Part 2 Microeconomic Analysis of Finance ???????? Chapter 4 Household Finance ?????

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Part 2 Microeconomic Analysis of Finance
????????Chapter 4 Household Finance ?????

• Naotsugu HAYASHI ? ??
• Professor of Economics ?????
• Faculty of Business Administration ????
• Hosei University ????

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1.Flow of Funds of Household ???????
?sources of funds income (rewards to production
factors such as labor, capital and land)
increase in financial debt sales of financial
assets ??????(??????????????) ??????????
???? ?ways to spend funds consumption real
investment increase in financial assets
repaying of financial liabilities
???????????????????????? Since ??, a flow of
funds equation for household is income-consumption
real investmentnet increase in financial
assets real
investmentfinancial investment investment
????????? ??-?????????????-???????
???????????????????? ??
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2.Definition of Savings ?????
savings income consumption a residual of
the income that was not consumed ? investment
real investment financial investment
????-????????????????
??????????? consumption/income
savings/income average propensity to consume
average propensity to save 1
??/????/??????????????1 additional
consumption/additional income additional
savings/additional income marginal propensity
to consume marginal propensity to save 1
????/????????/???? ????????????1
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3.Savings and Portfolio Selection ???????
Portfolio Selection a judgment of asset
selection whether I have savings in the form of
cash, deposits, stocks and other financial assets
or real assets ????????????????????????????
??????????????? Income consumption
savings real investment financial
investment ??-????????????????(????) If
savings real investment, then financial assets
do not increase. ????????????????????
Surplus unit savingsgtreal investment ? net
increase in financial asset Deficit unit
savingsltreal investment ? net increase in
financial liabilities ??????gt???? ?
???????? ??????lt???? ? ????????
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4-1. Indifference Curve of Consumption and
Savings (Future Consumption) ???????????
utility function UU(C1,C2) 2-period model of
present and future times ???? UU(C1,C2)
??????2????? indifference curve the
combinations of present and future consumption
goods that have the same level of
utility ?????????C1???????C2? ??????????????????
????????? Indifference map a group or map of
indifference curves ??????(indifference
map)????????
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4-2. Indifference Curve of Consumption and
Savings (Future Consumption) ???????????
Marginal Rate of Substitution MRS The ratio
of increased future goods against decreased
present goods to maintain the same level of
utility MRS-dC2/dC1 (?U/?C1)/(?U/?C2)
marginal utility of present consumption /
marginal utility of future consumption ? a slope
of the tangent line of indifference curve
????? ????C1?1????????????????
???????????????????????C2??? MRS-dC2/dC1
(?U/?C1)/(?U/?C2) ?????????/????????? ?
???????????
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5.Time Preference ????
Time Preference a tendency to prefer to
consume goods at present rather than in the
future. It is because the quality of goods tends
to deteriorate, a shortage of goods may happen
and other uncertain matter may occur in the
future ????????????????? ????????????????????
??????????????? ???????????? Marginal Rate of
Time Preference MRTP the marginal rate of
substitution on the point C minus unity
????????????45?????????????MRS-1 Marginal Rate
of Time Discounting MRTD the marginal rate of
substitution between present and future
consumptions - 1 ???????????????????????-1
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6-1. Inter-temporal Budget Constraint
Consumption Possibility Set ?????????(??????)
?When saving is impossible Present
ConsumptionC1?Present Income Y1 ?Future
ConsumptionC2 ?Future Income Y2 ?Consumption
Possibility Set rectangle that is enclosed by
the Origin O, present income Y1 , future income
Y2, and the point A. ????????????
??????????????????? ??O?????Y1?????Y2???A???????
?? ? When saving is possible Present
ConsumptionC1?Present Income Y1 ?Future
ConsumptionC2 ?(Present Income-Savings) Future
Income Y2 ?Consumption Possibility Set
trapezoid that is enclosed by the Origin O,
present income Y1, present income Y1 future
income Y2 (point B), and the point A
???????????? ?????????? ?????(????-??)???? ??O??
???Y1? ????Y2????Y1 (B?)? ??A????????
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6-2. Inter-temporal Budget Constraint
Consumption Possibility Set ?????????(??????)
? When lending is possible Present
ConsumptionC1?Present Income Y1 ?Future
ConsumptionC2 ?(Present Income-Savings)(1Interest
Rate r) Future Income Y2 ?Consumption
Possibility Set trapezoid that is enclosed by
the Origin O, present income Y1, (1Interest
Rate r) present income Y1 future income Y2
(point C), and the point A Market Interest Rate
Discounting Rate for Calculating Present Value
????????????? ????C1?????Y1 ?????C2
?(????-??)(1??? r) ???? Y2 ??O?????Y1?????Y2?
???Y1(1???r)???A???????? ??????r???????????
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6-3. Inter-temporal Budget Constraint
Consumption Possibility Set ?????????(??????)
? When lending and borrowing are possible
(1Interest Rate r)( Present Income Y1- Present
Consumption C1)Future Income Y2Future
Consumption C2 ? C1C2/(1r)Y1Y2/(1r)
Present Consumption Discounted value of Present
Consumption Future Income Discounted value of
Future Income Consumption Possibility Set
triangle that is enclosed by the Origin O, the
point Z, and the point C ???(??????)??????????
