Macroeconomics マクロ経済学

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1 Macroeconomics マクロ経済学
Chap.14 Determination of Equilibrium Income and Multiplier Mechanism 第14章　均衡所得の決定と乗数機構

2 Consumption expenditure + investment expenditure = total demand of society = Total supply of national product ⇒ Equilibrium national product and balanced national income are determined Adjustment mechanism in product market ⇒ 2 types, (1) Elastic Price Adjustment Mechanism ... Equilibrium income and equilibrium price are determined at the same time by classical school, neoclassical school, Walrasian price adjustment, Walrasian stability condition (2) Quantity adjustment mechanism under a fixed price economy ... Keynesian school, equilibrium income is decided under given fixed price by Marshallian quantity adjustment, Marshallian stability condition Stable equilibrium even with excess supply left = under-employment equilibrium ⇒ Analysis of 45 °line which is the idea of Samuelson is effective Full employment equilibrium without excess supply = full-employment equilibrium, classical school full employment equilibrium Ripple effects on equilibrium income when independent expenditures etc. change= multiplier effect 消費支出＋投資支出＝社会の総需要 ＝国民生産物の総供給　⇒　均衡国民生産物、均衡国民所得が決定 生産物市場での調整機構⇒2種類、 （１）伸縮的な価格調整機構…古典派や新古典派、ワルラス的な価格調整により均衡所得と均衡物価とが同時に決定、ワルラス的な安定条件 （２）固定価格経済のもとでの数量調整機構…ケインズ派、マーシャル的な数量調整により所与の固定価格の下で均衡所得が決定、マーシャル的な安定条件 超過供給を残したままでも安定均衡＝不完全雇用均衡⇒サミュエルソンの発案になる45°線の分析が有効 超過供給のない完全雇用の均衡＝完全雇用均衡、古典派の完全雇用均衡 独立支出などの与件が変化した場合、均衡国民所得に波及的に及ぼす効果＝乗数効果

3 １．Equilibrium Income in Flexible Price Economy 伸縮価格経済における均衡所得
Aggregate demand AD＝aggregate expenditure AE= private consumption C + private investment I + government expenditure G + net export (X - M) For simplification, to discrete government part and overseas part, or to divide consumption and investment even in government and overseas parts ⇒ aggregate demand AD = consumption C + investment I Aggregate supply AS = national product Y Flexible price economy ⇒ Price is flexible and price adjustment works well. Aggregate demand AD and aggregate supply AS both respond to changes in price P, and are a function of price P. 総需要（aggregate demand）AD＝総支出（aggregate expenditure）AE＝民間消費C＋民間投資I＋政府支出G＋純輸出（X－M） 単純化のために政府部門と海外部門を捨象するか、政府部門と海外部門も消費と投資に分離⇒　　総需要AD＝消費C+投資I 総供給（aggregate supply）AS＝国民生産物Y 伸縮価格経済（flexible price economy）⇒価格が伸縮的で価格調整（price adjustment）が十分に働く、総需要ADも総供給ASも物価Pの関数、物価Pの変化に反応

4 １B．Equilibrium Income in Flexible Price Economy 伸縮価格経済における均衡所得
Aggregate demand AD is the downward curve of aggregate demand, aggregate supply AS is the upward curve of aggregate supply. Aggregate demand AD = aggregate supply AS ⇒ AD = C + I = Y　 at equilibrium point E. Equilibrium price level P *, equilibrium national product (national income) Y *, are determined as shown in the figure. Prices P '> P * ⇒ Excess supply ES, prices decline, supply drops, demand increases, equilibrium E is restored. Prices P '' <P * ⇒ Excess demand ED, prices rise, supply increases, demand decreases, equilibrium E is restored. ⇒ Equilibrium E is Walrasian equilibrium. The demand and supply rule is satisfied ⇒ The excess demand function ED (P) = AD (P) - AS (P) is downward to the right and the Walrasian stability condition is satisfied. Classical school (J. P. Say) "Supply always produces its own demand” = Say's law 総需要ADは右下がりの総需要曲線、総供給ASは右上がりの総供給曲線 総需要AD＝総供給AS　⇒均衡点Eでは、AD＝C＋I＝Y 均衡物価水準P*、均衡国民生産物（国民所得）Y*、が図のように決定 物価P’＞P*⇒超過供給ES、物価が下落、供給は減り、需要は増え， 均衡点E　が回復 物価P’’＜ P*⇒超過需要ED、物価が上昇、供給は増え、需要は減り、 均衡Eをワルラス均衡（Walrasian equilibrium） 需要・供給の法則が満たされる⇒超過需要関数ED(P)＝AD(P)－AS(P)は 右下がりで、ワルラスの安定条件は満たされる 古典派のセイ（J. P. Say） 「供給は必ず自らの需要を作り出す」 セイの法則（Say’s Law）　　 　　　　14－1図　価格調整による均衡所得　　

