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Slutsky Theorem and Derivation of the Demand Curve
The Slutsky theorem is a key result in consumer theory that breaks down the effect of a price change into two components:
- Substitution Effect – The consumer substitutes the cheaper good for the more expensive one while maintaining the same utility.
- Income Effect – The consumer’s purchasing power changes due to the price change, affecting overall consumption.
This theorem helps in understanding how the demand curve is derived and how consumers react to price changes.
1. Slutsky Theorem: Breaking Down the Price Effect
When the price of a good changes, the total change in demand (ΔQ\Delta Q) is split into: Total Price Effect=Substitution Effect+Income Effect\text{Total Price Effect} = \text{Substitution Effect} + \text{Income Effect}
Mathematical Expression
dXdP=dXsdP+dXmdM⋅dMdP\frac{dX}{dP} = \frac{dX^s}{dP} + \frac{dX^m}{dM} \cdot \frac{dM}{dP}
where:
- dXdP\frac{dX}{dP} = Total effect of a price change on demand
- dXsdP\frac{dX^s}{dP} = Substitution effect (movement along the indifference curve)
- dXmdM\frac{dX^m}{dM} = Income effect (change in demand due to purchasing power change)
- dMdP=X\frac{dM}{dP} = X (since income changes by the amount of expenditure on the good)
Thus, Slutsky’s equation is: dXdP=dXsdP−X⋅dXmdM\frac{dX}{dP} = \frac{dX^s}{dP} – X \cdot \frac{dX^m}{dM}
This equation shows that the total price effect consists of a pure substitution effect and an income effect adjusted by consumption of the good.
2. Graphical Representation of Slutsky Theorem
Step 1: Initial Equilibrium
- The consumer starts at equilibrium where the budget constraint is tangent to the highest possible indifference curve.
Step 2: Price Change and Substitution Effect
- If the price of a good falls, the budget line rotates outward.
- The substitution effect moves the consumer along the same indifference curve to a new point where the marginal rate of substitution (MRS) matches the new price ratio.
Step 3: Income Effect
- The real income (purchasing power) increases because the consumer can buy more of the good with the same budget.
- The budget line shifts outward to reflect this increased purchasing power, leading to a new equilibrium point.
🔹 For Normal Goods: Both substitution and income effects increase demand.
🔹 For Inferior Goods: The income effect works against the substitution effect.
🔹 For Giffen Goods: The income effect is stronger, leading to a decrease in demand despite the price drop.
3. Derivation of the Demand Curve Using Slutsky Theorem
Step 1: Understanding the Demand Function
The Marshallian Demand Function expresses quantity demanded as a function of price and income: X=f(P,M)X = f(P, M)
where PP is the price of the good and MM is income.
Step 2: Applying Slutsky’s Equation
Using Slutsky’s theorem, the change in demand due to a price change is: dXdP=dXsdP−X⋅dXmdM\frac{dX}{dP} = \frac{dX^s}{dP} – X \cdot \frac{dX^m}{dM}
- The substitution effect is always negative (dXsdP<0\frac{dX^s}{dP} < 0), meaning when price drops, demand increases.
- The income effect depends on whether the good is normal or inferior.
Step 3: Constructing the Demand Curve
- As price decreases, the demand for a normal good increases due to both substitution and income effects.
- For an inferior good, demand may increase or decrease depending on the strength of the income effect.
- Plotting different price levels against corresponding demand gives the downward-sloping demand curve.
4. Special Cases of Slutsky’s Theorem
| Type of Good | Substitution Effect | Income Effect | Overall Effect on Demand |
|---|---|---|---|
| Normal Good | Negative (more demand) | Positive (more demand) | Price drop increases demand |
| Inferior Good | Negative (more demand) | Negative (less demand) | Depends on which effect is stronger |
| Giffen Good | Negative (more demand) | Strongly Negative (less demand) | Price drop decreases demand |
🔹 Giffen Goods Exception: If the income effect is stronger than the substitution effect, the demand curve can slope upward.
5. Conclusion
Slutsky’s theorem helps us break down how price changes impact consumer behavior. The substitution effect always increases demand for the cheaper good, while the income effect can vary depending on the type of good. This principle is essential in understanding consumer demand and deriving the downward-sloping demand curve in most cases.