Comparative Statics: Analysis of Individual Demand and Labor Supply

1. Comparative Statics: Analysis of Individual Demand and Labor Supply Chapter 4

2. Introduction • Rational households are never quite able to find local bliss • Ever-changing prices and income require households to continuously adjust their commodity bundle • Can study these changes by comparing one equilibrium position to another • Comparative statics analysis • Investigates a change in some parameters holding everything else constant • Called ceteris paribus • With preferences held constant, individual indifference curves remain fixed • Comparative statics is concerned with sensitivity of a solution to changes in parameters • Will derive a household’s demand curve for each commodity

3. Introduction • Will investigate a change in income, holding all prices constant • Develop Engel curves and Engel’s Law associated with income changes • Based on shapes (slopes) of demand and Engel curves • Commodities are generally classified as normal, luxury, or inferior goods in terms of income change • Ordinary or Giffen goods for price change • Discuss how Slutsky equation considers total effect of a price change as shown to be sum of substitution effect and income effect • Illustrate theoretical possibility of a positively sloping demand curve • Giffen’s Paradox • Discuss use of Slutsky equation to measure compensated price changes and Laspeyres Index to measure Consumer Price Index (CPI) • Extend Slutsky equation to changes in price of another commodity • Develop concepts of gross and net substitutes and gross and net complements

4. Chapter Objective • To derive a household’s demand functions for commodities it purchases and its labor supply function • Quantity demanded should generally decline as price of a commodity increases • Demand should generally increase with a rise in income • We investigate underlying determinants for this response of quantity demanded

5. Introduction • Determinants of a household’s supply of labor given a rise in wages may result in supply increasing, declining, or remaining unchanged • Can develop aggregate (market) demand and supply • Market supply and demand functions will provide a foundation for investigating efficient allocation of society’s resources • Applied economists estimate consumer demand and labor supply functions • To determine how responsive consumers and labor are to changes in prices, wages, incomes, sales promotion, and various government programs • Consumer demand is a very large area in economics • Labor economics is also a large area • Impacts of government welfare programs, working conditions, exploitation, and unions

6. Derived Household Demand • With indifference curves representing preferences and budget lines as income constraints, • Can derive a theoretical relationship between price and a person’s quantity demanded • Consider case of a price change in one of two commodities x1 and x2 • Budget line is • I = p1x1 + p2x2 • Where I, p1, and p2 represent income, per-unit price of commodity 1, and per-unit price of commodity 2, respectively • Figure 4.1 is a graph of budget line • With an x1-intercept of 5 units and an x2-intercept of 10 units

7. Figure 4.1 Derived demand for a decrease in p1

8. Derived Household Demand • A household maximizes its utility for a given level of income • At a point on budget line tangent with an indifference curve (commodity bundle A) • Equilibrium bundle (at a given income level and prices) corresponds to one point on household’s demand curve (point a) • Additional points on household’s demand curve for x1 are obtained • By changing price of commodity x1 while holding income and price of x2 constant • For example, decreasing price of x1 from 2 to 1 (p1 = 1) results in budget line tilting outward • With this new price for p1, commodity bundle C represents new equilibrium level of utility maximization

9. Derived Household Demand • Further changes in p1 will result in additional tangencies of a budget line • With an indifference curve and a corresponding point on household’s demand curve • Connecting points results in household’s demand curve for commodity x1 • Each point on demand curve corresponds with a tangency point between indifference curve and budget line • Household is maximizing utility for a given income level and market price of commodity x1 • Illustrates how much of the commodity a household is willing and able to purchase at a given price • As price of p1 declines household’s MRS(x2 for x1) declines • How much it is willing to pay for an additional unit of x1 also declines • A decline in price results in an increase in a household’s level of utility • Household’s purchasing power has increased as a result of this price decline

10. Shift in Demand versus a Change in Quantity Demanded • Convenient to graph x1 as a function of its own price • With understanding that income and all other prices are being held constant • As illustrated in Figure 4.1, assuming two commodities and considering a change in p1, then x1 = x1(p1|p2, I) • Where p2 and I are being held constant • By varying p1, price consumption curve traces out locus of tangencies • Between budget line and indifference curve • Negatively sloped demand curve can be derived from this price consumption curve • Decrease in p1 will result in an increase in quantity demanded (a movement along demand curve) • A change in p2 or I will shift demand curve