5 ２．Equilibrium Income in Fixed Price Economy 固定価格経済における均衡所得
Fixed price economy= Price is fixed and price adjustment does not work, quantity adjustment　works. Prices P are constant, and the nominal and real values are the same. Aggregate demand is insufficient in comparison with aggregate supply. As there is excess supply, so prices do not rise and quantity adjustment works. ⇒ Aggregate supply AS is under a fixed price up to full employment level YF. So aggregate supply curve AS is horizontal. Since prices P rise after reaching the full employment level YF , AS goes up to the right. Downward rigidity of prices continue until the full employment. 固定価格経済（fixed price economy）＝価格が固定的で 価格調整が働かずに、数量調整（quantity adjustment）。 物価Pが一定で、名目値と実質値は同じ。 総需要が総供給に対して不足、超過供給があるので物価P は上がらず数量調整 ⇒総供給ASは完全雇用水準YFまでは固定価格で水平の 総供給曲線AS 完全雇用水準YFにいたってから物価Pが上昇するので右上がり 完全雇用に至るまでは価格の下方硬直性 （downward price rigidity）があるという。 4－2図　数量調整による均衡所得

6 ２B．Equilibrium Income in Fixed Price Economy 固定価格経済における均衡所得
Production volume is lower than that at equilibrium point E like point A ⇒ Marshallian excess demand price ADA, stock removal and quantity adjustment work to recover equilibrium point E. Production volume is higher than that at equilibrium point E like point B ⇒ Marshallian excess supply price BDB, inventory buildup and quantity adjustment work to recover E. Non-Walrasian equilibrium, Keynesian equilibrium, Marshallian stability condition Stable convergence to equilibrium point E by quantity adjustment in fixed price economy, stable condition is satisfied. Equilibrium national income Y * under equilibrium price P * ≠ full-employment national income YF From the aspect of price adjustment, excess supply (YF－Y*) remains under P* Supply and demand are unbalanced. This excess supply is an excess supply of labor = unemployment Keynes called an under-employment equilibrium, point F is a full-employment equilibrium 生産量がA点のように均衡点Eより少ない水準⇒マーシャル流の超過需要価格ADA、在庫取り崩し、数量調整が働いて均衡点Eを回復 生産量がB点のように均衡点Eより多い水準⇒マーシャル流の超過供給価格BDB、在庫積み増し、数量調整が働いてEを回復 非ワルラス的均衡（non-Walrasian equilibrium）、ケインズ均衡（Keynesian equilibrium）、マーシャルの安定条件 固定価格経済での数量調整により均衡点Eに安定的に収束、安定条件は満足 均衡物価P*のもとでの均衡国民所得Y*は≠完全雇用国民所得YF 価格調整の側面から見ると、P*のもとで超過供給（YF－Y*）が残ったまま 需給は不均衡。この超過供給は労働の超過供給＝失業 ケインズは不完全雇用均衡（under-employment equilibrium）、F点は完全雇用均衡（full-employment equilibrium）

7 ３．Flexible Price Economy vs. Fixed Price Economy 伸縮価格経済か固定価格経済か
Keynesian assumption = price adjustment by flexible prices does not work sufficiently during recession, and prices are downward rigid at least in the short run, so quantity adjustment works under fixed price economy In the age when monthly data of consumer price index can not be obtained, it can not be verified. Figure 14-3 shows the rate of change in monthly consumer price index since 1971 compared with the same month in the previous year in Japan. There is no time when the inflation rate is zero percent and it can be regarded as a fixed price in the short run within at least one year in the past 40 years. Fig Consumer price inflation rate (compared with the same month of the previous year) 1971 – 2011 ケインジアンの仮定＝不況時は伸縮価格による価格調整 が十分に働かず、少なくとも短期的には価格が下方硬直 性を持つので、数量調整の固定価格経済 消費者物価指数などの月次のデータが入手できない時 代には、検証できない 日本の月次の消費者物価指数を調べる、1971年以降の 消費者物価指数の前年同月比の変化率を図示、14-3図。 過去40年間で少なくとも1年以内の短期において物価上 昇率がゼロ％で固定価格と見なせるような時期は存在し ない。 　14-3図　消費者物価上昇率（前年同月比）1971～2011年

