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Consumption & Savings Romer Chapter 7 Topics 1. What is savings? 2. Consumption, savings and income 3. Savings and the Interest Rate 4. Uncertainty and Savings The Data • Data on Expenditure Categories are typically obtained from the National Income and Product Accounts gathered by the statistical authorities. • USA: Bureau of Economic Analysis, Dept. of Commerce – The national income and product accounts provide an aggregated view of the final uses of the Nation's output and the income derived from its production; two of its most widely known measures are gross domestic product (GDP) and gross domestic income (GDI). BEA also prepares estimates of the Nation's stock of fixed assets and consumer durable goods. Data • In HK, data is collected by the Census and Statistics Department: NIPA Tables • The U.N. maintains statistical databases for a wide variety of countries UN Main Aggregates Database Consumption in HK • Four consumption Consumption Shares in HK categories 140 1. Food 120 100 2. Non-Durables: Clothes, 80 Toys 60 3. Durables: White Goods, 40 Electronics 20 4. Services: Health, Rental 0 1970 1975 1980 1985 1990 1995 2000 FOOD NONDURABLES DURABLES SERVICES Source: CEIC Database 2005 Gross domestic product....... 12455.8 Categories of Personal consumption expenditures. 8742.4 Durable goods................... 1033.1 Spending Motor vehicles and parts...... Furniture and household 448.2 equipment.................... 377.2 Other......................... 207.7 Nondurable goods................ 2539.3 Food.......................... 1201.4 Clothing and shoes............ 341.8 Gasoline, fuel oil, and other energy goods................. 302.1 Other......................... 694.0 Services........................ 5170.0 Housing....................... 1304.1 BEA NIPA Table 2.3.5 Household operation........... 483.0 Electricity and gas......... 199.8 Other household operation... 283.2 Transportation................ 320.4 Medical care.................. 1493.4 Recreation.................... 360.6 Other......................... 1208.4 HK Short-term: Year to year growth 0.4 0.3 0.2 0.1 Durables NonDurables 0 GDP 1967 1969 1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 -0.1 -0.2 -0.3 Theory of consumption best explains non-durables, services and food consumption. HK NIPA Table 038 Savings • Output which is not devoted toward current consumption Gross Savings = Income – Personal Consumption Expenditure – Government Consumption Expenditure 2005 Personal consumption expenditures 8742.4 BEA NIPA Tables Government Expenditure and gross investment 2372.8 Less Government Investment 397.1 Gross Consumption 10718.1 Gross domestic product 12455.8 Less Gross Consumption 10718.1 Gross Savings 1737.7 As a share of GDP 14.0% 2004 2005 Personal income 9731.4 10239.2 Personal Compensation of employees, received Proprietors' income 6665.3 7030.3 911.1 970.7 Savings Rental income Personal income receipts on assets 127.0 72.8 1427.9 1519.4 Personal current transfer receipts 1426.5 1526.6 vs. Less: Personal current taxes Equals: Disposable personal income 1049.8 1203.1 8681.6 9036.1 Gross Less: Personal outlays Personal consumption expenditures 8507.2 9070.9 8211.5 8742.4 Durable goods 986.3 1033.1 Savings Nondurable goods 2345.2 2539.3 Services 4880.1 5170.0 Personal interest payments\1\ 186.0 209.4 Personal current transfer payments 109.7 119.2 What’s Equals: Personal saving 174.3 -34.8 Missing? Personal saving as a percentage of disposable personal income 2.0 -0.4 BEA NIPA Tables Retained Earnings and Depreciation are not counted in Personal Savings Gross Saving 2005 Gross saving 1612 Net saving 7.2 Net private saving 319.7 Personal saving -34.8 Undistributed corporate profits 354.5 Net government saving -312.5 Consumption of fixed capital 1604.8 Private 1352.6 Government 252.2 Gross domestic investment, 1683.1 capital account transactions, and net lending, Bureau of Economic Analysis 1,550 1,600 1,650 1,700 1,750 1,800 1,850 1,900 1,950 19 50 19 52 19 54 19 56 19 58 19 60 19 62 19 64 19 66 19 68 19 70 19 72 19 74 19 76 19 