(1r)(Y1-C1)Y2C2 ? C1C2/(1r)Y1Y2/(1r)
??????????????????????????????
??O?????Y1(1-???r)?????Y2????Y1(1???r)
????????
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7.Optimal Decision of Consumption and Savings
??????????
Utility Maximization under a budget
constraint ?to select the optimal combination of
present and future consumption goods ?choose a
combination of goods at the point where the
budget line comes into contact with the
indifference curve ? the slope of the
indifference curve MRS the slope of the budget
line (1r) the marginal rate of time discount
MRDT 1 ?MRDT the market rate of
interest ????????????? ??????????????????????? ???
?????????????????? ? ????????MRS
????????(1r) (??????? MRTD1) ? MRTD?????
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8.The Role of Money and Financial markets
??????????
? Function as a store of value ?to play a role in
carrying over the value of present income into
the future ? to enlarge a consumption possibility
set from the rectangle OY1AY2 to the trapezoid
OY1AB ? to enhance the utility level from A to E
????????????????????????????????????OY1AY2????O
Y1AB?????????A??E???? ?When lending is possible
in the financial market ?to enlarge a
consumption possibility set from the trapezoid
OY1AB to the trapezoid OY1AC ? to enhance the
utility level from E to E'
???????????????????????OY1AB????OY1AC?????????E??E
'???? ? When lending is possible ?to enlarge a
consumption possibility set from the trapezoid
OY1AC to the triangle OZC ? to enhance the
utility level from A to F ???????????????????
????OY1AC???????OZC?????????A??F????
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9. Interest Income and Capital Gains ?????????
Revenue of or return on financial assets
interest income (dividend income in case of
stock) capital gain Capital gain sale price
purchase price Rate of returninterest
raterate of capital gain ?????????? ??????(?
???????)???? ???? ????-???? ??????(????-???
?)/????
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10-1. Standard Deviation as a Measure of Risk
?????????????
a rate of return x the sum of the rates
of return S S i0nxi the mean of the
rates of return m S / n the deviation of
the rate of return D xi m the sum of
squared deviations SSD S i0n(xi -m) the
variance the mean of squared deviations s2
SSD/n the standard deviation the root of the
variance s vs2 an average size of risk.
???x ?????? SS i0nxi ?????? m
S / n ??????? Dxi -m ?????????? SSD
S i0n(xi -m). ??s2SSD/n ??????????s
vs2 ???????????
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10-2. Safe Asset and Risky Asset ?????????
Safe Asset asset whose earnings are certain
(ex. deposits and fixed-interest-bearing
securities) and whose standard deviation is zero
Risky Asset asset whose earnings are not
certain (ex. stocks, mutual funds (investment
trust)) and whose standard deviation is not
zero ???????????????(??????????)
??????? ?????????????(????????)
??????????
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11. Marginal Utility and Risk Attitude
??????????
Assets held increase by one unit ? a variance
or standard deviation tends to increase risk-avert
er a consumer who experiences an additional
decrease in incremental utility or a decrease in
marginal utility when assets held
increase risk-lover a consumer who experiences
an increase in marginal utility when assets held
increase risk-neutral a consumer who
experiences a constant marginal utility when
assets held increase ?????1???? ?
???????????????? ??????????1??????????????????? ?
?????????1??????????????????? ??????????1???????
???????????
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12. Mean-variance and Risk Attitude ??????????
Assets held increase by one unit ? a variance
or standard deviation (risk) tends to
increase risk-averter a consumer who requires
higher rate of return when risk increases by one
unit risk-lover a consumer who allows lower
rate of return when risk increases risk-neutral
a consumer who do not care a constant rate of
return when risk increases ?????1???? ?
?????????(??)??????? ?????????1??????????????????
??????? ?????????1??????????????????????? ??????
???1???? ??????????????
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13. Expected Utility Theory ??????
a contract that has different conditions
depending upon uncertainties of the situation
contingent contract the goods whose conditions of
transaction are different depending upon
uncertainties of the situation contingent
goods ??????????????????? ??????
??????????????????????? Prize of lottery
inexpensive one x1, expensive one x2, winning
probability of x1 be p1, winning probability of
x2 be p2, (p1p21) utility obtained from the
prize X is uu(X) expected utility in the case of
winning is v(X)p1u(x1)p2u(x2) von Neumann -
Morgenstern theory of expected utility
maximization Under uncertainty, people maximize
an expected utility v(X) not utility u(X)
??????????x1??????x2??????????p1????p2
?p1p21???x???????? uu(X) ??????????
v(X)p1u(x1)p2u(x2) ???????????????????????
????????????????????????
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14-1. Expected Utility and Risk Preference
??????????
a consumer who gets larger utility u (X) obtained
by a certain prize X than an expected utility v
(X) in case of winning u(x1)Point A,
u(x2)Point B, v(X)Point V, u Point U
v(X)ltu ?prefer point U than point V ?risk
averter ? insurance premium premium that he
may pay instead of not buying a lottery
negative risk premium u(x1)A??u(x2)B??v(X)V??u
U?? v(X)ltu ?V???U???? ?????? ???????? ??
?????????? ??????????? ?????????
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14-2. Expected Utility and Risk Preference
??????????
v(X)gtu ?prefer point V than point U ?risk
lover ? risk premium premium that he may pay
if he can buy a lottery v(X)gtu
?U???V?????????? ???????? ?????????? ???????????