8 ３B．Flexible Price Economy vs. Fixed Price Economy 伸縮価格経済か固定価格経済か
The lowest rate of change in consumer price index is Its 12 months average rate of change is 0.029%, almost zero%. However, monthly rates of change can not be regarded as fixed price, because they changed every month between -0.5% and 0.8% Fig. Consumer price inflation rate (year-on-year comparison) 2004 消費者物価変化率が最も小さい2004年、12ヶ月の平均物価変化率は0.029％とほぼゼロ％。しかし月ごとの変化率は、－0.5％から0.8％の間で毎月変化、固定価格と見なすことはできない。 　14-4図　消費者物価上昇率（前年同月比）2004年

9 ４. Determination of Equilibrium Income by 45 degree Line 45度線による均衡所得の決定
Equilibrium in macro product market 　　 AD＝C＋I＝Y In the absolute income hypothesis, the consumption function is C＝C（Y）＝a＋cY The marginal propensity to consume c is 0 <c <1, the slope of the consumption function is less than 1. Investment I is independent investment that does not depend on national income Y ⇒ Horizontal investment curve I Aggregate demand that is a sum of consumption and investment= Aggregate demand curve AD which shifts consumption curve C by the amount of independent investment I On 45 degree line devised by P. A. Samuelson, always aggregate demand AD = aggregate Supply AS=Y Therefore, at the intersection point E of aggregate demand curve AD and the 45 degree line, aggregate demand AD = aggregate supply Y ⇒ equilibrium national income = equilibrium aggregate supply Y * 14-5 Figure 45, Equilibrium Income by 45 degree Line マクロの生産物市場における均衡　　AD＝C＋I＝Y 絶対所得仮説では消費関数は 　　C＝C（Y）＝a＋cY 限界消費性向cは、0＜c＜1、消費関数の勾配は1より小。 投資Iは国民所得Yに依存しない独立投資（independent investment）⇒水平の投資曲線I 消費と投資を合計した総需要＝消費曲線Cを独立投資Iの分だけ 上方へシフトさせた総需要曲線AD サミュエルソン（P. A. Samuelson）が考案した45°線（45 degree line）上では常に総需要AD＝総供給AS=Y よって総需要曲線ADと45°線との交点Eで総需要AD＝総供給Y ⇒均衡国民所得（equilibrium national income）＝均衡総供給Y* 14－5図　45°線による均衡所得

10 ４B． Determination of Equilibrium Income by 45 degree Line 45度線による均衡所得の決定
Aggregate demand increases and equilibrium point shifts to A ⇒ equilibrium national income = equilibrium aggregate supply = YA When demand side determines equilibrium income (demand - determined ), its aggregate demand is called effective demand, Determination of income based on effective demand= the principle of effective demand Aggregate supply Y is larger than aggregate demand AD at point A⇒ Excess supply causes remaining unsold ADA , it causes increase in inventory, firms decrease production to restore equilibrium E. Aggregate demand AD is larger than aggregate supply Y at point B ⇒ Excess demand causes shortage of products BDB, it causes decrease in inventory, firms increase production to restore equilibrium E. Marshallian quantity adjustment mechanism rather than price adjustment 14-6 Figure Savings = equilibrium income from investment 総需要が増えて均衡点がA点に上方シフト⇒ 均衡国民所得＝均衡総供給はYA、 需要側が決定因となって（demand-determined）均衡所得を 決める場合、その総需要を有効需要（effective demand） 有効需要による所得決定を有効需要の原理 （principle of effective demand） 総供給Yが総需要ADより大きいA点⇒超過供給によって売れ残りADA、 売れ残りは在庫積み増し、企業は生産量を減らして均衡点Eにまで戻す 総供給Yより総需要ADが大きいB点⇒超過需要によって品不足BDB、 品不足は在庫の取り崩し、企業は生産量を増やして均衡点Eにまで戻す 価格調整ではなくマーシャル的な数量調整機構 　14-6図　貯蓄＝投資による均衡所得