11. Shift in Demand versus A Change in Quantity Demanded • Figure 4.2 illustrates difference in a change in quantity demanded versus a shift in demand • At bundle A, a decrease in p2 results in a movement from bundle A to B • Shifts demand curve • Shift depicted as a shift from point a on demand curve x1( p1| p2°, I) to point b on x1( p1| p'2,I ) • A decrease in p1 at bundle A • Results in a movement from bundle A to C • Causes a movement along demand curve x1( p1|p2°, I ) from point a to c • Change in quantity demanded • Change in either of the variables on axes causes a movement along a curve • Whereas change in any factor not on one of the axes causes a shift in curve • For example, a change in income or preferences will shift a demand curve

12. Inverse Demand Curves • Demand functions depicted in Figures 4.1 and 4.2 are sometimes called inverse demand functions • Dependent variable is on vertical axis and independent variable is on horizontal axis • Price as dependent variable states what level of quantity demanded for a commodity would have to be for household to be willing to pay this price per unit • Inverse demand functions represent price as a function of quantity demanded • As opposed to quantity demanded as a function of price

13. Figure 4.2 Shift in demand versus change in quantity demanded

14. Generalizing For K Commodities • In general we can solve for optimal levels of x1*, x2*, … , xk* and * as functions of all parameters (prices and income) • Quantities of x1, x2, … , xk demanded by the household will depend on • Shape of utility function (consumer preferences) • p1, p2, … , pk and I • Demand functions state how much a household is willing and able to consume of a commodity at given prices and income • Mathematically, demand functions are represented as

15. Homogeneous of Degree Zero Demand Functions • In many developing countries price and income indexing occurs • Due to high rates of inflation • For example, if inflation is running at 10% annually, incomes are automatically adjusted (indexed) upward by 10% • Keeps households’ purchasing power the same • Does not change their demands for commodities • Assuming no money illusion • When all prices and income change proportionately, optimal quantities demanded would remain unchanged • Slope and intercepts of budget constraint do not change • Illustrated in Figure 4.3

16. Figure 4.3 Homogeneity of demand functions

17. Homogeneous of Degree Zero Demand Functions • Generally, if prices and income are multiplied by some positive constant a, same budget constraint remains • Given no change in budget constraint from multiplying prices and income by  > 0 • Quantity demanded by a household will also not change • Called homogeneous of degree zero • In this case, consumer demand functions are homogeneous of degree zero in all prices and income. • Household demands are not affected by pure inflation

18. Numeraire Price • Result of homogeneous of degree zero demand functions • Can divide all prices and income by one of the prices • Demand for a commodity depends on • Price ratios (called relative prices) • Ratio of money income to a price (called real income) • Picking any price, say p1, and multiplying demand function by 1/p1 gives • xj = xj(p1, p2,…, pk, I) = xj(1, p2/p1,…, pk/p1, I/p1) • Where a is 1/p1 • Setting p1 = 1 • Which is relative price to which all other prices and income are compared • Called numeraire price

19. Changes in Income • A college graduate’s income will generally substantially increase upon landing that first professional job • Results in a change in purchasing power • An increase in income results in an expected increase in purchases • Represented by parallel shifts in budget lines • Only I has changed, so price ratio remains constant • Income consumption path, or income expansion path • Curve intersecting all points where indifference curves are tangent with budget lines (locus of utility-maximizing bundles) • Every point on path represents demanded bundle at that level of income

20. Engel Curves • From income expansion path, can derive a function that relates income to demand for each commodity at constant prices • Represented by Engel curves as illustrated in Figure 4.4 • As income rises from I1 to I2 and then to I3 • Demand for x1 increases from x11 to x12 and then to x13 • Plotting this increased demand with rise in income yields an Engel curve • Illustrate a relationship between demand for a commodity and income • In Figure 4.4, Engel curve has a positive slope • However Engel curves can have either positive or negative slopes • Positively sloped Engel curves are called normal goods • An increase in income results in more of commodity being purchased

21. Figure 4.4 Income expansion path and Engel curve …

22. Homothetic Preferences • If income expansion paths, and thus each Engel curve, are straight lines (linear) through origin • Household will consume same proportion of each commodity at every level of income • Assumes prices are held fixed • Homothetic preferences • Preferences resulting in consuming same proportion of commodities as income increases • Commodities are scaled up and down in same proportion as income changes

