Slutsky Equation - PDF

# Chapter 8: Slutsky Equation Economists are interested in how a consumer's behavior changes in response to changes in the economic environment. This chapter focuses on how a consumer's choice of goods responds to changes in a good's price. It is natural to assume that when the price of a good rise...

# Chapter 8: Slutsky Equation Economists are interested in how a consumer's behavior changes in response to changes in the economic environment. This chapter focuses on how a consumer's choice of goods responds to changes in a good's price. It is natural to assume that when the price of a good rises, the demand for it will fall. However, as we saw in Chapter 6, it is possible to construct examples where the optimal demand for a good decreases when its price falls. A good with this property is called a **Giffen good**. Giffen goods are a theoretical curiosity, but other situations exist where changes in prices might have "perverse" effects. For example, we normally assume that people will work more if they get a higher wage. However, if someone's wage increased from $10/hour to $1000/hour, would they really work more? It is plausible that they would decide to work fewer hours and use their money to do other things. If their wage was $1,000,000/hour, would they work at all? Another example involves apples. Normally, if the price of apples goes up, demand for apples will fall. Consider a family who grows apples to sell. If the price of apples goes up, their income might increase so much that they can afford to consume more of their own apples. In this case, an increase in the price of apples might lead to an increase in the family's consumption of apples. ## The Substitution Effect When the price of a good changes, two effects take place. 1. The rate at which one good can be exchanged for another changes. 2. The total purchasing power of one's income changes. If good 1 becomes cheaper, one needs to give up less of good 2 to purchase good 1. This changes the rate at which the market allows one to substitute good 2 for good 1. The trade-off between goods has changed. Simultaneously, if good 1 becomes cheaper, one's money income can purchase more of good 1. Purchasing power has increased (though the number of dollars remains the same). * The change in demand caused by the change in the rate of exchange between goods is called the **substitution effect**. * The change in demand caused by the change in purchasing power is called the **income effect**. To further analyze these effects, we can decompose price movements into two steps: 1. Change relative prices and adjust money income to hold purchasing power constant. 2. Change purchasing power while holding relative prices constant. This can be illustrated graphically. If the price of good 1 decreases, the budget line rotates around the vertical intercept m/p2 and becomes flatter. This movement can decomposed into a **pivot** and a **shift**. * **Pivot:** The budget line pivots around the original demanded bundle. This is where the slope changes, but the purchasing power of income does not. * **Shift:** The budget line is shifted outward to the new demanded bundle. Here, the slope remains constant, but purchasing power changes. The pivot gives the substitution effect, and the shift gives the income effect. ### Calculating the Substitution Effect We can calculate the substitution effect by first calculating how much income needs to change to keep the old bundle just affordable. Let m' represent this change. We can use the following formula: $Am = x_1 \Delta p_1$ where: * $\Delta p_1$ is the change in price 1 * $x_1$ is the original amount of good 1 consumed The pivoted budget line is the budget line at the new price with the income adjusted by $\Delta m$. The change in income necessary to keep purchasing power fixed will be negative when the price of good 1 decreases, and positive when the price of good 1 increases. The substitution effect represents how the consumer substitutes one good for another when a price changes but purchasing power remains constant. ### Calculating the Income Effect The income effect is the change in demand when income changes while holding prices constant. This can be calculated by changing income from m' to m, while holding the price of good 1 (p1) fixed. To find the income effect: $\Delta x_1 = x_1(p_1,m) - x_1(p_1,m')$ ### The Sign of the Substitution Effect The income effect can be positive or negative, depending on whether the good is a normal or inferior good. The substitution effect is a bit more specific. When the price of good 1 decreases, the change in demand due to the substitution effect must be non-negative. This can be proven using the following logic. * Consider the pivoted budget line and the points on this line where the amount of good 1 consumed is less than at the original bundle X. All these bundles were affordable at the old prices, but were not purchased. * Since the consumer is choosing the optimal bundle at the original budget line, X must be preferred to the bundles that lie underneath the original budget line and on the pivoted budget line. * The optimum choice on the budget line must be either X, or a point to the right of X. * This indicates that the substitution effect will involve consuming at least as much of good 1 as originally. Since the substitution effect moves opposite the price movement, we say that it **is negative**. ### The Total Change in Demand The total change in demand ( $\Delta x_1$ ) is the change in demand due to the change in price while holding income constant. This is the Slutsky identity, which states that the total change in demand equals the sum of the substitution effect and the income effect: $\Delta x_1 = \Delta x_1^s + \Delta x_1^m$ where: * $\Delta x_1^s$ is the substitution effect * $\Delta x_1^m$ is the income effect This is an identity, meaning it is true regardless of the values of the variables. To summarize, the substitution effect must always be negative, opposite the change in price, while the income effect can be either positive or negative. Therefore, the sign of the total change in demand can be positive or negative, depending on the signs of the substitution and income effects. **The Law of Demand:** If the demand for a good increases when income increases, then the demand for a good will decrease when its price increases. Said another way, normal goods have downward-sloping demand curves. ### Examples of Income and Substitution Effects This chapter provides examples of how to decompose price changes for specific utility functions. Cases discussed include: * Perfect complements * Perfect substitutes * Quasilinear preferences The impact of a tax on gasoline is also considered, along with the impact of a rebate. ### The Hicks Substitution Effect The Hicks substitution effect keeps utility constant, as opposed to the Slutsky substitution effect which keeps purchasing power constant. In the Hicks substitution effect, income is adjusted to bring the consumer back to their original indifference curve, as opposed to their original consumption bundle. The Hicks substitution effect is also negative, but is not identical to the Slutsky substitution effect. They are similar for small changes in price. ### Compensated Demand Curves We can graph the relationship between price and quantity demanded while holding one of the following constant: * income * purchasing power * utility Each of these produces a different demand curve: * **Standard demand curve:** income is fixed. * **Slutsky demand curve:** purchasing power is fixed. * **Hicks demand curve:** utility is fixed. Both the Slutsky and Hicks demand curves always have a negative slope. The Hicksian demand curve is sometimes called the compensated demand curve. It is particularly useful in advanced economics courses. ## Summary * When the price of a good decreases, there are two effects: a substitution effect and an income effect * The substitution effect is the change in demand due to relative price changes and holds purchasing power constant. * The income effect is the change in demand due to changes in purchasing power while holding prices constant. * The Slutsky equation states that the total change in demand equals the sum of the substitution effect and the income effect. * The Law of Demand states that normal goods have downward-sloping demand curves.