11 ５．Balance of Savings and Investment and Equilibrium Income 貯蓄・投資の均衡と均衡所得
Equilibrium in macro product market, AD＝C＋I＝Y National product = national income Y,　Y＝C＋S Equilibrium of product market and balance of savings and investment are equivalent (⇔) AD＝AS　⇔　S＝I Savings function of absolute income hypothesis S（Y）＝Y－（a＋cY）＝－a＋（1－c）Y　　　0＜1－c＜1 The savings curve S whose intercept is -a and whose gradient is marginal propensity to save = 1 – c Investment is independent investment ⇒ Horizontal investment curve I Point E achieves the product market equilibrium that Savings S = Investment I Point A where Savings S is greater than Investment I ⇒ Points left unused (or insufficient investment) due to excess supply of savings. Companies reduce production and return to equilibrium point E Saving S is smaller than Investment I Point B ⇒ Insufficient savings (or unsatisfied investment) due to excess demand of savings. The company increases the production volume and returns to the equilibrium point E マクロの生産物市場における均衡、　AD＝C＋I＝Y 国民生産物＝国民所得Y、　Y＝C＋S 生産物市場の均衡と貯蓄・投資の均衡は同値（equivalent：⇔）　　AD＝AS　⇔　S＝I 絶対所得仮説の貯蓄関数　S（Y）＝Y－（a＋cY）＝－a＋（1－c）Y　　　0＜1－c＜1 切片が－a、勾配が限界貯蓄性向s＝1－c の貯蓄曲線S 投資は独立投資⇒水平の投資曲線I 貯蓄S＝投資Iの生産物市場均衡を達成するE点 貯蓄Sが投資Iより大きいA点⇒貯蓄の超過供給によって貯蓄の使い残し（ないし投資不足）AIA。企業は生産量を減らして均衡点Eにまで戻る 貯蓄Sが投資Iより小さいB点⇒貯蓄の超過需要によって貯蓄不足（ないし投資の未充足）BDB。企業は生産量を増やして均衡点Eにまで戻る

12 ６．Inflation Gap and Deflation Gap インフレ・ギャップとデフレ・ギャップ
Economy is in a state of full employment ⇒ Aggregate demand curve AD has an intersection with a 45 °line at an equilibrium point F.＝Full-employment national income YF In the case that AD’ curve becomes higher than AD curve⇒ Equilibrium point E‘ is unachievable (infeasible) ⇒ Difference AF between aggregate demand AD' and full employment national income YF is an inflationary gap ⇒ Adjustment through price increases Keynes called it a true inflation. In the long run, firms increase employment capacity and capital equipment to increase full employment national income YF . Figure 14-7 Inflationary gap and deflationary gap 経済が完全雇用の状態⇒総需要曲線ADは均衡点Fで45° 線と交点＝完全雇用国民所得 （full-employment national income） YF AD’曲線のようにAD曲線より上に来る場合⇒均衡点はE’は 達成不可能（infeasible）⇒総需要AD’と完全雇用国民所得 YFとの差額AFを、インフレーション・ギャップ（inflationary gap）、インフレ・ギャップ⇒物価上昇による調整 ケインズは真性インフレーション（true inflation） 長期では企業は雇用能力や資本設備を拡大して、完全 雇用国民所得YFを増大 14-7図　インフレ・ギャップとデフレ・ギャップ

13 ６B．Inflation Gap and Deflation Gap インフレ・ギャップとデフレ・ギャップ
In the case that AD'' curve becomes lower than AD curve⇒ Equilibrium point is E'', equilibrium national income Y* The difference BF between full employment national income YF and aggregate demand AD'' , the deficiency of aggregate demand is a deflationary gap. In a fixed price economy with price downward rigidity, As unsold inventories occur, inventories are added up. In a flexible price economy, price adjustment mechanism works, the prices P fall Figure 14-7 Inflation gap and deflation gap AD’’曲線のようにAD曲線より下に来る場合⇒均衡点はE’’、均衡国民所得Y* 完全雇用国民所得YFと総需要AD’’との差額BF、総需要の不足分を、デフレーション・ギャップ（deflationary gap）、デフレ・ギャップ 価格の下方硬直性がある固定価格経済では、 売れ残りが生じて在庫の積み増し 価格が伸縮的な経済では価格調整 機構が働き、物価Pが下落