78 19 USA 80 19 82 19 84 Hours per Worker 19 86 19 88 19 90 19 92 19 94 19 96 19 98 20 00 20 02 20 04 20 06 20 08 20 10 Two Consumption Theories • Keynesian: Consumption is dependent on current income. • Permanent Income Theory: Consumption decision is a savings decision so households take into account future income as well as outstanding financial wealth. Keynesian Consumption Function • Consumption Function C = A + mpc×[GDP – TAX] – C = Household Consumption Expenditure – A = Autonomous Consumption { Consumption not dependent on current income} – mpc = Marginal propensity to consume • {Fraction of extra income will be spent on consumption} • mpc will be smaller than consumption to GDP ratio if A is positive. Why do Chinese Save so Much? Why do Americans Save so Little? 45.00% 40.00% 35.00% 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 China USA UN Main Aggregates Data Base East Asian Savings Rates • As a region, East Asia has high savings rates. These high savings rates have helped finance high rates of capital accumulation and growth. • Why have East Asian savings rates been so high? Culture? Luck? Period Saving • Will it last? 5 Macao SAR of China(Patacas) 2003 55.02% 9 Singapore(Singapore Dollars) 2003 44.89% 12 China(Yuan Renminbi) 2003 42.48% 13 Malaysia(Ringgit) 2003 42.34% GDP C G s 22 23 Thailand(Baht) Republic of Korea(Wons) 2003 2003 33.27% 33.02% GDP 25 Dollars) Hong Kong SAR of China(Hong Kong 2003 31.92% 30 Vietnam(Dong) 2003 28.21% 41 Japan(Yen) 2003 25.49% UN Main Aggregates 55 Canada(Canadian Dollars) 2003 24.30% Data Base 68 Germany(Euros) 2003 21.43% 108 United States(Dollars) 2003 13.50% Cultural Reasons • mpc simply depends on cultural factors and not economic factors. • Hayashi, 1989 Japan's Saving Rate: New Data and Reflections • Japan: 1960-1990 Savings Rate averaged about 30% • Japan 1880-1935 Savings Rate average less than 15%! Japanese Gross Saving Rate 1994-2004 Source: CEIC Database 0.32 0.3 0.28 0.26 0.24 0.22 0.2 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Income and Savings Present Discounted Value • Life cycle consumption functions assume that households consider not just the current flow of income but the present value of lifetime income. • Consider a stream of income received over time {y0, y1, …, yT}. This is equivalent in value to a certain amount of current income, pvy < y0+ y1+ …+yT. • Funds available today are worth more than equivalent funds which are not available until the future. Present value • Reason: Today can earn interest. – Q: How much do you need today to have yt in t periods. y t • Answer: (1 r ) t • A future payment discounted by the interest rate raised by the number of periods that must be waited until the payment is made is called the present value. Present value of a stream of payments • Households earn a stream of income over their lifetime. {y0, y1, …, yT}. • Present value of an income stream is the sum of the present values of each payment. y1 y2 yT pv y y0 .... (1 r ) (1 r ) 1 2 (1 r ) T Consumption, Savings, and Future Consumption • The decision of the household to spend money on goods is a simultaneous decision not to save this money in the form of financial assets. • A decision not to save money for the future is simultaneously a decision not to have that wealth available in the future to purchase consumption goods. • The consumption decision is based on a trade-off between the welfare gained from consumption today and welfare from consumption based in the future. Why do People Save? • Life Cycle Motives – Income is Not Smooth Across Time. Households save, in part, to transfer income from high income periods to low income periods. • Precautionary Motives – Households like to achieve a buffer stock of wealth in the case of a possible bad outcome. If households have a buffer stock of saving, bad outcomes in terms of income don’t result in really bad outcomes in terms of consumption. Household born in period 0 and lives until period T. (T+1 period lives) Household begins with real financial wealth F • Present value of consumption equals present value of financial & human wealth B0 Y0 F0 C0 B1 Y1 C1 B1 Y1 (1 r ) B0 C1 B0 1 r 1 r 1 r B2 Y2 B1 C2 B2 Y2 (1 r ) B1 C2 1 r 2 1 r 2 1 r 1 r 2 B3 Y3 B2 C3 B3 Y3 (1 r ) B3 C3 1 r 1 r 1 r 1 r 3 3 2 3 Combine the period-by-period savings equations. • Present value of consumption equals present value of financial & human wealth C1 C2 C3 CT C0 ... W HW F 1 r 1 r 1 r 2 3 1 r T Y1 Y2 Y3 YT HW Y0 ... 