Income and Substitution Effects PDF

7 Income and Substitution Effects It may not be clear at this point why two different concepts of demand curves are useful. Both show a downward sloping demand curve if we plot price vs quantity, so why do we need two? It turns out that Marshallian and Hicksian demands behave slightl...

7 Income and Substitution Effects It may not be clear at this point why two different concepts of demand curves are useful. Both show a downward sloping demand curve if we plot price vs quantity, so why do we need two? It turns out that Marshallian and Hicksian demands behave slightly differently for changes in the price of one of the goods. These notes will take a deeper look at how price changes affect consumer demand. Concepts Covered Effect of a Price Change on Marshallian Demand Decomposing Price Changes The Slutsky Equation Elasticity 7.1 Effect of a Price Change on Marshallian Demand We will stick to a utility function over two goods U = U (x, y) We have shown that we can solve for the Marshallian demand for x and y as a function of income and prices x∗m = x(px , py , I) ∗ ym = y(px , py , I) Here we are interested in how these demands react to a change in one of the prices. We will focus on how a change to px affects the demand for good x. In other words, assume the price of x jumps from px to some p0x. Then we are interested in x∗m (p0x , py , I) − x∗m (px , py , I) We could also represent the rate of change (change for a very small change in price) in demand with respect to price using the derivative of Marshallian demand with respect to price. ∂x∗m ∂px We will call this change in the Marshallian demand the total effect of a price change. 1 7.2 Decomposing Price Changes Although the analysis above gave us the total effect on the quantity demanded when we change price, we can go a little bit further to explain why this change occurs. There are essentially two components to the total price change in Marshallian demand, which we will call the substitution effect and the income effect. Substitution Effect First, since the relative price of x to y has changed, the consumer will be better off substituting towards the cheaper good. Even if we kept the overall level of utility fixed, the consumer would prefer to choose a different point on the indifference curve since relative prices have changed. The graphs below plots this idea using a utility function U = x1/2 y 1/2 with Ū = 4, an initial income of 16, and an increase in the price of x from 1 to 4 (price of y is fixed at 4 throughout). 10 10 8 8 6 6 y y B 4 4 A A 2 2 4 4 0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 x x We can see from the graph on the left (or by plugging into Marshallian demand), that the optimal consumption before the price change occurs at the bundle x∗ = 8, y ∗ = 2 (note that these are also the Hicksian demands when Ū = 4). This point is marked as point A on the graph above. When we change the price of x from 1 to 4, the relative price of x to y changes from 1/4 to 1, so the slope of the budget constraint gets steeper. However, let’s assume we want to stay on the same indifference curve, so we will look for an expenditure line that has a slope of 1, but is still tangent to our indifference curve (i.e. keeping utility at the maximum fixed at 4). Since the price of x went up, staying on the same indifference curve in the least expensive way possible means consuming less x and more y, which results in the point (4, 4), marked by B on the graph. Let’s think about the exercise we just did. We held utility fixed at 4. Then, given prices, we calculated the quantity of x and y that would give us the U = 4 in the cheapest possible way, which is simply calculating Hicksian demand. In other words, what we have done is to find x∗h (px = 4, py = 4, Ū = 4) − x∗h (px = 1, py = 4, Ū = 4) 2 We will call this effect the substitution effect since it captures the change in quantities that comes from substituting x for y while keeping utility constant. More generally, we could write the substitution effect as xh∗ (px0 , py , Ū ) − xh∗ (px , py , Ū ) Where px is the original price of x and px0 is the new price and Ū is set to the value of the indirect utility function at the initial income and prices. If the price change were small enough, we could have calculated it as ∂xh∗ ∂px Note that while this derivative will be in terms of Ū , we should substitute in the indirect utility function for Ū to get it in terms of I instead. Income Effect Notice that the allocation we got above is significantly more expensive than the original. At the old allocation with the old prices, the consumer was spending px x + py y = (1)(8) + 4(2) = 16 and at the new allocation with the new prices, they were spending (4)(4) + (4)(4) = 32. To stay at the same utility, the consumer would need to spend twice as much as they did before. However, the consumer only has an income of 16. It is not feasible for them to remain at the same utility after the price change. Their income relative to the prices of the two goods has fallen, so the total utility they receive falls. In this case, if we solved for the utility at the optimal consumption, it would be U = 2. On the graph, we could show this change as 10 10 8 8 6 6 y y B B 4 4 A C A 2 2 4 4 0 0 2 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 x x Intuitively, if goods are becoming more expensive, we will not be able to maintain the same utility level. While point B shows us the optimal ratio of x and y given the new prices, it is not feasible given the initial income. The best the consumer can do is instead point C. We call this change the income effect. 