14 ７．Multiplier Effect乗数効果
Analyze the effect of affecting equilibrium income when given conditions change. Independent investment increases, investment curve shifts upward from I0 to I1 ⇒ Aggregate demand AD0 shifts up to AD1 ⇒ The equilibrium point of the product market shifts from E0 to E1 ⇒ Equilibrium national income also shifts from Y0からY1. Y1－Y0＝(C(Y1)―C(Y0))＋(I1－I0) If the change is represented by differentiation, dY＝C’(Y)dY＋dI　 C’(Y) is the marginal propensity to consume c, ∴ dY＝（1／（1－c））dI　 （1－c） is the marginal savings propensity s ⇒ When independent investment increases by dI, income increases by 1 / (1 - c) = 1 / s times 1 / (1-c) = 1 / s is multiplier, the increment dI is multiplicand, this effect is multiplier effect, The greater the marginal propensity to consume c is, the smaller the marginal propensity to save s is, the larger the multiplier becomes. 与件が変化した場合に均衡所得に影響を及ぼす効果を比較静学で分析 独立投資が増加、投資曲線がI0からI1へ上方シフト⇒総需要AD0はAD1へと上方シフト⇒生産物市場の均衡点はE0点からE1点へとシフト⇒均衡国民所得もY0からY1へとシフト、 　　Y1－Y0＝(C(Y1)―C(Y0))＋(I1－I0) 変化分を微分で表せば　dY＝C’(Y)dY＋dI　C’(Y)は限界消費性向c、 　　dY＝（1／（1－c））dI　（1－c）は限界貯蓄性向s 　⇒独立投資がdIだけ増加すると、1／（1－c）＝1／s倍だけの所得 増加dY, 倍率1／（1－c）＝1／sを乗数（multiplier）、増分dIを 被乗数（multiplicand）、この効果を乗数効果（multiplier effect） 限界消費性向cが大きいほど、限界貯蓄性向sが小さいほど、乗数は 大きくなる

15 ７B．Multiplier Effect 乗数効果
　The equilibrium relationship of savings and investment can also explain multiplier effect. Independent investment increases, investment curve shifts upward from I0 to I1 ⇒ Equilibrium point of savings=investment shifts from E0 to E1 ⇒ equilibrium national income also shifts from Y0 to Y1 　 Y1－Y0＝(C(Y1)―C(Y0))＋(I1－I0), dY＝C’(Y)dY＋dI, dY＝(1/(1－c)) , dI＝(1/s)dI <Numerical example> Increase in independent investment by 100 million yen dI, income of people engaged in the investment related sector increase by 100 million yen. The marginal propensity to consume c is 0.6, 6 billion yen is their new consumption expenditure ⇒ Increase in income of people engaged in consumer goods related sector by 600 billion yen → 360 billion yen is their new consumption expenditure Increase in independent expenditure dI → Increase in income dY → Increase in consumption expenditure dC → Increase in income dY → these successive process = multiplier process, multiplier mechanism 乗数効果は貯蓄・投資の均衡関係を使っても同様に説明できる.　独立投資が増加、投資曲線がI0からI1へ上方シフト⇒貯蓄＝投資の均衡点はE0点からE1点へとシフト⇒均衡国民所得もY0からY1へとシフト Y1－Y0＝(C(Y1)―C(Y0))＋(I1－I0), dY＝C’(Y)dY＋dI 　　dY＝(1/(1－c)) , dI＝(1/s)dI ＜数値例＞1億円の独立投資の増加dI、その投資関連部門に従事 している人々の所得は1億円増。限界消費性向cを0.6、6000億円が 彼らの新たな消費支出⇒消費財関連部門に従事している人々の所得 は6000億円増⇒3600億円が彼らの新たな消費支出　独立支出の 増加dI→所得の増加dY→消費支出の増加dC→所得の増加dY→ ……と続くプロセス＝乗数過程（multiplier process）、乗数機構 （multiplier mechanism）