1 r 1 r 1 r 2 3 1 r T Algebra Trick • If x ≠ 1, then 1 x x 2 ....xT 1 1 xT 1 1 x – If x = 1, 1+ x +…+ xT = T+1 • If x ≠ 1 x x x ....x 2 T x 1 xT 1 x – If x = 1, x+ x +…+ xT = T Annuities & Annuity Value • Just as any stream of future payments has a present value, so does any current sum have an annuity value. • An annuity is an asset that makes a constant payment every period, for a number of years, T. Such an asset has a present value. • The annuity value of any current amount is the annuity payment generated by an annuity whose present value is equal that current amount. Present Value of an Annuity Payment: Annuity Value of Present Wealth • The real present YP YP YP YP Vt Y T P ... 1 r 1 r 1 r 1 r 2 3 T value of an annuity with 1 1 1 1 payment YP. Y 1 P ... 1 r 1 r 1 r 1 r 2 3 T • Off-the-shelf formula for geometric sum • 1 Solve for present 1 1 r value of an T 1 annuity Y Vt T Y P 1 1 1 r Annuity Value of a Present Value • If you have some current lump sum, PV, 1 payment and you want 1 to buy a annuity for T Y P 1 r W 1 periods. 1 1 r T 1 • Q: How big an annuity payment Y can you get. • A: Invert Equation 5) Permanent Income • We define a households, permanent income as the annuity value of its wealth, W. 1 r 1 W YP 1 r W 1 r 1 1 1 1 1 r 1 r T 1 T 1 • Conceptually, if the household borrowed on all of its future income and added that to its financial wealth, it could buy an annuity generating perfectly smooth income. Permanent Income and Average Income • If FW = 0, and r = 0, then YP = W/T W HW Y0 Y1 Y2 ... YT Y P T T T • If r > 0, then Annuity Value is a weighted average of lifetime income with larger weights on current income than on income in the far future. Permanent Income and Current Income If Y grows at constant rate • Yt = (1+g)tY0 (1 g )Y0 (1 g ) 2 Y0 (1 g )3 Y0 (1 g )T Y0 HW Y0 ... 1 r 1 r 1 r 1 r 2 3 T 1 1 g T 1 Y 1 1 T 1 1 1 r 1 g 1 Y0 Y P 1 r 1 g 1 1 r 1 r 1 1 g 1 r 1 1 T 1 1 r 0 Permanent Income and Current Income If Y is mean reverting Yt Y Yt BC , Yt BC Yt 1 tY0BC BC W r 0, C0 T 1 HW T 1 Y Y0BC ( )Y0BC ( ) 2 Y0BC ( )3 Y0BC ... ( )T Y0BC HW T 1 Y 1 1 1 T 1 Y0 Y Y P 1 T 1 1 1 1 T 1 Y0 Intratemporal Utility Function • A household will exist for t = 0,…,T periods then expire. • Household will enjoy a stream of consumption spending {c0, c1, c2,….cT} • Households preferences over this stream can be defined by a utility function U = U(c0, c1, c2,….cT) • Often a utility function is represented as a weighted sum of utility in each period (called felicity functions). Example: Felicity • Agents get the same utility from consumption in each period. • Households lifetime utility is a weighted sum of the felicity that they receive in each period. • The per-period utility of the household is called the felicity function, u(ct). • Felicity displays diminishing returns from consumption u’(C) > 0, u’’(C) < 0 Felicity Function u(c) u’(c) c Example: Time separable utility • Weights are higher in earlier period due to households impatience. Households discount future utility. U = u(c0) + β u(c1) + β2 u(c2) + β3u(c3)+…. • Rate at which the household discounts future utility is time discount rate. Maximize Discounted Utility • Maximize max u (C0 ) u (C1 ) u (C2 ) ... u (CT ) 2 T T t u (Ct ) t 0 C1 C2 CT s.t. C0 ... HW F 1 r 1 r 2 1 r T Lagrangian Penalty • Assume that there is some utility cost λ of overspending the budget constraint. Maximize utility including this cost and set λ as small as necessary so that people exactly hit their budget constraint. T T Ct max u (Ct ) { t W} 1 r t t 0 t 0 First-Order Conditions • Budget Constraint Holds C1 C2 C3 CT C0 ... HW F 1 r 1 r 1 r 2 3 1 r T • For each period, discounted marginal utility equals discounted cost of spending one more good over the limit. u '(C0 ) , u '(C1 ) , (1 r ) u '(C2 ) 2 , u '(C3 ) 3 ,..., u '(CT ) T (1 r ) 2 (1 r ) 3 (1 r )T Euler Equation • The marginal utility of consumption in one period is equal to the marginal benefit of waiting one period which is the consuming the good plus interest times the extra utility gained from extra future consumption discounted by impatience. u '(Ct ) (1 r ) u '(Ct 1 ) Permanent Income • Permanent Income Hypothesis: β(1+r)=1 then c0=c1 • The permanent income theory says that households keep consumption smooth consuming the annuity value of their financial wealth, F, plus the present value of lifetime income, W. 1 1 C 1 r 1 [ HW F ] Y P 1 1 r T 1 Example • The fraction is referred to as the propensity to consume out of wealth. 1 11 r T 1 1 1 1 r • A household lives for = 40 periods and the real interest rate is .02. In every period they would consume a fraction of their wealth equal to 1 11.02 .0353 1 11.02 41 Applications: Wealth Effect • Changes in asset prices will change the current value of financial wealth. • The effect of an increase in financial wealth on consumption is called the wealth effect. • According to the PIH, a one dollar increase in the value of a stock portfolio should lead to an increase in consumption equal to the propensity to consume out of wealth. • Econometricians estimate that the wealth effect to be less than $.05 consistent with our theory. Application: Life Cycle of Saving • Permanent Income Hypothesis suggests that households like to keep a constant profile of consumption over time. • Age profile of income however is not constant. Income is low in childhood, rises during maturity and reaches a peak in mid-1950’s and drops during retirement. • This generates a time profile for savings defined as the difference between income and consumption. Time Path of Savings C,Y S>0 C S<0 S<0 Y time East Asian Demographics • Due to plummeting birth rates, East Asia had a plummeting ratio of youths as a share of population Change in Age Shares %Below 15 % Prime Age 20-59 • This put a large share of 1950-1990 2005-2025 population in high savings China -13.56 0.41 years. Hong Kong -20.64 NA • Share of prime age adults Indonesia Japan -7.26 -16.72 5.52 -4.03 has hit its peak in most South Korea -18 -4.12 Asian countries and will fall Malaysia -7.7 7.5 over the next half century. Singapore -20.22 8.35 Taiwan -18.82 NA Thailand -14.74 0.25 East Asian Demographics • During last 25 years, East Asian Nations had a sharp decrease in their ‘dependency ratio’. • Dependency ratio is the % of people in their non-working years (children & seniors. • Dependents are dis-savers and non- dependents are savers. Applied Consumption Function • Optimal consumption is a linear function of human wealth and financial wealth. Both growth part and cyclical part of human wealth is proportional to current income. C aHW bFW a d Y b FW * t • Dynamics of consumption expenditure self correct to the optimal level Ct 0 1 Ct 1 ... j Ct j f Ct 1 Ct*1 g Ct* j t Interest Rates and Savings Two period problem • Intuition from this problem can be derived using simplest version of the theory in which T = 1. U = U(c0, c1) – Thus, there is only a current period (t = 0) and a future period (t = 1). • Preferences can be represented with 2 dimensional indifference curves. Preferences • People prefer some combinations of present and future consumption. – More is better. If two combo’s have equal future consumption, choose the combo with more present consumption. – Smooth over time. Households have diminishing returns to consumption in any period. – Consumption is a normal good If income goes up, ceteris parabis, consumption goes up in all periods. • Preferences are represented by indifference curves – Smooth sets of combo’s amongst which the household is indifferent. Indifference Curves c1 I3 I2 I1 c0 Savings and the Budget Constraint • Agents start out with a certain amount of financial wealth fw, that they carried over from the previous period. • In each period, household will earn income from producing goods yt. Households will also have a fixed tax obligation taxt. Household