EXPLORE · ADJUST · EXPLAIN

One price change, two effects

Separate the response to relative prices from the response to purchasing power.

Price experiment

Good whose price changes

Type of this good

Price direction

Price change · %

Up to +200% or −80%, in 10-percentage-point steps.

Other-good price · P

Original selected-good price: p₀ = 0.5P = 1.

Income · m

Preference share · α

Here s = x is the selected good and t = y is the other. Switching x/y rotates the stylized preferences so the selected good has the chosen type.

Original → compensated → final

Thick solid lines = budgets. Bold dotted curves = indifference curves.

A · original budgetB · compensated budgetC · final budgetu₀ · original utilityu₁ · final utility

For a normal good, the income and substitution effects reinforce each other.

Calculated from your parameters

Prices, income, and compensation

\begin{aligned}p_0&=1\quad\longrightarrow\quad p_1=1.5\\P&=2,\qquad m=14\\ m^H&=e(p_1,P,u_0)\\&=17.1464\end{aligned}

Bundle	x	y
A · Original	7	3.5
B · Compensated	5.7155	4.2866
C · Final	4.6667	3.5

Read each effect as a change in quantity

\begin{aligned}\mathrm{SE}_{x}&=5.7155-7=-1.2845\\\mathrm{IE}_{x}&=4.6667-5.7155=-1.0488\\\mathrm{TE}_{x}&=4.6667-7=-2.3333\\\underbrace{\mathrm{SE}_{x}}_{A\to B}+\underbrace{\mathrm{IE}_{x}}_{B\to C}&=\underbrace{\mathrm{TE}_{x}}_{A\to C}\end{aligned}

What is held fixed?

Hicks: adjust income until the consumer can just reach the original utility. B lies on the original indifference curve.

Slutsky: adjust income so the old bundle A remains affordable. The compensated budget passes through A. Re-optimizing can give higher utility than at A.

Walrasian, or Marshallian, demand is ordinary demand at given prices and income. Its purchasing-power decomposition is conventionally called the Slutsky decomposition.

Good x	Hicks	Slutsky
Substitution	-1.2845	-1.1667
Income	-1.0488	-1.1667
Total	-2.3333	-2.3333

For finite changes, the two methods generally give different components. They agree on the total effect and coincide to first order.

Utility, demand, and the compensation formula

Ordinary (Marshallian) demand

Normal-good example

\begin{aligned}U(s,t)&=s^\alpha t^{1-\alpha}\\d_s(p,P,m)&=\frac{\alpha m}{p}\\d_t(p,P,m)&=\frac{(1-\alpha)m}{P}\end{aligned}

Hicks demand & Slutsky compensation

Minimum spending to reach utility u

\begin{aligned}e(p,P,u)&=u\left(\frac p\alpha\right)^\alpha\left(\frac P{1-\alpha}\right)^{1-\alpha}\\h_s&=\frac{\alpha e}{p},\qquad h_t=\frac{(1-\alpha)e}{P}\end{aligned}

Two different compensation rules

\begin{aligned}u_0&=U\!\left(d(p_0,P,m)\right)=4.9497\\q^H&=h(p_1,P,u_0)\\m^S&=p_1s_0+Pt_0\\q^S&=d(p_1,P,m^S)\end{aligned}

For either compensated bundle qᶜ: SE = qᶜ−q⁰, IE = q¹−qᶜ, and TE = q¹−q⁰. Apply these identities separately to both coordinates.