# Demand. Lecture 3. August Reading: Perlo Chapter 4 1 / 58

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1 Demand Lecture 3 Reading: Perlo Chapter 4 August / 58

2 Introduction We saw the demand curve in chapter 2. We learned about consumer decision making in chapter 3. Now we bridge the gap between the two concepts. In this lecture, we see how the demand curve is derived. 2 / 58

3 Outline Deriving demand curves - Use consumer theory to see how a change in price causes a movement along demand. E ects of an increase in income - How does an income change a ect demand? E ects of a price increase - How does the change in price a ect demand? Cost-of-Living Adjustment - How do policy makers measure price changes? Revealed preferences - We can recover an individual s preferences by observing their demand. 3 / 58

4 Deriving the Demand Curve The demand curve plots quantity demanded against the price. If we want to draw my demand curve for beer, we need to nd my optimal consumption of beer for di erent prices. We can do this derive demand graphically or analytically. 4 / 58

5 Deriving the Demand Curve We do it graphically rst. The optimal bundle occurs where the indi erence curve is tangent to the budget constraint. The intercepts of the budget constraint for goods q 1 and q 2 are Y p 1 and Y p 2. If the price of a good 1 decreases, the budget constraint will rotate out and we have a new optimal bundle. 5 / 58

6 Deriving the Demand Curve The price consumption curve traces the optimal consumption bundles for di erent prices. 6 / 58

7 Deriving the Demand Curve I drink 26.7 gallons of beer when the price of beer is \$12, 44.5 gallons when it is \$6, and 58.9 gallons when it is \$4. Just plot this points in price of beer and gallons of beer consumed space and we have demand. 7 / 58

8 Deriving the Demand Curve 8 / 58

9 Deriving the Demand Curve EXAMPLE Let s graphically derive the demand curve for wine (the good on the vertical axis). 9 / 58

10 Deriving the Demand Curve We can do the same thing using math. We want to nd the optimal consumption of a good for di erent prices. 10 / 58

11 Deriving the Demand Curve If we have a Cobb Douglas utility function U(q 1, q 2 ) = (q 1 ) a (q 2 ) 1 α Knowing that the optimal bundle occurs where MRS = p 1 p 2 nd the demand curves. The marginal rate of substitution is we can MU 1 = U/ q 1 = a (q 1) a 1 (q 2 ) 1 α MU 2 U/ q 2 (1 a) (q 1 ) a (q 2 ) α = a q 2 1 a q 1 11 / 58

12 Deriving the Demand Curve Set this equal to the price ratio to nd the optimal consumption ratio. a q 2 = p 1 1 a q 1 p 2 ) a 1 a q 2 q 1 p 2 = p 1 Plug this into the budget constraint Y = q 1 p 1 + q 2 p 2 and solve for q 2. q 2 = (1 a) Y p 2 We would solve for p 2 and plug that into the budget constraint to nd demand for q / 58

13 EXAMPLE Find the demand curve for good x 1 for the utility function U = x 1 x / 58

14 EXAMPLE Find the demand curve for good x 1 for the utility function U = ln (x 1 ) + x 2. What does this tell you about quasi-linear utility functions? 14 / 58

15 EXAMPLE Find the demand curve for x 1 for the utility function U = minfx 1, x 2 g. 15 / 58

16 EXAMPLE Find the demand curve for x 1 for the utility function U = maxfx 1, x 2 g. 16 / 58

17 E ect of an Increase in Income If the price of beer changes, we move along the demand curve for beer. If something other than the price of beer changes (which a ects how much beer you drink) the demand curve shifts. Now lets look at what happens when our income changes 17 / 58

18 E ect of an Increase in Income a change in income leads to a parallel shift in the budget constraint (the intercepts are Y p 1 and Y p 2, leading to a new optimal bundle). The income consumption curve plots the optimal consumption levels of beer and wine for di erent levels of income. 18 / 58

19 E ect of an Increase in Income Our income is higher. At every price we will now demand more beer so our demand curve shifts right. 19 / 58

20 E ect of an Increase in Income It is also useful to draw the Engel curve, which shows the relationship between quantity demanded of a single good and income. 20 / 58

21 E ect of an Increase in Income We mentioned income elasticity of demand before, this tells us the shape of the income consumption curve and the Engel curve. Income elasticity of demand ξ tells us the percentage change in quantity demanded to a given percentage change in income. ξ = % Q % Y = Q/Q Y /Y = Q Y Y Q and as Q and Y go to zero we just use calculus ξ = Q Y Y Q 21 / 58

22 E ect of an Increase in Income EXAMPLE Find the income elasticity of demand for the demand curve where Y = 20 is income and q 2 = q 2 = 100Y 10p 1 22 / 58

23 E ect of an Increase in Income Is this always positive? That is, does any increase in income always lead to an increase in quantity demanded? Of course not, there are some goods you only consume when you are poor. 23 / 58

24 E ect of an Increase in Income If ξ > 0, a good is a normal good, we demand more of it when income rises. If ξ < 0, it is an inferior good, we demand less of it when income rises. if ξ > 1, we say it is a luxury good, quantity demanded rises more in proportion to a person s income. If 0 ξ 1 it is a necessity. 24 / 58

25 E ect of an Increase in Income Books ξ = 1.44 Restaurant Meals ξ = 1.40 Tobacco ξ =.64 Public Transportation ξ =.36 Automobiles ξ = / 58

26 E ect of an Increase in Income The shape of the income consumption curve is determined by income elasticities. Income increases, do we consume more or less of a good? 26 / 58

27 E ect of an Increase in Income EXAMPLE Is x 1 a normal good or an inferior good if the demand for x 1 is x 1 = (1 a) Y p 1 27 / 58

28 E ect of an Increase in Income EXAMPLE Draw the income consumption curve and the Engel curve for a good which is normal for low levels of income and inferior for high levels of income. 28 / 58

29 E ect of an Increase in Income It is not possible for all goods to be inferior. If both goods are inferior, the consumer would purchase less of both goods as income goes up. This new basket would like in her original budget set, this violates the axiom of revealed preferences. 29 / 58

30 E ects of a Price Increase What exactly happens when prices change? We can break a price change down into the income e ect and the substitution e ect. Income e ect + substitution e ect = total e ect of a price change. 30 / 58

31 E ects of a Price Increase Suppose we have apples and oranges, and the price of apples decreases. Apples are now relatively cheaper than oranges. If we were to hold utility constant, the consumer will always substitute towards the relatively cheaper good. The substitution e ect always works in this direction. 31 / 58

32 E ects of a Price Increase BUT, the change in price changes the consumers purchasing power. If the price of apples went down, it is just as if I am richer. I can use the leftover money to purchase more apples or more bananas. The change in quantity demanded as a result of the change in purchasing power, holding relative prices constant, is the income e ect. We don t always know which way this works. 32 / 58

33 E ects of a Price Increase Recall that if a good is inferior, an increase in income will cause you to consume less of that good. If a good is normal, an increase in income will cause you to consume more of that good. 33 / 58

34 E ects of a Price Increase Notice the income and substitution e ect go the same way... the good below is normal. 34 / 58

35 E ects of a Price Increase With an inferior good, you eat less of it when your income goes up (and more when it goes down). The income and substitution e ects work in opposite directions. 35 / 58

36 E ects of a Price Increase EXAMPLE Lets show the income and substitution e ects when a good is inferior. 36 / 58

37 E ects of a Price Increase Suppose the price of q 1 decreases. The substitution e ect will cause you to consume more q 1. If q 1 is inferior, the income e ect will cause you to consume less q 1. If the income e ect causes you to consume less of q 1 than you did initially, this good is a Gi en good. 37 / 58

38 E ects of a Price Increase That is, a decrease in the price will cause you to consume less of that good if it is Gi en. Gi en goods violate the law of demand (their demand curves slope upwards). Potatoes during the Irish potato famine were an example of a Gi en good. The price of potatoes went up and people started consuming more potatoes. 38 / 58