23. Luxury and Necessary Goods • Can further divide normal goods into luxury and necessary goods • If income expansion path bends toward one commodity or the other • Figure 4.5 illustrates income expansion path bending toward commodity x1 • Making x1 a luxury good • x1 is a luxury good if (x1/I)I/x1 > 1 • As income increases, household spends proportionally more of its income on x1 • Examples: fine wines and silk suits • Necessary good • If, as income increases, household spends proportionally less of its income on a commodity • Examples are gasoline and textbooks

24. Figure 4.5 Income expansion path and Engel curve for a luxury good, x1

25. Luxury and Necessary Goods • As household receives more income, it wishes to consume more of both types of commodities • But proportionally more of luxury good than of necessary good • For two-commodity case, if one commodity is a luxury good • Other must be a necessary good

26. Inferior Goods • Negatively sloped Engel curve is associated with a backward-bending income expansion path • With an increase in income, a household actually wants to consume less of one of the commodities • In Figure 4.6, as income increases • Consumption of x1 declines • Called an inferior good and is defined as x1/I < 0 • Examples include cheap wine and used books

27. Figure 4.6 Income expansion path and Engel curve for an inferior good x1

28. Engel’s Law • Relationship between income and consumption of specific items has been studied since 18th century • Engel was first to conduct such studies • Developed generalization about consumer behavior • Proportion of total expenditure devoted to food declines as income rises • Has been verified in numerous subsequent studies • Engel’s Law appears to be such a consistent empirical finding that some economists have suggested proportion of income spent on food might be used as an indicator of poverty • Families that spend more than say 40% of their income on food might be regarded as poor

29. Changes in Price • If p1 is allowed to vary holding p2 and I fixed, budget line will tilt • Illustrated in Figure 4.7 • Locus of tangencies will sweep out a price consumption curve • Curve connecting all tangencies between indifference curves and budget lines for alternative price levels

30. Figure 4.7 Price consumption curve and demand curve …

31. Ordinary Goods • Price consumption curve for an ordinary good is illustrated in Figure 4.7 • Where xj ÷ pj < 0 defines an ordinary good • Demand curve derived from price consumption curve has a negative slope • Indicates inverse relationship between a commodity’s own price and quantity consumed

32. Giffen Goods • Slope of a demand curve could be positive • xj ÷ pj > 0 • Defines a Giffen good • Decrease in p1 results in a decrease in demand for x1 • Shown in Figure 4.8

33. Figure 4.8 Price consumption curve for a Giffen good

34. Substitution and Income Effects • Determinants of whether a commodity is ordinary good or Giffen good • Depend on direction and magnitude of substitution and income effects • Figure 4.9 shows effects for case of an own price change where p1 decreases • Initial budget constraint • I = p1x1 + p2x2 • New budget constraint • I = p‘1x1 + p2x2 • Where p'1 < p°1 • Price decrease in p1 results in increased quantity demanded of x1 • Shown in Figure 4.9 • Increase in quantity demanded is total effect of price decline • Total effect = x1/p1 < 0 • Can be decomposed into substitution and income effects

35. Figure 4.9 Substitution and income effects …

36. Substitution Effect • In 2002, automobiles in Canada generally cost from 20% to 35% less than in US • With fall of trade barriers and harmonizing of environmental and safety regulations • Only major differences between new cars made for sale in Canada and those made for sale in U.S. • Speedometers and odometers

37. Substitution Effect • U.S. automobile buyers attempt to substitute Canadian cars for U.S. ones • Illustrates substitution effect (also called Hicksian substitution) • As an incentive for consumers to purchase more of a lower-priced commodity (Canadian cars) and less of a higher-priced commodity (U.S. cars) • Given a change in price of one commodity relative to another • To determine substitution effect • Hold level of utility constant at initial utility level, U° • Consider price change for x1 • If a household were to stay on same indifference curve • Consumption patterns would be allocated to equate MRS to new price ratio

38. Substitution Effect • Represented in Figure 4.9 by a budget line parallel to new budget line • But tangent to initial indifference curve • Point B, illustrates a household’s equilibrium for a level of utility U° with p1' as the price of x1 • Bundle A represents consumer equilibrium for the same level of utility as bundle B • Decrease in p1 results in consumer purchasing more of x1 and less of x2 with level of utility unchanged • Movement from bundle A to bundle B is substitution effect

