CONSUMPTION Carl-Johan Dalgaard Department of Economics University of Copenhagen
INTRODUCTION So far: The average propensity to save has been exogenous Albeit our analysis showed that di ff erence in “ s ” matters to long-run prosperity, we so far as avoided asking why s might di ff er So far: Only changes in current income matters to total savings. This chapter: Optimal savings — > Which factors determine savings if consumers are forward looking? 2
OVER VIEW OF THE CHAPTER A . The intertemporal optimization problem of the consumer: Private consumption . Business cycle fact: Consumption is less volatile than GDP. Why? B . Government consumption What is the impact of taxation on consumption/savings? Does it matter whether the government fi nances its spending via bonds or taxes? 3
A) PRIVATE CONSUMPTION Assume households are equipped with preferences over consumption today ( c 1 ), and tomorrow ( c 2 ). Speci fi cally σ − 1 σ − 1 ½ σ 1 σ σ σ σ − 1 c + σ − 1 c , if σ > 0 , 6 = 1 1 1 1+ φ 2 U ( c 1 , c 2 ) = u ( c 1 )+ 1 + φu ( c 2 ) = 1 log ( c 1 ) + 1+ φ log ( c 2 ) , if σ = 1 . For per period felicity we assume u 0 ( c i ) > 0 , u 00 ( c i ) < 0 for i = 1 , 2 . φ is the rate of time preference So: (i) positive marginal utility from consumption today or tomorrow, (ii) diminishing marginal utility. Note: c 1 and c 2 are normal goods. 4
A) PRIVATE CONSUMPTION 1 Intertemporal utility function: U ( c 1 , c 2 ) = u ( c 1 ) + u ( c 2 ) 1+ φ Figure 1: Slope ? MRS: Di ff the utility function = − (1 + φ ) u 0 ( c 1 ) 1 + φu 0 ( c 2 ) dc 2 = 0 ⇔ ∂c 2 1 u 0 ( c 1 ) dc 1 + u 0 ( c 2 ) ∂c 1 5
A) PRIVATE CONSUMPTION Period 1 budget constraint . Income: Born with wealth V 1 , work, Y L 1 , pay taxes: T 1 . Expenditure: consumption, c 1 , and savings, s. In sum c 1 + s = V 1 + Y L 1 − T 1 In period 2 the consumer also works, Y L 2 , and pay taxes T 2 . In addition, there might be income from savings s + rs = (1 + r ) s ≡ V 2 |{z} |{z} period 2 wealth interest earnings Amount saved Savings tranf c 1 into c 2 ; (1 + r ) is therefore the marginal rate of trans- formation (MRT) P eriod 2 constraint is c 2 = (1 + r ) s + Y L 2 − T 2 6
A) PRIVATE CONSUMPTION Substitute for s , and we can consolidate the two constraints: The in- tertemporal budget constraint . 1 − T 1 + Y L 2 − T 2 c 2 = V 1 + Y L c 1 + ≡ V 1 + H 1 |{z} 1 + r 1 + r | {z } | {z } "human wealth" Lifetime consumption Life time income Slope (MRT) ? E is the endowment point. Lender vs. Borrower. Figure 2: 7
A) PRIVATE CONSUMPTION Optimal consumption. Choose highest attainable utility, given the in- tertemporal budget constant. Figure 3: Optimal consumption therefore implies MRS = MRT (also often re- ferred to as “the Keynes-Ramsey rule”) − (1 + φ ) u 0 ( c 1 ) u 0 ( c 2 ) = − (1 + r ) ⇔ u 0 ( c 1 ) u 0 ( c 2 ) = 1 + r 1 + φ. 8
A) PRIVATE CONSUMPTION Experiment 1: Temporary increase in income. Period 1 income in- creases. What is the impact on c 1 , c 2 ? Figure 4: Hence, since c 1 , c 2 are normal goods - > consumption in period 1 al- ways increases by less than income; some of the gain is passed on to tomorrow. “Consumption smoothing”. 9
PRIV ATE CONSUMPTION Do people smooth consumption? That is, follow ( u 0 ( c 1 ) u 0 ( c 2 ) = 1+ r 1+ φ ) income consumption adjusted consumption -1.6 -1.8 -2 -2.2 30 40 50 60 age consumption and income profiles for couples in UK (born 1935-1939) Figure 5: 10
A) PRIVATE CONSUMPTION Experiment 2: Permanent increase in income. Period 1 and period 2 income rises. Suppose to same extent dY L 1 = dY L 2 . What is the impact on c 1 , c 2 ? Figure 6: 11
SUMMARY OF INCOME CHANGES Temporary changes in income (e.g., unemployment) - > less than pro- portional changes in consumption. Consumption smooting. The identi- fi cation of the consumption smoothing e ff ect is due to Modigliani - the life cycle theoryt of consumption. Key explanation for low volatility of consumption relative to income (Cf BC facts) . Permanent changes in income has more dramatic e ff ect on consumption. In theory we might fi nd 1:1. The hypothesis that permanent income changes has a larger impact on consumption is due to Milton Friedman - The permanent income hypothesis). NOTE: Income change can either be due to Y or T ! Permanent tax cuts more important! 12
A) PRIVATE CONSUMPTION Experiment 3: increasing r . Twisting the budget constraint in the endowment point Figure 7: Notice the di ff erence between ex ante being a lender, or a borrower! 13
