 = 60. Trip 4 fails: 360 - 540 = -180. So 3 trips run.

Worked steps

MC after the cut of 0: 300, 380, 460, 540, 620
Net increments: 600, 320, 60, -180, -400
Last trip that passes: trip 3, net 60
Trip 4 fails with net -180

Use the idea

When someone argues for one more unit to use an earlier investment, ask whether that unit's own benefit exceeds its own cost.

Where the conclusion applies

Benefits and costs per trip are known and ordered, and the depot cost cannot be recovered either way. A trip that preserves a licence or lowers future costs would add a forward benefit.

Check your understanding: With a cost cut of 300, how many trips run?

Net increments are 900, 620, 360, 120 and -100, so four trips run and the fifth still fails.

Chapter 2 source: section "Marginal analysis".

Splitting a price rise into substitution and income

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Compensating vs equivalent variation

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

When is an inferior staple Giffen?

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Add trips until marginal benefit falls below marginal cost

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies