Lecture: Demand. Demand

Transcription

1 Lecture:

2 Uncompensated By de nition demand tells us how much a consumer is willing to buy at a given price, holding constant other factors (such as tastes and preferences, income, and prices of complements and substitutes). Take the utility maximization problem analyzed earlier. We solved for the optimal quantities as functions of prices and income. That is, we have found the consumer s uncompensated demand functions for these goods: q 1 = Z(p 1, p 2, Y) q 2 = B(p 1, p 2, Y). In the Cobb-Douglas example, the uncompensated demands were q 1 = αy/p 1, q 2 = (1 α)y/p 2. The two goods are neither complements nor substitutes: the demand depends only on the good s own price.

3 Uncompensated (Continued) Suppose that we double all prices and income. What will happen to uncompensated demand? The utility maximization problem becomes max q 1,q 2 U(q 1, q 2 ) s.t. 2p 1 q 1 + 2p 2 q 2 = 2Y. Note that the new budget constraint is equivalent to the original one. So the two problems will have identical solutions. If we multiply all prices and income by some number, uncompensated demand will not change! Consider our Cobb-Douglas example: q 1 = α2y 2p 1 = αy (1 α)2y (1 α)y, q 2 = =. p 1 2p 2 p 2

4 Constructing the Uncompensated Curve Now we use a graphical approach to construct the uncompensated demand curve. If we increase a good s price while holding other prices, tastes and income constant, the budget constraint will rotate. Thus, the optimal consumption bundle will change as the budget constraint rotates. If we connect all optimal bundles in this graph, we will get the price consumption curve. From the price consumption curve we can obtain the demand curve.

5 Constructing the Uncompensated Curve (Cont.) On the graph below, we vary the price of good 1 (beer) Connecting the equilibrium bundles e 1, e 2, e 3 yields the price consumption curve.

6 Constructing the Uncompensated Curve (Cont.) We can translate the price consumption curve from a (q 1, q 2 )-diagram to a (q 1, p 1 )-diagram. This gives us the demand for beer!

7 Constructing the Uncompensated Curve (Cont.) For some preferences the demand curve can be upward sloping! Such goods are called Gi en goods. Good y U 3 Price consumption curve U 2 U 1 P 1 I P x1 P x falls I P x2 I P x3 Good x Price of x P 2 P 3 x 2 x 3 x 1 curve x

8 How Income Changes Shift Uncompensated In an earlier lecture, we argued that a change in income causes a shift of the (uncompensated) demand curve. Now we examine how consumer behavior changes when income changes, while keeping prices and tastes constant. A rise in income leads to a parallel outward shift of the budget line. Thus, as income changes, the optimal bundle will change. If we connect the optimal bundles for all income levels, we get the income-consumption curve.

9 How Income Changes Shift (Continued) On the graph below, we vary the consumer s income. Connecting the equilibrium bundles e 1, e 2, e 3 gives us the income-consumption curve.

10 How Income Changes Shift (Continued) We can plot the optimal quantity of beer in (q 1, p 1 )-space. The change in income shifts the demand curve.

11 How Income Changes Shift (Continued) We can plot the income-consumption curve in a (q 1, Y)-diagram. This will give us the Engel curve.

12 How Income Changes Shift (Continued) It is possible that the income-consumption curve is downward-sloping, and hence the Engel curve could be downward-sloping. y Income consumption curve D C U 4 B U 3 A U 2 BL 2 BL 3 BL 4 BL 1 U 1 x I 4 D Weekly income I I 3 I 2 I 1 C B A Engel Curve x

13 Consumer Theory and Income Elasticities Remember the formula for income elasticity of demand: ξ = Q/Q Y/Y = Q Y Y Q. Some goods have negative income elasticities: ξ < 0. As income increases, we consume less of them. They are called inferior goods. Example of inferior goods: potatoes. As a person s income rises, she tends to consume less of them. Potatoes are replaced with more nutritious food. If ξ > 0, the good is called a normal good. If ξ > 1, the good is called a luxury good. If 1 > ξ > 0, the good is called a necessity.

14 Income-Consumption Curve and Income Elasticities If the income-consumption curve is upward-sloping, all goods are normal. If the income-consumption curve is downward-sloping, one good is normal and the other good is inferior.

15 Compensated Alternatively, we could derive compensated demand functions. Compensated demand shows how the quantity demanded changes as the price rises, holding the utility constant. In contrast with the uncompensated demand, here we vary the consumer s "real" income, so he can maintain a constant level of utility as the price changes. Hence the term compensated (sometimes called Hicksian) demand. The compensated demand system is the solution to the expenditure minimization problem. Its form is q 1 = H(p 1, p 2, Ū) q 2 = M(p 1, p 2, Ū). In our Cobb-Douglas example, the compensated demand is α (1 1 α α)p1 αp2 q 2 = Ū, q 1 = Ū. αp 2 (1 α)p 1

16 Deriving Compensated Graphically Suppose that we vary p x. To maintain the same level of utility, we must also vary Y accordingly! Plot the curve in a (x, p x )-diagram to get compensated demand. p x p x A p x B x 1 x 2 Compensated x

17 E ects of a Price Increase Consider an increase in the price of good 1. This has two e ects on the uncompensated demand of good 1. substitution e ect: the change in the quantity of a good that a consumer demands when the good s price rises, holding other prices and the consumer s utility constant. income e ect: the change in the quantity of a good that a consumer demands due to the change in his "real" income, holding prices xed. The total e ect on uncompensated demand is the sum of the income and the substitution e ects. The substitution e ect always goes in one direction: buy more of the good that becomes cheaper. The income e ect can go either way, depending on whether the good is normal or inferior.

