Consumer theory · A tool for lessons 9 to 16
The Consumer Workbench
One consumer, seven kinds of tastes, every object in the module. Pick tastes, set prices and income, and move the new price. The page computes the demand, the utility and the expenditure function, the elasticities, both splits of a price change, the three money measures of welfare, and then checks the identities between them against the numbers it has just found.
○ old bundle ● new bundleRed line and dot: Hicks compensationDotted line, gold dot: Slutsky compensation
Demand and utility
At (px, py, m) = (2, 1, 100) she buys (25, 50) and reaches utility v = 35.355. The least she could spend to reach it is e = Rs 100.
Elasticities of x
Own price −1, cross price 0, income 1. For y: own −1, cross 0, income 1.
The price of x goes from Rs 2 to Rs 1
Beforex = 25
After Slutsky compensation (old bundle affordable)x = 37.5; substitution +12.5
After Hicks compensation (old utility)x = 35.355; substitution +10.355
After the price changex = 50; total +25; income effect: Slutsky +12.5, Hicks +14.645
Money measures of the change
Hicks CV = −29.289 (income change at the new prices that restores the old utility). Hicks EV = +41.421 (income change at the old prices that gives the new utility). Change in consumer surplus = +34.657. Slutsky versions: CV −25, EV +50.
Utility before and after
v goes from 35.355 to 50. She is better off, so CV is negative and EV positive.
Identities that tie the objects together, computed from the numbers above
e(p, v) = m: the cheapest way to reach her utility costs her incomee = 100 against m = 100 ✓
Hicksian demand at v is her chosen bundle(25, 50) against (25, 50) ✓
Shephard's lemma: ∂e/∂p_x = Hicksian x25 against 25 ✓
Roy's identity: x = −(∂v/∂p_x)/(∂v/∂m)25 against 25 ✓
Slutsky equation: ∂x/∂p = ∂h/∂p − x ∂x/∂m−12.5 against −12.5 ✓
Own + cross + income elasticities of x = 00 against 0 ✓
Homogeneity: doubling prices and income leaves demand alone(25, 50) against (25, 50) ✓
Walras' law: p·x = m100 against 100 ✓
Things to try
- Cobb–Douglas. Leave the preset (u = xy, income 100, price of x falls from 2 to 1) and compare the Hicks and Slutsky columns: 10.36 and 12.5 for the substitution effect, 41.42 and 29.29 for EV and CV.
- Perfect complements. Choose perfect complements and move the new price: the substitution effect is zero and the whole change is the income effect.
- √x + y. Choose quasi-linear tastes and raise income: the demand for x does not move, the income effect is zero, and CV, EV and the change in consumer surplus are the same number.
- CES. Slide ρ from 0.8 to −3. Elasticity of substitution 1/(1 − ρ) falls from 5 to a quarter, and the own-price elasticity of x changes with it.
- Shifted Cobb–Douglas. Give x a positive shift of 3 and watch its income elasticity fall below 1 (a necessity); give it a negative shift and it rises above 1 (a luxury).
Signs follow the convention of Varian and of the JNU papers: the compensating variation is the change in income, at the new prices, that restores the old utility, so it is negative after a price fall; the equivalent variation is the change in income, at the old prices, that gives the new utility, so it is positive after a price fall. The change in consumer surplus is shown as a gain.
The tool runs on the same engine as the lessons, and every number in them was checked against it. The lessons start at the module page; duality is Lesson 14 and the money measures are Lesson 15.
Read next
All resources →MicroeconomicsHicksian demand, the expenditure function and duality: Shephard's lemma and Roy's identityExpenditure minimisation as the dual of utility maximisation, Hicksian (compensated) demand, the expenditure function and its properties, Shephard's lemma, the indirect utility function, Roy's identity and the Consumer Workbench.MicroeconomicsCompensating and equivalent variation: measuring welfare change in rupeesCompensating and equivalent variation from the expenditure function, their Slutsky versions, the JNU Anand questions, and how CV, EV and the change in consumer surplus compare for normal goods.MicroeconomicsConsumer theory: preferences, budget, demand and the Slutsky equationA module in seventeen lessons: preferences, utility and the MRS, the budget, utility maximisation, corners, taxes and subsidies, labour supply and saving, demand, elasticity and Engel curves, the Slutsky equation, revealed preference, duality, compensating variation, the properties of demand, and a lesson of Delhi School problems. With labs, a Workbench, worked questions and graded past papers.