3  We can calculate the income effect by taking the difference between the total effect and the substitution effect (x∗m (p0x , py , I) − x∗m (px , py , I) − (x∗h (p0x , py , Ū )) − x∗h (px , py , Ū )) Where again px is the original price of x and p0x is the new price and Ū is the value of the indirect utility function However, note that at the initial point, Marshallian demand and Hicksian demand are equal to each other (since we chose Ū as the utility level that corresponds to the maximum utility given our initial income). In other words x∗m (px , py , I) = x∗h (px , py , Ū ) so we can simplify the income effect to x∗m (p0x , py , I) − x∗h (p0x , py , Ū ) Which is just the Marshallian demand at the new prices minus the Hicksian demand at the new prices (but again, importantly, the Ū here must be chosen based on the utility received at the original prices). For a small change in price, we can again use a derivative to represent the income effect. The derivation requires some subtlety, so we will just present it as a definition as −∂x∗m ∗ x ∂I m Intuitively, when prices increase, it can be interpreted as a decrease in the consumer’s income (which explains the negative sign in front). Note that it is theoretically possible for x to change in either direction as income falls. If quantity of x rises as income increases, we call x a normal good and if the quantity of x falls as income increases, we call x an inferior good. Summary We decomposed the total effect of a price change into two components: the income effect and the substitution effect. To sum up what we said above Total Effect: The change in consumption when price changes and income stays constant ∂x∗m x∗m (p0x , py , I) − x∗m (px , py , I) or, for a small change in price, ∂px Substitution Effect: The change in consumption when price changes and utility stays constant ∂x∗h x∗h (p0x , py , Ū ) − x∗h (px , py , Ū ) or, for a small change in price, ∂px Income Effect: The difference between the total and substitution effects ∂x∗ x∗m (p0x , py , I) − x∗h (p0x , py , Ū ) or, for a small change in price, − m x∗m ∂I 4 7.3 The Slutsky Equation The Slutsky equation says that Total Effect = Substitution Effect + Income Effect ∂x∗m ∂x∗h ∂x∗m ∗ = − x ∂px ∂px ∂I m Let’s prove the Slutsky equation works for our utility function U = x1/2 y 1/2. First, recall that Marshallian and Hicksian demand are given by 1 I x∗m = 2 px  1/2 py x∗h = Ū px The total effect is ∂x∗m 1 I =− 2 ∂px 2 px The substitution effect is ∂x∗h 1 = − Ū p1/2 p−3/2 ∂px 2 y x And the income effect is ∂x∗m ∗ 1 I − xm = − 2 ∂I 4 px We need one more step to show the relationship. Since the substitution effect depends on Ū , but the others depend on income, we will substitute in the indirect utility function for Ū. The indirect utility function (shown in the last notes) is 1 V = Ipx−1/2 p−1/2 y 2 Plug this into the substitution effect result to get 1 1 1 I − ( Ipx−1/2 p−1/2 y )p1/2 y px −3/2 =− 2 2 2 4 px Adding the income and substitution effects we get 1 I 1 I 1 I − + − = − 4 p2x 4 p2x 2 p2x Which is exactly the same as our original total effect. 5 7.4 Elasticity One final way that we can analyze the effect of price changes is to measure how large these price changes are. Formally, we define the elasticity of a good with respect to its own price (the “own-price elasticity”) as ∂x∗m px ex,px = ∂px x∗m Often this elasticity is reported as a positive number (absolute value) even though it is usually negative (if price of a good goes up, the quantity purchased goes down). Elasticity is a useful way of helping to standardize the size of the effect of a price change. For example, let’s say that an increase in price of $1 causes a decrease in consumption of 10 units. Is that a big response? To answer that question, we need to know two things. First, how large is a price increase of $1? If the initial price of the good was $2, then a $1 increases represents a 50% increase, a pretty big change. But if the initial price was $1000, then a $1 increase is only a 0.1% increase, not much change at all. Similarly, we need to know whether 10 units is a large decrease or not. Again, if the initial consumption is small (say 20 units) this could be a big decrease in quantity, but if the initial consumption is high (1000 units), it could be a relatively small change. Elasticity solves this problem by using the initial values to standardize the elasticity. To correct for the initial price and quantity, we multiply the change by the ratio pxx. Making this standardization also makes the elasticity unitless so the number itself does not have a tangible meaning. Instead, we mostly want to use elasticity as a relative concept. Higher numbers mean quantity responds a lot when prices change. When a good changes a lot in response to price, we say it is elastic and when it changes very little it is inelastic. We can also define the cross price elasticity as the sensitivity to changes in the prices of other goods ∂x∗m py ex,py = ∂py x∗m And the income elasticity as the change in quantity with respect to income ∂x∗m I ex,I = ∂I x∗m 6