Part 2 Microeconomic Analysis of FinancePart 2 Microeconomic Analysis of Finance　　　　　　　　金融のミクロ分析金融のミクロ分析

Chapter 4 Household FinanceChapter 4 Household Finance 　　　　　　　　　　　　　　家計の金融家計の金融

Naotsugu HAYASHI 林 直嗣Professor of Economics 経済学教授

Faculty of Business Administration 経営学部Hosei University 法政大学

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１． Flow of Funds of Household 　家計の資金循環

①sources of funds ＝ income (rewards to production factors such as labor, capital and land) + increase in financial debt + sales of financial assets

資金源泉＝所得（賃金、利子、地代等の要素報酬）＋金融負債の増加＋金融資産の売却

②ways to spend funds = consumption + real investment + increase in financial assets + repaying of financial liabilities

資金使途＝消費＋実物投資＋金融資産の増加＋金融負債の返済Since ① ＝② , a flow of funds equation for household is

income−consumption ＝ real investment ＋ net increase in financial assets

＝ real investment ＋ financial investment = investment

家計の資金循環式は　所得−消費＝実物投資＋金融資産の増加 金融負債の増加−　　　　　＝実物投資＋金融資産の純増加＝実物投資＋金融投資 = 投資　　

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２． Definition of Savings 貯蓄の定義

　 savings ＝ income – consumption = a residual of the income that was not consumed ⇒ investment = real investment + financial investment

　貯蓄＝所得−消費＝所得のうち消費しなかった残余 ⇒投資＝実物投資＋金融投資

consumption/income + savings/income = average propensity to consume + average propensity to save = 1 消費／所得＋貯蓄／所得＝平均消費性向＋平均貯蓄性向＝１

additional consumption/additional income + additional savings/additional income

= marginal propensity to consume + marginal propensity to save = 1 追加消費／追加所得＋追加貯蓄／追加所得

＝限界消費性向＋限界貯蓄性向＝１　

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3 ． Savings and Portfolio Selection 貯蓄と資産選択

　 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

　資産選択＝貯蓄を現金、預金、株式等の金融資産で保有するか、　　　　 実物資産に投資するかという判断　　Income　–　consumption　=　savings　=　real　investment　+　financial　

investment 所得 消費＝貯蓄＝実物投資＋金融資産の純増加（金融投資）− If　savings　=　real　investment,　then　financial　assets　do　not　

increase. 貯蓄＝実物投資ならば、金融資産は増加しない

　　Surplus　unit　…　savings＞ real　investment　⇒ 　net　increase　in　financial　asset

　　Deficit　unit　　…　savings＜ real　investment　⇒ 　net　increase　in　financial　liabilities　　

黒字主体…貯蓄＞実物投資　⇒　金融資産の純増加 赤字主体…貯蓄＜実物投資　⇒　金融負債の純増加

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４ -1 ． Indifference Curve of Consumption and Savings (Future Consumption) 消費と貯蓄の無差別曲線

utility function U ＝ U(C １ ,C ２ )

…2-period model of present and future times

効用関数　 U ＝ U(C １ ,C ２ )

…現在と将来の２期間モデル

indifference curve ＝ the combinations of

present and future consumption goods

that have the same level of utility

無差別曲線＝現在消費 C １と将来消費＝貯蓄 C ２の

組合わせで、同じ効用水準をもたらす組み合わせを表す曲線

Indifference map = a group or map of indifference curves

無差別曲線群（ indifference map ）＝無差別曲線の一群

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４ -2 ． Indifference Curve of Consumption and Savings (Future Consumption) 消費と貯蓄の無差別曲線

Marginal Rate of Substitution; MRS

= The ratio of increased future goods against

decreased present goods to maintain

the same level of utility

MRS ＝− dC ２／ dC １

＝ (∂U ／∂ C １ ) ／ (∂U ／∂ C ２ )

= marginal utility of present consumption

/ marginal utility of future consumption

⇔ a slope of the tangent line of indifference curve

限界代替率＝現在消費 C １を１単位減らすときに、同じ効用水準に

　とどまるために、増やさなければならない将来消費 C ２の割合

MRS ＝− dC ２／ dC １ ＝ (∂U ／∂ C １ ) ／ (∂U ／∂ C ２ )

　＝現在消費の限界効用／将来消費の限界効用　⇔　無差別曲線の接線の勾配

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５． Time Preference　　 時間選好

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 時間選好＝同じ財でも将来より現在消費 することを好む傾向、消費を将来に回すと、　待忍の必要、品質低下、入手困難　など不確実なことが起こる Marginal Rate of Time Preference; MRTP = the marginal rate of substitution

on the point C minus unity 限界時間選好率＝原点からの 45 度線と無差別曲線との交点の MRS−１Marginal Rate of Time Discounting; MRTD = the marginal rate of substitution

between present and future consumptions － 1

限界時差割引率＝現在消費と将来消費との限界代替率 １−

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６ -1 ． Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約（消費可能集合）①When saving is impossible Present Consumption Ｃ１≦ Present Income Y

１ 、 Future Consumption Ｃ２ ≦ Future Income Y ２ 。 Consumption Possibility Set = rectangle that is enclosed by the Origin O, present income Y １ , future income Y2, and the point A.