after tax income is yt . • Balance is any funds that are left over after consumption bt = yt +fwt - ct • Financial wealth will just be balance plus some goods interest rate, fwt = (1+r)bt-1 Intratemporal Budget Constraint • The savings in period 1, the last period of life will be 0. We can write this as a budget constraint for each period. Assume fw0 = 0 – Time 0: c0 = y0 + fw0 – b0 – Time 1: c1 = y1 +(1+r)b0 Intertemporal Budget Constraint • We can combine these budget constraint into one intertemporal budget constraint. – Divide time 1 budget constraint by (1+r)1 and add up the budget constraints. c1 y1 c0 y0 fw hw fw 1 r 1 r fw 0 c1 y1 0 y0 hw 1 r 1 r Intuition of Budget Constraint • The intertemporal budget constraint says that the present value of consumption must be equal to the present value of after-tax income plus initial financial wealth. • The present value of after-tax income could be referred to as human wealth as it is the value of the households ability to produce goods in the future. Budget Constraint • There is a trade-off between consumption today and consumption and the future which can be represented geometrically. • If the household has zero future consumption, it can consume c0= hw0. • If the household has zero consumption today it can consume c1 = (1+r)(hw0). • For each good given up in period 0, the household can get an extra (1+r) in period 1. • The s0 point is on the budget constraint. Define au0 = y0 and au1 = y1 Budget Constraint c1 (1+r)w 1+r au1 c0 au0 w Preferences • Principles describe consumer preferences. 1. More is better: Higher indifference curves are preferred. 2. Diminishing returns to consume in any period. The slope of the indifference curves is decreasing. Consumption in every period is a normal good. Increases in income increase consumption of a normal good. 2U 2U , 2 0 c0 c1 2 Optimum • Optimal consumption choice is: – on an indifference curve tangent to the budget constraint (so the slope of the indifference curve is equal to 1+r). – Where marginal rate of substitution is equal to real interest rate U MU 0 c0 MRS (1 r ) MU1 U c1 Optimum • The optimal choice is also the solution to a maximization problem. c1 y1 max U (c0 , c1 ) s.t. c0 y0 c0 , c1 1 r 1 r y1 c1 max U ( y0 , c1 ) c1 1 r 1 r dU 1 U U 0 U1 U 2 (1 r ) dc1 1 r c0 c1 • Marginal utility of consumption at time t is marginal felicity discounted by the discount factor MUC,t = βt × u’(ct) • Marginal rate of substitution of consumption between two periods is the ratio of marginal utility. MU t j u '(ct j ) MRS j MU t u '(ct ) Specific Utility • Felicity function is the natural logarithm, u(ct) = ln(ct) MRS c1 c0 1 1 c • Two period case. c1 0 c1 max ln(c0 ) ln(c1 ) s.t. c0 w 1 r c1 1 1 1 1 r 1 max ln( w ) ln(c1 ) 0 c1 1 r c1 1 r w c1 c1 c0 1 r Optimal Consumption: Lender c1 (1+r)w c1* au1 c0 c0* au0 W Optimal Consumption: Borrower c1 (1+r)W au1 c1* c0 * c0 au0 W Implications • Current and future income affect optimal consumption only through their affect on fw. An increase in fw results in a parallel shift in budget constraint. • Normal Good: An increase in w results in an increase in both present and future consumption. – A temporary increase in current income alone will lead to an increase in w, an increase in current consumption and saving. – The expectation of an increase in future income will lead to an increase in w, an increase in current consumption and a decrease in savings. – A permanent increase in income will increase both current and future consumption as well as current and future income and therefore have negligible impact on consumption and savings. Increase in Current Income c1 Current Autarky (1+r)(w’) income increases. Future autarky does (1+r)w {c0**,c1**} not. Savings must rise. c1 ** c1* c0 c0 ** w w’ c0* Increase in Future Income c1 Future Autarky (1+r)(w’) income increases. Current autarky does (1+r)w {c0**,c1**} not. Savings must rise. c1 ** c1* au1 c0 c0 ** w w’ c0* Consumption Smoothing • Because consumption faces diminishing returns in any period, consumers have an incentive to allocate temporary increases in income to all