39 E ects of a Price Increase EXAMPLE Lets draw the income and substitution e ects for a Gi en good. 39 / 58

40 E ects of a Price Increase At the start of the lecture, we derived the Marshallian demand. The Marshallian demand curve shows the total e ect of a price change (both the income and substitution e ect). Compensated (or Hicksian) looks at the change in demand from a price change resulting only from the substitution e ect. Utility is constant at all points on the compensated demand curve. We cannot observe compensated demand because we cannot observe utility levels. 40 / 58

41 E ects of a Price Increase 41 / 58

42 E ects of a Price Increase The law of demand must hold for compensated demand curves. Why? If the good is normal, the uncompensated demand curve will be shallower because the income e ect reinforces the substitution e ect. If the good is inferior, the uncompensated demand curve will be steeper because the income e ect and substitution e ect work in opposite directions. 42 / 58

43 E ects of a Price Increase EXAMPLE Graphically derive the compensated and uncompensated demand curves for an inferior good. 43 / 58

44 E ects of a Price Increase EXAMPLE Will the compensated demand curve of a Gi en good slope downwards? Why? 44 / 58

45 E ects of a Price Increase We saw price changes broken down into income and substitution e ects on a graph. Now lets do it with math. This is called the Slutsky equation. 45 / 58

46 E ects of a Price Increase We want to know how demand changes when the price changes. We are looking for ε, the price elasticity of demand. The price elasticity of demand can be broken down into the income e ect and the substitution e ect. 46 / 58

47 E ects of a Price Increase ε the substitution elasticity of demand, the percentage change in demand for a given increase in price compensating the consumer to keep utility constant. (substitution e ect) ξ θ, which is the income elasticity of demand ξ times the share of budget spent on the good θ. ε = ε + ( θ ξ) The larger is the θ, the larger is the income share of the good. does this tell you about Gi en goods? What 47 / 58

48 E ects of a Price Increase EXAMPLE You spend 3 4 of your income on potatoes. demand for potatoes is 4. The substitution elasticity of demand is 1. What can you say about potatoes? The income elasticity of 48 / 58

49 Cost of Living Adjustments The price of goods rise over time, and we have seen what consumers do when the price of goods rises. It is important to have an accurate measure of price changes. Think of long-term employment contracts which tract some measure of price changes. The government frequently uses the consumer price index (CPI), which measures the cost of a given bundle of goods over time. 49 / 58

50 Cost of Living Adjustments The consumer price index identi es a "typical" basket of goods. For example, the basket of goods can be 3 bananas, 3 oranges and a car. The CPI measure the price of this basket of goods over time. 50 / 58

51 Cost of Living Adjustments Suppose the price of oranges increases much faster than the price of bananas. A contract which tracks the CPI would compensate you for the increase in the price of bananas... but ignores the fact that people will substitute towards oranges. The CPI overcompensates people for in ation by ignoring the fact that substitution towards cheaper goods occurs. 51 / 58

52 Cost of Living Adjustments We need a cost-of-living adjustment that takes the substitution into account. Utility is what we want to keep constant over time. Should give people just enough more money such that their utility remains the same. 52 / 58

53 Revealed Preferences It is never possible to exactly see people s preferences. Preferences can be recovered, however, by observing purchasing behaviour of individuals. Recovering preferences from observed behaviour is known as the theory of revealed preferences. 53 / 58

54 Revealed Preferences For example. Suppose bundle a is more expensive than bundle b. If the consumer purchases bundle a, we can say that this individual prefers bundle a to bundle b. Both a and b were available, and the consumer picked a. As we observe the consumer s behaviour in more situations, we can learn more about her preferences. 54 / 58

55 Revealed Preferences 55 / 58

56 Revealed Preferences EXAMPLE The consumer faces budget line BL1 yesterday and chose bundle a. The consumer faced budget line BL2 today and chose bundle b. Use the concept of revealed preferences to show she is better o today than yesterday. 56 / 58

57 Summary How do you derive the demand curve? What is the income consumption curve and the price consumption curve? What is the Engel curve? What is the substitution e ect and income e ect? 57 / 58

58 Summary What is an inferior good? What is a Gi en good? What is the di erence between compensated and uncompensated demand? What is the Slutsky equation? Why does the CPI overcompensate for price changes? What are revealed preferences? 58 / 58

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