　貯蓄がない場合の予算制約 現在消費≦現在所得、将来消費≦将来所得　　原点 O と現在所得 Y １、将来所得 Y ２、及び A 点で囲まれる四角形②　When saving is possible Present Consumption Ｃ１≦ Present Income Y １

、 Future Consumption Ｃ２ ≦ (Present Income-Savings) +Future Income Y２ 。 Consumption Possibility Set = trapezoid that is enclosed by the Origin O, present income Y １ , present income Y １ + future income Y2 (point B), and the point A

　貯蓄がある場合の予算制約 現在消費≦現在所得、将来消費≦ ( 現在所得－貯蓄 ) ＋将来所得原点 O と現在所得 Y １、将来所得 Y ２＋現在所得 Y １　 (B 点 ) 、及び A 点で囲まれる台形

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６ -2． Inter-temporal Budget Constraint : Consumption Possibility

Set 異時点間の予算制約（消費可能集合） ③ When lending is possible Present Consumption Ｃ１≦ Present Income Y １

、 Future Consumption Ｃ２ ≦ (Present Income-Savings)(1+Interest Rate r) +Future Income Y ２ 。 Consumption Possibility Set = trapezoid that is enclosed by the Origin O, present income Y １ , (1+Interest Rate r) present income Y １ + future income Y2 (point C), and the point A

Market Interest Rate = Discounting Rate for Calculating Present Value

貸付が可能な場合の予算制約　現在消費Ｃ１≦現在所得 Y １ 、将来消費Ｃ２ ≦ ( 現在所得 - 貯蓄 )(1+ 利子率 r) + 将来所得 Y ２

　原点 O と現在所得 Y １、将来所得 Y ２＋現在所得 Y １ ×(1 ＋利子率 r) 、及びA 点で囲まれる台形

∴市場利子率 r ＝現在価値を求める割引率

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６ -3． Inter-temporal Budget Constraint : Consumption Possibility

Set 異時点間の予算制約（消費可能集合） ④ When lending and borrowing are possible (1 ＋ Interest Rate r)( Present

Income Y １ − Present Consumption C １ ) ＋ Future Income Y ２＝ Future Consumption C ２ ⇔　 C １＋ C ２／ (1 ＋ r) ＝ Y １＋ Y ２／ (1 ＋ r)

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

金融が（貸付も借入も）可能な場合の予算制約　 (1 ＋ r)(Y １− C １ ) ＋ Y ２

＝ C ２ ⇔　 C １＋ C ２／ (1 ＋ r) ＝ Y １＋ Y ２／ (1 ＋ r)　現在消費＋将来消費の割引現在価値＝現在所得＋将来所得の割引現在価値 　原点 O と現在所得 Y １ ×(1−利子率 r) 、将来所得 Y ２＋現在所得 Y １ ×(1

＋利子率 r) で囲まれる三角形

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７． Optimal Decision of Consumption and Savings 消費と貯蓄の最適決定

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 (1+r)

= the marginal rate of time discount MRDT + 1

∴MRDT = the market rate of interest

予算制約のもとで効用最大化

⇒現在消費と将来消費との最適な組み合わせを選択

⇒予算制約線と無差別曲線との接点で最適決定

⇒ 無差別曲線の勾配 MRS

＝ 予算制約線の勾配 (1 ＋ r)

＝ ( 限界時差割引率 MRTD ＋ 1)

∴ MRTD= 市場利子率

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８． The Role of Money and Financial markets 貨幣と金融市場の役割

① 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 OY １ AY ２ to the trapezoid OY １ AB ⇒ to enhance the utility level from A to E

貨幣の価値貯蔵機能⇒現在所得の価値を将来に持ち越す⇒消費可能集合を四角形 OY １ AY ２から台形 OY １ AB へ拡大⇒効用水準を A から E へ高める

②When lending is possible in the financial market

⇒ to enlarge a consumption possibility set from the trapezoid OY １ AB to the trapezoid OY １ AC ⇒ to enhance the utility level from E to E'