18 Decomposing the E ects Graphically Suppose that the price of good 1 has increased. Thus, the budget constraint rotates inward around the vertical axis. We decompose the total e ect as follows. 1 Find the optimal bundle before the price change. 2 Find the optimal bundle after the price change. 3 Now draw an "imaginary" budget line which is parallel to the new budget line, but tangent to the old indi erence curve. The change between the two parallel lines is the income e ect. The change due to rotating the budget line around the old indi erence curve is the substitution e ect.

19 Decomposing the E ects Graphically (Continued) The e ects are illustrated on the graph below:

20 Numerical example Suppose that the consumer s utility is U(q 1, q 2 ) = q q We know that the corresponding uncompensated demands are q 1 = 0.6 Y p 1, q 2 = 0.4 Y p 2. Assume that p 1 = 15, p 2 = 20, Y = 300. Then the price of good 1 increases to p 0 1 = 30. Initially the consumer buys /15 = 12 units of good 1 and /20 = 6 units of good 2. After the price increase she buys /30 = 6 units of good 1 and 6 units of good 2. Her consumption of good 1 decreases by 6 units (from 12 to 6). How much of this change is due to the income e ect and how much due to the substitution e ect?

21 Numerical example Before the price increase, the consumer s utility was Ū = After the price increase, how much income Y does the consumer need to have to attain the old level of utility Ū? This income would solve the equation 0.6Y 30 Solving for Y yields Y = Y 0.4 = If the consumer had income Y = 450, at the new price p1 0 he would buy /30 = 9 units of good 1. The decrease from 12 to 9 is due to the substitution e ect. We have maintained the "real" income constant, so the consumer s level of utility is unchanged. The remaining change is due to the income e ect.

22 Price Changes and Normal Goods Suppose the price of a normal good increases (as in slide 19). This good would become relatively more expensive. Thus, the substitution e ect says that we should consume less of it. Furthermore, the consumer s real income decreases (the old consumption bundle becomes una ordable). Since the good is normal, the income e ects says that we should consume less of it. Thus, both e ects work in one direction: consume less. The demand curve of a normal good is necessarily downward-sloping!

23 Price Changes and Inferior Goods Next suppose that the price of an inferior good increases. Then this good would become relatively more expensive. Thus, the substitution e ect says that we should consume less of it. Furthermore, the consumer s real income decreases (the old consumption bundle becomes una ordable). Since the good is inferior, the income e ects says that we should consume more of it. The two e ects work in di erent direction! The overall e ect depends on which e ect is stronger.

24 Price Changes and Inferior Goods (Continued) Suppose that the substitution e ect is stronger. Then demand will still be downward-sloping! y B A U 1 C BL d Income effect BL 1 x B x C x A BL 2 U 2 x

25 Price Changes and Inferior Goods (Continued) What if the income e ect is stronger? Then we have a Gi en good: demand is upward-sloping! All Gi en goods are inferior. But the converse is not true! y U 1 Income effect U 2 x

26 Changes in Compensated versus Uncompensated s Note that the substitution e ect is actually the change in compensated demand we get if we x the utility at the old level! If we set the consumer s income to be equal to the minimized expenditure, the compensated demand will be the same as the uncompensated demand. Next consider a decrease in the price of good 1. If the good is normal, the income e ect will increase consumption. Therefore, the compensated demand will increase by less than the uncompensated (will be steeper). If the good is inferior, the income e ect will decrease consumption. The compensated demand will increase by more than the uncompensated (will be atter).

27 Compensated vs Uncompensated s (Continued) Illustration of a normal good:

28 The Slutsky Equation The relationship between the total change in demand, the income and the substitution e ects can be demonstrated mathematically. This relationship is known as the Slutsky equation: D p 1 H p 1 D = H p 1 p 1 D Y q 1. is the change in the uncompensated demand (the total change). is the change in the compensated demand (the substitution e ect) D Y q 1 is the income e ect. We can rewrite the Slutsky equation as ε = ε where ε is the price elasticity of demand, ε is the elasticity of compensated demand, ξ is the income elasticity of demand, and θ = p 1 q 1 /Y is the expenditure share. θξ,

29 Remember the numerical example with U(q 1, q 2 ) = q q0.4 2, p 1 = 15, p 0 1 = 30, p 2 = 20, Y = 300. The uncompensated demands are q 1 = 0.6 Y p 1, q 1 = 0.4 Y p 2. The elasticity of uncompensated demand for good 1 is The income elasticity of good 1 is ε = q 1 p 1 = 0.6 Y p 1 p 1 p 1 q 1 p Y = 1. 1 ξ = q 1 Y = Y p 1 Y q 1 p Y = 1. The compensated demands are q 1 = 3p2 0.4, Ū 2p q1 1 = 2p Ū 3p 2 The elasticity of compensated demand for good 1 is ε = q 1 p p2 3p = 2 p Ū 1 2p1 = 0.4. p 1 q 1 2p 1 3p 2 The market share is θ = = 0.6. The Slutsky equation holds! 2p 2 1