Chapter 4 – Households 1/35 Consumption The objective in this chapter is to apply what we have learned about feasible sets and preferences to more general settings. Consider Marı́a, whose income is equal to 56. Everything is spent on clothing and food. I The price of food is 1.75 If she spends all on food she can afford 56/1.75 = 32units I The price of clothing is 1.12 If she spends all on clothing she can afford 56/1.12 = 50units 2/35 4.1. CONSUMPTION food I/pf = 32 • B pc /pf • C A • I/pc = 50 clothing FIGURE 4.1 Maria’s budget set and budget line 3/35 Denote by I Marı́a’s income and I pf = price of food, pc = price of clothing I F the quantity of food Marı́a buys, C the quantity of clothing Marı́a buys. Then, if (F , C ) is affordable we have pf × F + pc × C ≤ I (1) If Marı́a spends all her income on food and clothing then we have pf × F + pc × C = I (2) Marı́a’s feasible set is given by all the combinations (C , F ) that satisfy Eq. (1). The frontier of the feasible set is a straight line given by Eq. (2). 4/35 When talking about a person or a household’s feasible set with respect to consumption we use the term budget set (instead of feasible set). The boundary of the budget set —Eq. (2)— is the budget line. We can rewrite the equation of the budget line as the equation of a line in the (C , F ) space, f = pc I − c pf pf (3) The absolute value of the slope of this line is MRT = pc pf 5/35 In Chapter 3 we have seen that the optimal choice is such that MRT = MRS So, in the case of Marı́a, we have when her choice is optimal MRS = pc pf In English: Marı́a’s optimal consumption level is such that How much food she’s willing to give up for an extra unit of clothing = ration of price(pc /pf ) 6/35 Changes in income If we increase an individual’s income the budget line will shift upward. But what about the optimal consumption? I Consumption of both goods increases? I Consumption of one goods increases, and decrease for the other good? That depends. . . 7/35 I If the consumption of a good increases when income increases, then the good is a normal good. Examples: Uber, leisure, housing I If the consumption of a good decreases when income increases, then the good is an inferior good. Examples: public transportation, junk food. 8/35 4.1. CONSUMPTION y I2 py b2 I1 py b1 Case when x is a normal good U2 U1 y2 y1 • C1 x1 y U1 U2 • C2 x x2 I1 px I2 px Case when x is an inferior good 9/35 x x1 y U1 I2 py y2 I1 py x2 I1 px I2 px Case when x is an inferior good U2 b2 b1 y1 • C2 • C1 x2 x1 x I1 px I2 px FIGURE 4.2 Effect of an increase in income 10/35 Income and substitution effects of a price change Consider a consumer with income I and choosing how much to consume of good x and of good y . I px = price of x I py = price of y . So the budget line intersects I The x axis at I (she spends all of her income on x). px I The y axis at I py 11/35 Suppose now the price of x changes from px to px0 (with px0 > px ). Hence, the budget line moves and now intersects I The x axis at I (a lower point than before) px0 I The y axis at I py So the optimal choice of the consumer changes: We see that the consumption of x drops (not surprising, x is now more expensive). 12/35 y UC I py U A bA bC yA yC • C • A x xC I p0x xA I px Figure 1: Income eﬀect (arrow I) and substituti 13/35 The change from A to C can be decomposed in two parts: I A substitution effect; I An income effect. Intuition I px increases. So the real income decreases: Real income: capture what we can actually buy. We can buy less of x, so the real income decreases. I If px increases, then the relative price of x (relative to y ) increases: For 1 additional unit of x we need to sacrifice more units of y than under px . 14/35 I Substitution effect The change of consumption due to x being relatively more expensive than y . A change in x cause by a change in relative prices, keeping real income constant I Income effect The change due to the fact that the real income is lower. A change in x cause by the change in real income (i.e., keeping relative prices constant). 15/35 y I+ I py UC U A bB I py bA bC • yB yA yC • C B I • A S x xC xB I p0x xA I px Figure 1: Income eﬀect (arrow I) and substitutio 16/35 Sign of income/substitution effect I If x is a normal good, the income effect has the opposite sign of price change: Price increases, income effect is negative (consumption of x decreases). I If x is an inferior good, the income effect has the same sign of price change: I The substitution effect has always the opposite sign of price change: this is the law of demand. 