periods. • Consumption will be smoother than income at given interest rate. This matches the reality. However, quantitatively, consumption not smooth enough. – Interest rate moves endogenously – Borrowing constraints. Rise in Interest Rate c1 (1+r)w au1 c0 au0 w Rise in Interest Rate • A change in the interest rate results in a pivot in the budget constraint around the no-savings point. • Two basic affects of a change in the interest rate. – Substitution Effect: The real interest rate is the relative price of consuming today (relative to future consumption). – Income Effect: The real interest rate affects the budget opportunities available to agents. Before interest rate rise, optimal c0 = au0 After interest rate rise, steeper budget constraint crosses indifference curves along for higher utility to left of au0. Optimal consumption drops, savings rise. au0,au1 Before interest rate rise, optimal c0 > au0 After interest rate rise, new steeper budget constraint is to the left of previous consumption level. Household must cut back on current consumption just to get to affordable consumption combination. Savings rise. au0,au1 Before interest rate rise, optimal c0 < au0. After interest rate rise, new steeper budget constraint is to the right of previous consumption level. New set of affordable consumption combinations which make the household better off, some of which can involve less consumption in period 1. Savings may rise or fall. Income Effect • Change in interest rates changes the value of your savings. – For savers, (i.e. consumption to left of autarky point), a rise in interest will increase the future value of those savings increasing lifetime income – For debtors, (i.e. consumption to right of autarky point) a rise in the interest rate will increase future costs of paying debts reducing lifetime income. • If income goes up, you will have a tendency to consume more in both periods. If income falls, you will have a tendency to consume less in both periods and savings will rise. Substitution Effect • A rise in the interest rate will make consumption today more expensive relative to consumption in the future. • A rise in the real interest rate will lead to a reduction in consumption today relative to consumption in the future. How strong is the substitution effect? • Constant Elasticity Intertemporal Substitution Utility Function 1 1 1 1 c u (c ) , 0 u '(c) c 1 1 1 1 c0 c0 1 r c1 1 r c1 • When ψ = 1, the CEIS felicity is natural log for all intents and purposes. Natural log felicity is sometimes referred to as unit elasticity of intertemporal substitution. Point Elasticities • Elasticity is the % change in one variable caused by a % change in another variable. • Elasticity of substitution is the % change in the demand for one variable relative to another. • Functions with constant elasticities are log linear. Income Effect: Borrowers • For borrowers, households to the right of au, an increase in the interest rate offers a lower budget constraint, which allows less present consumption if we keep future consumption constant. • Substitution effect and income effect work the same way. Present consumption drops relative to future consumption but at any given future income, affordable present consumption will drop. – Present consumption of borrowers will drop if real interest rate rises. Income Effect Lenders • For lenders, households to the left of au0, an increase in the interest rate offers a higher budget constraint and allows higher present consumption if we keep future consumption constant. • Income and substitution effects will work in opposite ways. A rise in the interest rate reduces current consumption relative to future consumption, but at any given future consumption a higher level of present consumption is affordable. – Effect of an increase in the interest rate on consumption is ambiguous. Effect of Interest Rate on Savings • Empirically, opinion on the effect of interest rate on savings varies in a range from zero to mildly positive. The Effect of Interest-Rate Changes on Household Saving and Consumption: A Survey Douglas W. Elmendorf 1996-27 Uncertainty