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14-3. Expected Utility and Risk Preference
??????????

• v(X)u ?indifferent between U and V
• ? risk neutral
• v(X)u ?U???V??????
• ??????
• Arrow Pratts degree of risk-aversion
• by using derivatives of utility function.
• Absolute risk aversion
• u (X)''/u '(X)
• 2nd derivative of utility/1st derivative of
  utility
• Relative risk aversion Xu''(X)/u '(X)
• X2nd derivative of utility/1st derivative of
  utility
• ?????????????
• ????????u(x)/u(x)???2???/???1???
• ????????xu(x)/u(x)x???2???/???1???

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15. Mean-variance Approach ??????

• the rate of return on risky assets be i1 in boom
  and i2 in recession
• Probability of boom and recession be p1 and p2
• Average rate of return u p1i1p2i2
• Variance v2p1 (i1-u)2p2 ( i2-u)2
• Average rate of return on risky assets A and B
  uA, uB
• Holding ratio of them a, b ( 1 b )
• Average rate of return the two assets µ auAbuB
• Variance of rates of return s2a2sA2b2sB22ab?sAs
  B
• Where ? correlation coefficient, sA standard
  deviation of A, sB standard deviation of B
• ?????????????i1??????i2?
• ????????p1????p2
• ?????up1i1p2i2?
• ???????v2p1 (i1-u)2p2 ( i2-u)2
• ????A?B??????uA?uB??????????a?b(1-a)
• ????????????????µauAbuB
• ??????s2a2sA2b2sB22ab?sAsB
• ??????????sAA??????sBB?????

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16-1. Investment Opportunities and Effective
Frontier ?????????????
Effective Frontier a set of points that bring
about a maximum rate of return with the same
variance among feasible investment
opportunities (1) the average return µ on risky
assets R and Q on the vertical axis, its
variance v2 on the horizontal axis and the
correlation coefficient ? the investment
opportunity line of their portfolio is ? when ?
1 ? a straight line that connects points R and
Q ? When ? -1 ? a polygonal line that connects
points R, Q and S ? When -1 lt? lt1 ? a curve that
connects points R and Q The upper part of the
convex curve Effective Frontier ????????
????????????????????????????????????? (1)????R?Q?
??????u????? ????v2????????????????? ??1????
R?Q????? ??-1????R???????S?Q?????? ?-1lt?lt1????R?
Q????? ??????????????????????
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16-2. Investment Opportunities and Effective
Frontier ?????????????
(2) In the portfolio consisting of two risky
assets and one safe asset, Effective frontier
a tangent line that connects the point C and the
curve between A and B The optimum combination of
risky assets is determined by the tangent point
M. (2) 2??????R?Q????1??????S?????
??????????????????? ?S????RQ????ST
?????????????T????
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17-1. Optimal Portfolio and Separation Theorem
??????????????
In the portfolio consisting of two risky assets R
and Q and one safe asset S, the effective
frontier a tangent line from the point S and to
the curve between R and Q ? the optimum
combination of risky assets the tangent point
T That determines the optimal portfolio of risky
assets 2??????R?Q?1??????S??????????????? ??????
????S????RQ???? ?????T???????? ?????????????
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17-2. Optimal Portfolio and Separation Theorem
??????????????
The indifference curve of a risk-averter is
tangent to a point E and E is the optimal
portfolio of the whole The optimal point T of
risky assets is determined independently from the
determination of the whole optimal point between
safe assets and risky assets ? Tobin's separation
theorem ???????????????????????E????
?E?????????????? ?????????T??????????????
???E??????? ??????????