Slutsky Equation [Micro Economics]
Micro Economics
- Chapter 8. SLUTSKY EQUATION
The lecture covers the concepts of substitution and income effects in consumer theory, as well as the Slutsky equation. It begins by introducing the concepts of normal, inferior, ordinary, and Giffen goods based on consumer behavior in response to changes in income and prices. The lecture then delves into the Slutsky equation, which decomposes the total effect of a price change on demand into substitution and income effects. The Slutsky method involves compensating the consumer so that the initial bundle is just affordable at the new prices, while the Hicksian method compensates the consumer to maintain the initial utility level.
The lecture explains the differences between the two methods, provides examples, and discusses the implications for the law of demand. The key takeaways include the properties of substitution and income effects, the interpretation of the Slutsky equation, and the conditions under which goods are classified as normal, inferior, ordinary, or Giffen.
The lecture emphasizes the importance of practicing problem set questions to solidify the understanding of these concepts.
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Overview
1. Introduction to Normal, Inferior, Ordinary, and Giffen Goods
2. Slutsky Equation and Decomposition of Price Change Effects
3. Substitution and Income Effects for Normal and Inferior Goods
4. Interpretation of the Slutsky Equation and the Law of Demand
5. Examples and Special Cases
The lecture provides examples and discusses special cases, such as perfect substitutes and quasi-linear preferences. For perfect substitutes, the lecture explains how the substitution and income effects can lead to corner solutions, where the consumer switches from consuming one good to the other entirely. For quasi-linear preferences, the lecture mentions that there may be no income effect in certain cases, even with an interior solution.
Previous lecture
How consumer’s behavior changes in response to the changes in the economic environment.
– Income: Normal vs. Inferior goods
– Prices: Ordinary vs. Giffen Goods
Giffen goods might not be unreasonable.
Wage Example: Higher wage (per hour) → work more.
What if wage increases from $10 to $1.000/hour? Work more or leisure more? What if increases to $1.000.000/ hour?
: How does Commodity change → Demand function, effects?
(Reminder = Slope of BC should be -p₁/p₂)
8.1 The Substitution Effect
When the price of a good changes, there are two sorts of effects: the rate at which you can exchange one good for another changes, and the total purchasing power of your income is altered. If, for example, good 1 becomes cheaper, it means that you have to give up less of good 2 to purchase good 1. The change in the price of good 1 has changed the rate at which the market allows you to “substitute” good 2 for good 1. The trade-off between the two goods that the market presents the consumer has changed.
At the same time, if good 1 becomes cheaper it means that your money income will buy more of good 1. The purchasing power of your money has gone up; although the number of dollars you have is the same, the amount that they will buy has increased.
The first part—the change in demand due to the change in the rate of exchange between the two goods—is called the substitution effect. The second effect—the change in demand due to having more purchasing power—is called the income effect. These are only rough definitions of the two effects. In order to give a more precise definition we have to consider the two effects in greater detail.
The impact of ∆p₁ in x₁(p₁,p₂,M), with p₂ and M constant, can be seen as the sum of two effects:
– Substitution effect: effect of changing relative prices. This effect is always non-positive (to price change).
– Income effect: effect of changing real income. This effect can be positive or negative.
– Total effect in demand from a change in price is the algebraic sum of both impacts.
(= Sum of Substitution effect & Income effect)
Total Effect of a change in P₁:
Slutsky Substitution effect (SE): How demand changes when price changes but real income is kept constant(same purchasing power)
( I → F → S )
Suppose the consumer receives an extra income, enough to buy the initial bundle => The amount of income change is Compensation Variation(CV)
: Even if, the consumer receives an extra income, for enough to buy the initial bundle. They do not have a same affordable bundle. In the above case, x₁ decrease & x₂ increase.
**Slutsky Substitution Effect(SE) is always non-positive.
Slutsky Substitution Effect(SE) is a change from I to S.
A price increase induces a decrease in the quantity demanded. Substitution effect is always non-positive.
At (Pi,Mi), bundle (i)(=initial) was chosen, when bundles on the RHS of (i)(initial) on BCs(Pf,Ms) were affordable. So, bundle (i) should be preferred to those bundles and they won't be chosen at (Pf,Ms).
Δx₁se = xs₁(Pf,Ms) - xi₁(Pi,Mi)
: It moves from Xi to Xs.
8.2 The Income Effect
Slutsky Income effect (IE): How demand changes due to the change in the purchasing power.
( F → S )
Δx₁ne = xf₁(Pf,Mi) - xs₁(Pf,Ms)
+What is (NE)?
: NE means graphically from S to F. (Can be either positive or negative)
( S → F )
CV(Compensation Variation) implicate "Change in income and change in price will always move in the same direction"
: if p goes up, then we have to raise income to keep the same initial bundle affordable.