金融市場で貸付が可能な場合⇒消費可能集合を台形 OY １ AB から台形OY １ AC へ拡大⇒効用水準を E から E' へ高める

③ When lending is possible ⇒ to enlarge a consumption possibility set from the trapezoid OY １ AC to the triangle OZC ⇒ to enhance the utility level from A to F

金融市場で借入も可能な場合⇒消費可能集合を台形 OY １ AC から三角形台形 OZC へ拡大⇒効用水準を A から F へ高める

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９． Interest Income and Capital Gains 利子所得と資本利得

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　return ＝ interest　rate ＋ rate　of　capital　gain

　資産の保有・運用による収益＝利子所得（株式では配当所得）＋資本利得

　　資本利得 ＝売却価格 取得価格−収益率＝利子率＋（売却価格 取得価格）／取得価格−

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10-1. Standard Deviation as a Measure of Risk 危険の指標としての標準偏差

a rate of return = x

the sum of the rates of return = S = Σ i=0nxi

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 = Σ i=0n(xi － m)

the variance = the mean of squared deviations = σ2 = SSD ／ n

the standard deviation = the root of the variance = σ = √σ2

= an average size of risk.

収益率＝ｘ　　 収益率の総和＝ S=Σ i=0

nxi 　

収益率の平均＝ m = S / n

　各収益率の偏差＝ D ＝ xi － m

各収益率の偏差平方和＝ SSD = Σ i=0n(xi － m).

　分散＝ σ2 ＝ SSD ／ n 　　分散の平方根＝標準偏差＝ σ = √σ2 …リスクの平均的な大きさ

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10-2. Safe Asset and Risky Asset 安全資産と危険資産

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

安全資産＝収益が確定している資産 ( 例：預金や確定利付証券）　　　　　　標準偏差はゼロ 危険資産＝収益が不確定な資産（例：株式、投資信託）　　　　　　標準偏差はゼロでない

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11. Marginal Utility and Risk Attitude　　限界効用とリスク態度

Assets held increase by one unit

⇒ a variance or standard deviation tends to increase

risk-averter = 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

保有資産が 1 単位増加 ⇒ 分散ないし標準偏差が増加する傾向危険回避者＝保有資産が 1 単位増加するとき、限界効用が逓減する人危険愛好者＝保有資産が 1 単位増加するとき、限界効用が逓増する人危険中立者＝保有資産が 1 単位増加するとき、限界効用が不変の人

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12. Mean-variance and Risk Attitude 平均・分散とリスク態度

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

保有資産が 1 単位増加 ⇒ 分散ないし標準偏差（危険）が増加する傾向危険回避者＝リスクが 1 単位増加するとき、平均収益も増えなければいけな

い人危険愛好者＝リスクが 1 単位増加するとき、平均収益は低くなっても良い人危険中立者＝リスクが 1 単位増加するとき、平均収益が不変の人

　

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13. Expected Utility Theory 期待効用理論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

　不確実な事態に応じて取引条件が違う契約 =　 条件付き契約　不確実な事態に応じて取引条件が違う財＝条件付き財Prize of lottery = inexpensive one x1, expensive one x2,

winning probability of x1 be p1, winning probability of x2 be p2, (p １＋ p ２＝ 1)

utility obtained from the prize X is u ＝ u(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)

　宝籤の賞金＝低額賞金は x １、高額賞金は x ２、当選する確率＝低額が p １、高額が p ２　　、 p １＋ p ２＝１、賞金 x から得られる効用　 u ＝ u(X)

　当選の場合の期待効用 v(X) ＝ p １ u(x １ ) ＋ p ２ u(x ２ )

⇒ ノイマン＝モルゲンシュテルンの期待効用最大化説　　 不確実性の下では効用ではなく期待効用を最大化する

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14-1. Expected Utility and Risk Preference　　　　　 　期待効用とリスク選好

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(x １ )…Point A, u(x ２ )…Point B, v(X)…Point V, u …Point U

v(X) ＜ u ⇒prefer point U than point V

⇒ risk averter

γ ＝ insurance　premium = premium that he may pay instead of not buying a lottery = negative risk premium

u(x １ )…A 点、 u(x ２ )…B 点、 v(X)…V 点、 u　 　 … U 点、

　　v(X) ＜ u　 　　 ⇒ V 点より U 点を選好⇒ 危険回避者γ ＝保険プレミアム＝宝籤に参加しない代わりに　支払って良いプレミアム＝負の危険プレミアム

(x)̂(x)̂

(x)̂(x)̂

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14-2. Expected Utility and Risk Preference　　　　　 　期待効用とリスク選好

v(X) ＞ u 　　 ⇒ prefer point V than point U ⇒ risk lover

γ ＝ risk premium

＝　 premium that he may pay if he can buy a lottery

v(X)＞ u　 　　 ⇒ U 点より V 点を選好⇒危険愛好者γ ＝危険プレミアム＝宝籤に参加できるなら支払って良いプレミアム

(x)̂

(x)̂

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14-3. Expected Utility and Risk Preference　　　　　 　期待効用とリスク選好

v(X) ＝ u　 　　 ⇒ indifferent　between　U　and　V⇒ 　risk　neutralv(X) ＝ u　 　　 ⇒ U 点でも V 点でも無差別⇒ 危険中立的