17/35 Changes in income and prices We are now interested in two comparative statics problems: I The impact of changes in (non-labor) income I The impact of changes in the wage rate. One could think that the optimal choice about income–leisure is similar to the optimal choice about consumption (e.g., food v. clothing). It’s more complicated than that. . . 18/35 Income–leisure is I Similar to food–clothing: The more of, say food, the less of clothing (and vice-versa). I Different from food: There are two sources of income: I Labor income (so income–leisure is similar to food–clothing) I Non-labor income: a change of income does not come from less leisure. 19/35 consumption I + 24 w 24 w • I • c2 c1 leisure 24 Figure 1: Eﬀect of additional income 20/35 In the previous graph, an increase of (non-labor) income induced an increase of leisure ⇒ leisure is a normal good Things are different when the increase of income is from labor, i.e., the wage increases. 21/35 The income effect is smaller than the substitution effect consumption 24 w2 24 w1 • c2 • c1 leisure 24 Figure 1: Eﬀect of additional income 22/35 The income effect is greater than the substitution effect: The price of leisure (= the wage) increases, the chosen amount of leisure increases. consumption 24 w • c1 • c2 leisure 24 23/35 Labor supply The demand for leisure can be reinterpreted as labor supply. The supply of labor may react positively or negatively to a change in the wage rate depending on the relative magnitudes of the substitution and income effects. 24/35 Savings Saving decisions can be thought as a choice between consumption today and consumption in the future. Like leisure–grade or food-clothing or leisure–labor we can analyze the tradeoff between consumption today and consumption in the figure. The new concept we need is the interest rate, ra . 25/35 Let us simplify the problem (a model!): I There are two periods: I First period: working I Second period: retirement I The income in while working is I I No (labor) income while retired. If x is the amount saved in period 1, then in period 2 that amount is equal to x(1 + ra ). What if the interest rate changes from ra to rb , with ra > rb ? 26/35 The income effect is smaller than the substitution effect consumption during retirement (f ) I (1 + ra ) ba fa I (1 + rb ) • A bb fb • B savings ca cb consumption tod I 27/35 The income effect is greater than the substitution effect: The price of today’s consumption (= the wage) increases, the chosen amount of leisure increases. consumption during retirement I (1 + ra ) ba fa I (1 + rb ) fb • bb A B • savings cb ca consumption to I 28/35 Insurance Uncertainty is capture by assuming that the future is made up of multiple possible states of the world. For simplicity, let us assume that there are two states: I h, when the consumer is healthy; The consumer’s income is h1 I s, when the consumer is sick. The consumer’s income is s1 (= h1 − medical expenses). 29/35 The framework we used can be applied to insurance. Instead of choosing the amount of consumption (e.g., food & clothing) or labor supply (labor & leisure) the consumer choose a bundle consisting of I income when not sick (state h) I income when sick (state s) This is done by choosing the level of insurance 30/35 The idea of the insurance is very simple: I The agent pays (today) a premium p I If the agent is sick then she gets a payment q from the insurance company. If the initial situation is (h1 , s1 ) then when contracting the insurance the agent’s income is I h − p in stage h I s + q in stage s 31/35 When the indifference curves are about two states of the world (the consumer does not know which will prevail) the shape of indifference curves capture the agent’s risk aversion. If the agent is risk neutral, the indifference curves are straight lines. All the points on the same indifference curve yield the same expected utility 32/35 Suppose that the consumer has a probability ρ to be sick. Then the expected income is ρh + (1 − ρ)s 33/35 The insurance company proposes a set of contracts, which amounts to a set of combinations of income for states h and s. To each combination of incomes (one for each state) corresponds implicitly to a premium and a coverage q. This set of contracts acts like a budget constraints: the agent will maximize her preferences given the constraint offered by the insurance company. 34/35 sick 45 E2 • s2 s3 s4 • E4 • E3 U2 U3 U1 • s1 E1 healthy h4 h2 h3 h1 35/35