※ Essence of SE & IE :
8.4 The Total Change in Demand
Total change in demand: Δx₁ = Xf₁(Pf,Mi) - Xi₁(Pi,Mi) = Xf₁ - Xi₁
Δx₁ = Δx₁se + Δx₁ne
Δx₁ = Xs₁(Pf,Ms) - Xi₁(Pi,Mi) + Xf₁(Pf,Mi) - Xs₁(Pf,Ms) = Xs₁ - Xi₁ + Xf₁ - Xs₁
If price of good 1 increases and x₁ is a normal good) :
Δx₁ (-) = Δx₁se (-) + Δx₁ne (-)
If price of good 1 increases and x₁ is a inferior good) :
If positive IE(Income Effects) dominates the negative SE, then the demand for x₁ increases after its price increase, meaning that x₁ is a giffen good.
All giffen goods are inferior but an inferior good might not be giffen.
8.5 Rates of Change
Slutsky Equation:
Define the negative of the IE:
Δx₁NE = x₁F - x₁S (Δx₁IE = x₁S - x₁F = - Δx₁NE)
Δx₁IE = - Δx₁NE
Then, Δx₁ = Δx₁SE - Δx₁IE
Δm is as same as x₁Δp₁ (: Δm = x₁Δp₁ )
So, Dividing every term by Δp₁
Δx₁/Δp₁ = Δx₁SE / Δp₁ - Δx₁IE / Δp₁ = Δx₁SE / Δp₁ - x₁ Δx₁IE / Δm
So The Slutsky Equation is
Δx₁/Δp₁ = Δx₁SE / Δp₁ - x₁ Δx₁IE / Δm
:
Δx₁/Δp₁ = rate of change in demand as p changes and fixed m.
Δx₁SE / Δp₁ = rate of change in demand as p changes, adjusting income so as to keep the initial bundle just affordable.
x₁ Δx₁IE / Δm = how demand changes as income changes (after a price change to keep the initial bundle just affordable), times the original level of demand.
※Caution: If common good case, x₁ Δx₁IE / Δm = (-)
If inferior good case, x₁ Δx₁IE / Δm = (+)
8.6 The Law of Demand
The Law of Demand: If the demand for a good increases when income increases, then the demand for that good must decrease when its price increases.
8.8 Another Substitution Effect
Hicks Substitution Effect(SE): How demand changes when price changes but real income is kept constant (same utility).
( I → F → H )
Suppose the consumer receives an extra income to keep the initial utility => The amount of income change is Compensation Variation(CV) of Hicks
The Hicks Substitution Effect(SE) is the change from I to H. Price increase induces a decrease in the quantity demanded. Substitutioin effect(SE) is Always non-positive
( I → H )
※ The change in the quantity demanded must have the opposite sign from that of the price change
8.9 Compensated Demand Curves
The analysis of this chapter shows that the Slutsky and Hicks demand curves are always downward sloping curves. Furthermore the ordinary demand curve is a downward sloping curve for normal goods. However, the Giffen analysis shows that it is theoretically possible that the ordinary demand curve may slope upwards for an inferior good.
The Hicksian demand curve—the one with utility held constant—is some- times called the compensated demand curve. This terminology arises naturally if you think of constructing the Hicksian demand curve by ad- justing income as the price changes so as to keep the consumer’s utility constant. Hence the consumer is “compensated” for the price changes, and his utility is the same at every point on the Hicksian demand curve. This is in contrast to the situation with an ordinary demand curve. In this case the consumer is worse off facing higher prices than lower prices since his income is constant.
Hicks Substitution Effect(SE) is measured in the compensated demand curve, h(p₁,p₂,uI)
( I → H )
Hicks Income Effect(IE): How demand changes due to the change in the purchasing power. Graphically from H to F.
( H → F )
Dual Problems (& Solution method)
Utility Maximization | Expenditure Minimization | |
Purpose & how to | Max u(x₁,x₂) s.to p₁x₁ + p₂x₂ = m | Min p₁x₁ + p₂x₂ s.to u(x₁,x₂) = u₀ |
Solution | standard x₁*(p,m) & x₂*(p,m) | Compensated h₁*(p,u) & h₂*(p,u) |
Remark | u changes as p, m changes xi*(p,m) has both SE and IE | m changes as p, u changes hi*(p,u) has only SE |
(Hicks) Graphical phenomenon & Total Effect:
Case of (1)Normal Good, (2)Inferior Good, (3)Giffen Good
Summary
1. When the price of a good decreases, there will be two effects on consump- tion. The change in relative prices makes the consumer want to consume more of the cheaper good. The increase in purchasing power due to the lower price may increase or decrease consumption, depending on whether the good is a normal good or an inferior good.
2. The change in demand due to the change in relative prices is called the substitution effect; the change due to the change in purchasing power is called the income effect.
3. The substitution effect is how demand changes when prices change and purchasing power is held constant, in the sense that the original bundle remains affordable. To hold real purchasing power constant, money income will have to change. The necessary change in money income is given by Δm = x1Δp1.
4. The Slutsky equation says that the total change in demand is the sum of the substitution effect and the income effect.
5. The Law of Demand says that normal goods must have downward- sloping demand curves.
[Reference]
[1] Hal R. Varian - Intermediate Microeconomics_ A Modern Approach, 8th Edition -W.W. Norton & Co. (2010)
[2] Frank, chap. 4, pages 101-108
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