Arrow – Pratt’s 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)| = |X·2nd derivative of utility/1st derivative of utility |

アロー＝プラットの危険回避度絶対的危険回避度＝｜ u”(x)/u’(x)｜＝｜効用の２次微分／効用の１次微分｜

相対的危険回避度＝｜ xu”(x)/u’(x)｜＝｜ x ・効用の２次微分／効用の１次微分｜

(x)̂

(x)̂

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15. Mean-variance Approach 　平均・分散接近

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 = p １ i １＋ p ２ i ２

Variance v ２＝ p １ (i １− u) ２＋ p ２ ( i ２− u) ２

Average rate of return on risky assets A and B = u Ａ , u ＢHolding ratio of them = a, b ( = 1 – b )

Average rate of return the two assets μ = au Ａ＋ bu Ｂ

Variance of rates of return σ ２＝ a ２ σ Ａ２＋ b ２ σ Ｂ２＋ 2abρσ Ａ σ ＢWhere ρ = correlation coefficient, σ Ａ = standard deviation of A, σ Ｂ =

standard deviation of B

　危険資産の収益率＝好況時では i １、不況時では i ２、　起こる確率＝好況が p １、不況が p ２

　平均収益率 u ＝ p １ i １＋ p ２ i ２、　危険を表す分散 v ２＝ p １ (i １− u) ２＋ p ２ ( i ２− u) ２

　危険資産 A と B の平均収益率 u Ａ、 u Ｂ、それぞれの保有比率 a 、 b( ＝1−a)

　両者のポートフォリオの平均収益率 μ＝ au Ａ＋ bu Ｂ

　　収益率の分散 σ ２＝ a ２ σ Ａ２＋ b ２ σ Ｂ２＋ 2abρσ Ａ σ Ｂ ただし、 ρ＝相関係数、 σ Ａ＝ A の標準偏差、 σ Ｂ＝ B の標準偏差

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16-1. Investment Opportunities and Effective Frontier　　　　　 　投資機会と有効フロンティア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 = ρ

the investment opportunity line of their portfolio is

① when ρ = 1 ⇒ a straight line that connects points R and Q

② When ρ = -1 ⇒ a polygonal line that connects points R, Q and S

③ When -1 <ρ <1 ⇒ a curve that connects points R and Q

The upper part of the convex curve = Effective　Frontier　有効フロンティア　　＝実行可能な投資機会集合のうちで分散が同じなら最大

の収益率をもたらす点の集合（１）危険資産 R と Q の収益率の平均 u を縦軸に、　　　　　その分散 v ２を横軸にとった平面で、投資機会線は①ρ ＝１の場合⇒ R と Q を結ぶ直線②ρ ＝－１の場合⇒ R と分散がゼロの S” と Q を結ぶ折れ線③－１＜ ρ ＜１の場合⇒ R と Q を結ぶ曲線　　　　⇒この曲線の上方に凸の部分が有効フロンティア

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16-2. Investment Opportunities and Effective Frontier　　　　　 　投資機会と有効フロンティア

(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つの危険資産 R と Q 、および１つの安全資産 S とからなる　ポートフォリオでは、有効フロンティアは 点 S から曲線 RQへの接線 ST 危険資産の組合せはその接点 T で決まる

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17-1. Optimal Portfolio and Separation Theorem　　　　　 最適ポートフォリオと分離定理

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つの危険資産 R と Q 、１つの安全資産 S とからなるポートフォリオでは、

有効フロンティア⇒点 S から曲線 RQへの接線 ⇒その接点 T で危険資産だけの最適ポートフォリオが決まる

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17-2. Optimal Portfolio and Separation Theorem　　　　　 最適ポートフォリオと分離定理

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

　危険回避型の消費者の無差別曲線はその接線上の点 E で接する　⇒ E 点が全体の最適ポートフォリオ ∴危険資産の最適点 T は安全資産と危険資産の全体の　 最適点 E の決定とは独立　⇔トービンの分離定理

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