A) PRIVATE CONSUMPTION Suppose the consumer is a lender. 2 e ff ects from changing r. 1. Substitution e ff ect. It becomes more expensive to consume today - alternative cost. c 1 ↓ , c 2 ↑ 2. Income e ff ect. If you’re a lender, r ↑ implies an increase in the budget. c 1 ↑ , c 2 ↑ (normal goods). Thus: Net impact on period 1 consumption, c 1 , is ambigious. Suppose the consumer is a borrower. The income e ff ect works in reverse: c 1 ↓ , c 2 ↓ . An increase in r will always lower consumption for borrowers. 14
A) PRIVATE CONSUMPTION Geometry for the lending consumer Figure 8: 15
A) PRIVATE CONSUMPTION A few words about the formal analysis. The problem is to σ − 1 σ − 1 σ 1 σ σ σ max σ − 1 c + σ − 1 c 1 2 1 + φ c 1 ,c 2 S.t. c 2 c 1 + 1 + r = V 1 + H 1 Solve by substitution; maximize wrt c 1 . The fi rst order condition is MRS = MRT. Substitute for c 2 in the budget constraint. You fi nd c 1 = θ ( V 1 + H 1 ) ³ ´ − 1 1 − T 1 + Y L 2 − T 2 1 + (1 + r ) σ − 1 (1 + φ ) , and H 1 ≡ Y L where θ ≡ 1+ r . Observe that changes in r has an ambigious e ff ect on c 1 . σ > 1 - > substitution e ff ect dominates. 16
A) PRIVATE CONSUMPTION Finally à ! ³ ´ V 1 + H 1 Y L θY d ≡ ˆ c 1 = θ 1 − T 1 1 Y L 1 − T 1 where " # Y d V 1 1 ˆ 2 θ ≡ θ · + 1 + Y d Y d 1 + r 1 1 Y d c 1 = 1 . BUT: Y d θ constant ∂c 1 Observe that keeping ˆ 1 2 constant - > ∂Y d Y d 1 1 permanent change in income. Observe that changing Y d 1 only, leads to a smaller e ff ect. Temp. change. Y d Y d ∂c 1 1 1 = < 1 . ∂Y d c 1 V 1 + H 1 1 17
PRIV ATE CONSUMPTION These predictions are rather general. The speci fi c magnitudes, however, depends on the utility function. Hence, in general ⎛ ⎞ ⎝ Y d ⎠ C = C , g , r ( ± ) , V 1 1 (+) (+) (+) where g ≡ Y d 2 . Y d 1 Consumption increases with income; less than 1 for 1 if only a temporary change. Growing income ( g ) will also increase consumption (higher period 2 income, for period 1 unaltered) The e ff ect of changes in the real rate of interest is ambigious Higher fi nancial wealth V 1 , increases consumption. 18
PRIV ATE CONSUMPTION: EMPIRICS The association between ˆ θ and V 1 /Y d 1 : Average propensity to Wealth-income consume 1 ratio 2 1,1 6,5 Average propensity to consume 1,05 (left axis) 6 1 5,5 0,95 5 0,9 Wealth-income ratio (right axis) 4,5 0,85 0,8 4 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 Figure 9: Private consumption and the wealth/income ratio in Denmark 19
PRIV ATE CONSUMPTION: EMPIRICS With roughtly constant growth in income Y d 2 /Y d 1 is about constant. Then we expect c 1 /Y d 1 = ˆ θ which is constant for r constant and with V 1 /Y d 1 trendless (...remember Kaldor?). This seems to be true as well (for rich countries like US and DNK) Aggregate consumption / aggregate disposable household income 1,2 Denmark 1 USA 0,8 0,6 0,4 0,2 0 1929 1933 1937 1941 1945 1949 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 Year Figure 10: Average propensity to consume, Denmark and the US. 20
B . Government consumption: BACKGROUND The public sector is rather large in a place like Denmark Figure 11: Government expenditures as a fraction of GDP, 1988-2002: Denmark. 21
B . Government consumption: BACKGROUND Figure 12: Public sector surplus as a fraction of GDP, 1970-2002: Denmark .... and the government isn’t always running surpluses... and the same is true in many other countries.... 22
B . Government consumption: BACKGROUND Figure 13: Public debt as a fraction of GDP: Selected countries in 2002. 23
B . AN INTERTEMPORAL PERSPECTIVE ON THE PUB- LIC SECTOR Suppose the government lives for two periods (like other agents). Period 1: G 1 = T 1 + B where G 1 is public consumption, T 1 is tax revenue, and B represents net borrowing. Period 2: G 2 + (1 + r ) B = T 2 There is no "the day after tomorrow"; all debt is "retired" in period 2. The intertemporal budget constraint (substitute for B ) G 2 + (1 + r ) ( G 1 − T 1 ) = T 2 ⇔ G 2 T 2 1 + r + G 1 = 1 + r + T 1 24
BUDGET DEFICITS: A PROBLEM? It is often claimed that budget de fi cits are harmful Question 1: Why? Question 2: Will it always be harmful? Structure of argument: ( i ) What does the government do? ( ii ) What does the households do in response? ( iii ) Aggregate assessment: S = S p + S g = S p + T − G = I where the last equality represents a closed-economy assumption. 25
BUDGET DEFICITS: A PROBLEM? ( re: i) The current de fi cit G = T + B Experiment 1: T is decreased, and G kept constant. − dT = dB, dT < 0 i.e., increased debt . ( re: ii) Assume households follow a simple rule of thumb S p = s ( Y − T ) , s ∈ (0 , 1) As a result dS p = − sdT 26
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