Consumption by xiaopangnv

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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.

								
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