and Savings Taiwan, National Health Insurance • In 1995, Taiwan implemented a scheme providing national health insurance to all islanders. • This program raised coverage rates from 57% to 97% • Aggregate gross savings declined in Taiwan. • Careful study shows this to be concentrated among low income households who were not previously covered. National Health Insurance and precautionary saving: evidence from Taiwan Shin-Yi Chou , Jin-Tan Liu , James K. Hammitt Taiwan Gross Saving Rate (Taiwan National Income Accounts) 0.345 0.335 0.325 0.315 0.305 0.295 0.285 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Precautionary Savings • Decision making is taken under certainty. • Most saving is done under a cloud of uncertainty about the future. • Question: How does the uncertainty environment affect the willingness to save? • Return to Polonius. – Assume that β = (1+r) = 1 – If fw0 = tax0 = tax1 = 0, & y0 = y1 then c0 = y0 • Return to the two period life problem. Abstract from taxes and initial financial wealth. • When consumption decision is made, the household knows its current income, y0. However, second period income is a random variable. • Assume that there are two equally likely future outcomes, good and bad. If the outcome is good, the household will have income y0 + x. If the outcome is bad, the household will have income y0 – x. • Expected household income is .5 *(y0 + X) + .5* (y0 - X) = y0. Decision making under uncertainty • Most popular decision making paradigm is maximize expected utility subject to the budget constraint. – Pick three variables, c0, c1,GOOD cBAD • Expected utility is – U(c0) + .5 * u(c1,GOOD) + .5*u(c1,BAD) • Budget constraints – C1,GOOD = y1 + x + (y0-c0) – C1,BAD = y1 -x + (y0-c0) Maximization problem • Max u(c0) + .5 ∙ u(y1 + x + (y0-c0)) + + .5 ∙ u(y1 - x + (y0-c0)) • 0 = u’(c0) - .5 * u’(c1,GOOD)- .5*u’(c1,BAD) • u’(c0) = E[ u’(c1)] Under uncertainty, set marginal utility today equal to expected marginal utility tomorrow. Marginal utility function • Further assume utility is a diminishing function of consumption. • This is true if utility is a constant intertemporal elasticity of substitution function. • Expected value of marginal utility is greater than marginal utility of expected value. Expected Marginal Utility is greater than Marginal utility of expected value B: E[u’(y0)] A: u’(c0) u’(c) c y0 - x y0 y0 + x Precautionary Savings • If household sets marginal utility of consumption today equal to marginal utility of expected value tomorrow, this would be less than expected value of marginal utility tomorrow. • Must act to increase marginal utility today or reduce marginal utility in the future (i.e. shift income toward the future) • The household will shift income away from periods of certainty toward periods of uncertainty or save as insurance. Precautionary savers & spenders u’(c) Precautionary Savers B: E[u’(y0)] Precautionary A: u’(c0) Spenders Certainty Equivalent c Optimal Consumption: Borrowing Constraints c0 = au0 c1 (1+r)(w) au1 c1* c0 au0 c0 * w Buffer Stock Savings • Borrowing constraints and precautionary savings interact. • If short-term income falls sharply and borrowing constraints hold, then consumption in bad states may fall dramatically. • Expected marginal utility of consumption may be high due to this downside risk. • Precautionary savings should fall as income rises because high income people have a smaller chance of hitting liquidity constraint. Social Insurance, Financial Credit & Savings • Various government programs may reduce the uncertainty of income. • Social welfare or health insurance may reduce the individual unpredictability of insurance and reduce the need for precautionary savings. • A more smoothly operating financial system may also reduce the need for precautionary savings. MPC • Under certainty with perfect financial markets, the marginal propensity to consume out of temporary income must be very small (as shown by the wealth effect in stock markets. • Propensity to consume increases if large share of consumers face borrowing constraints or precautionary motives are large.