39. Compensated Law of Demand • In Figure 4.9, Strict Convexity Axiom makes it impossible for a tangency point representing the new price ratio (bundle B) to occur left of bundle A • If p1 decreases, implying p1/p2 decreasing • MRS also decreases • Only way for MRS to decrease is for x1 to increase and x2 to decrease • Thus, decreasing x1’s own price holding utility constant results in • Consumption of x1 increasing • Own substitution effect is always negative • Implying x1/p1|dU=0 < 0 • Known as Compensated Law of Demand • Where price and quantity always move in opposite directions for a constant level of utility

40. Income Effect • In general, a change in the price of a commodity a household purchases changes purchasing power of household’s income • Called income effect • For example, an increase in price of prescription drugs decreases ability to purchase both drugs and food • Decreased ability to purchase same level of commodities represents a decline in purchasing power • Has same effect as if household experienced a change in income • A price decline has effect of increasing a household’s purchasing power or real income

41. Income Effect • Figure 4.9 illustrates a price decline of p1 with utility remaining constant • Results in an increase in real income or purchasing power • Represented by a parallel outward shift in budget line associated with the new price • Equilibrium tangency shifts from point B to C, which is the measurement of the income effect • Mathematically, given the budget constraint I = p1x1 + p2x2 • Change in income from a change in p1, holding consumption of commodities x1 and x2 constant, is • I/p1 = x1 • Substitution effect will equal total effect if, given a decline in p1, income also falls by x1 • If income is not reduced, then this decline in p1 represents an increase in real income • Specifically, a decline in p1 results in an increase in IR of x1 • I/p1 = -x1 • Minus sign results from condition that a change in price and a change in real income move in opposite directions

42. Income Effect • Change in real income depends on how much x1 a household is consuming • If purchases of x1 are small, impact of a price change will be minor • Partial derivative x1/I may be either positive (normal good) or negative (inferior good) • Sign of income effect is indeterminate

43. Slutsky Equation • Combining equations for substitution and income effects yields • Called Slutsky equation • Mathematically defines substitution and income effects • Sum is total effect of a price change • Total effect is a movement from bundle A to bundle C or a movement along demand curve for a change in p1 (Figure 4.9) • Substitution effect defines a change in slope of budget line • Would motivate a household to choose bundle B if choices had been confined to those on original indifference curve • Income effect defines movement from B to C resulting from a change in purchasing power • p1 decreases • Implies an increase in real income

44. Slutsky Equation • If x1 is a normal good, a household will demand more of it in response to increase in purchasing power • Own substitution effect always holds • x1/p1|U=constant < 0 • Normal good results in a negative income and negative substitution effect • Total effect is sum of these two negative effects, so it also is negative • x1/p1 < 0, an ordinary good • Known as Law of Demand • Demand for a commodity will always decrease when its price increases • If demand increases with an increase in income

45. Slutsky Equation • Figure 4.10 represents income and substitution effects for a price increase in commodity x1 • Results in an inward tilt of budget line with a new equilibrium tangency point C • Substitution effect represents price increase holding level of utility constant • Bundle B is a tangency of a budget line, given new increase in price, with initial indifference curve • Movement from bundle A to bundle B is substitution effect • Increase in price of p1 results in a decrease in purchasing power or real income • Income effect measuring this decrease in real income is represented by a parallel leftward shift in budget line • Income effect is negative and reinforces negative substitution effect • Total effect is sum of substitution and income effect representing a movement from A to C

46. Figure 4.10 Substitution and income effects for an increase in p1

47. Slutsky Equation • Not all commodities are normal goods • Some commodities are inferior • Rise in income will yield a decrease in their consumption • Income effect is positive • Will partially or completely offset substitution effect • If income effect does not completely offset negative substitution effect • Total effect will still be negative

48. Slutsky Equation • Figure 4.11 illustrates these effects for inferior and ordinary goods • Movement from A to B is substitution effect • Decline in price results in an increase in consumption of x1 holding utility constant • Illustrates negative substitution effect • Income effect, movement from B to C, partially offsets negative substitution effect • If this positive income effect does not completely offset negative substitution effect, total effect is still negative • Results in an ordinary but inferior good

49. Figure 4.11 Substitution and income effects for an inferior and ordinary good, x1

50. Giffen’s Paradox • If x1 is an inferior good • Sign of total effect can be either positive or negative • Substitution effect is negative and income effect is positive • Positive income effect can be large enough to produce result illustrated in Figure 4.12 • Demand curve has a positive slope • Increase in price results in an increase in quantity demanded • English applied economist Robert Giffen claims to have observed effect in 19th-century Ireland • An increase in price of potatoes resulted in an increase in consumption of potatoes