16 ７C．Multiplier Effect 乗数効果
Infinite geometric series of 100 million yen million yen × million yen × 0.62＋… …= ¥ 100 million x 1 / (1-0.6) = ¥ 100 million / 0.4 = 250 million yen Increase in independent expenditure dI by 100 million yen increases income dY by 2.5 times increase, 1 / (1 - c) = 2.5 multiplier Japan‘s marginal propensity to consume c is about 0.6, the multiplier is 2.5, America’s marginal propensity to consume is about 0.8, the multiplier is 1 / (1-0.8) = 5 twice as large as Japan How do savings increase in the multiplier process? Marginal propensity to save is s = 0.4, of 40 million yen out of the initial income increase of 100 million yen, savings increased, while 24 million yen of the next 600 million yen income increase is an increase in savings. Infinite geometric series of 100 million yen x million ×0.6× million ×0.62× = 100 million yen x 1 / (1-0.6) x 0.4 = 100 million yen Savings equal to the initial incremental investment of 100 million yen are generated, dI＝dS The fact that the multiplier effect works is in the under-employment economy before full employment.　At full employment there is no multiplier effect, full employment constraint of multiplier effect 1億円＋1億円×0.6＋1億円×0.62＋…という無限等比級数＝1億円×1／（1－0.6）＝1億円／0.4＝2.5億円 1億円の独立支出の増加dIが2.5倍の所得の増加dY、1/(1－c)＝2.5が乗数 日本の限界消費性向cは約0.6で乗数は2.5、米国の限界消費性向は約0.8で、乗数は1/(1－0.8)＝5　日本の2倍 　乗数過程で貯蓄はどれだけ増えるか。限界貯蓄性向はs＝0.4、当初の1億円の所得増加のうち4000万円が貯蓄の増加となり、次の6000万円の所得増加のうち2400万円が貯蓄の増加、 1億円×0.4＋1億円×0.6×0.4＋1億円×0.62×0.4＋…という無限等比級数＝1億円×1/（1－0.6）×0.4＝1億円 　当初の独立投資の増加額1億円と同額の貯蓄が生成、dI＝dS 　乗数効果が働くのはあくまで完全雇用にいたる前の不完全雇用経済 　完全雇用では乗数効果はない、乗数効果の完全雇用制約

17 ８．Paradox of Savings 貯蓄のパラドックス
Balance of savings = investment, SI balance, S0＝I0 , equilibrium national income is Y0 ⇒ As the motivation to save becomes stronger, savings curve S0 shifts upward to S1 ⇒ The average propensity to save S/Y rises ⇒ equilibrium point shifts left from E0 to E1 ⇒ Equilibrium National Income shifts left from Y0 to Y1 As savings increase, consumption decreases, through its multiplier effect income is reduced ⇒ paradox of savings Investment also increases by the same amount as the increase in savings ⇒ To maintain the same equilibrium income Savings increase, capital supply increases in the capital market, market interest rate i falls down, investment increases from I0 to I1, equilibrium point shifts to E2 , maintaining original equilibrium income Y0 Figure Paradox of Savings 貯蓄＝投資の均衡、S0＝I0、均衡国民所得はY0 ⇒貯蓄意欲が強まって、貯蓄曲線S0がS1へと上方シフト ⇒平均貯蓄性向S/Yは上がる⇒均衡点はE0からE1へと 左方シフト⇒均衡国民所得はY0からY1へと左方シフト 貯蓄が増えると、消費が減って、その乗数効果を通じて 所得が減る⇒貯蓄のパラドックス（paradox of saving）、 節約のパラドックス 貯蓄の増加につれ投資も同額だけ増⇒同じ均衡所得を維持 貯蓄が増、資金市場で資金供給が増、市場利子率iが低下、 投資がI0からI1へと増、均衡点はE2へシフト、元の均衡所得Y0 を維持　　図　貯蓄のパラドックス

18 ９．Induced Investment and Complex Multiplier 誘発投資と複合乗数
Induced investment I(Y) = investment that changes according to the change in national income Y , I＝I(Y)＋I(－) = induced investment + independent investment Aggregate demand = aggregate supply, equilibrium condition Y＝C(Y)＋I(Y)＋I(－) Increase by differentiation dY＝C’(Y)dY＋I’(Y)dY＋dI(－) C ‘(Y) is the marginal propensity to consume, I’ (Y) is the marginal propensity to invest　　 dY＝（1／（1－C’(Y)－I’(Y)））dI(－) The multiplier (1 / (1 - C '(Y) - I' (Y))) is a complex multiplier. If C '(Y) = 0.6 and I' (Y) = 0.2, the complex multiplier is 1 / 0.2 = 5 When we set 1－C’(Y)＝S’(Y)= the marginal propensity to save, we obtain dY＝（1／（S’(Y)－I’(Y)））dI(－) . In order for the complex multiplier to have a positive value, S’(Y)＞I’(Y). 誘発投資（induced investment）＝国民所得Yの変動に応じて変化する投資I(Y) 　　I＝I(Y)＋I(－)＝誘発投資＋独立投資 総需要＝総供給の均衡条件式　　Y＝C(Y)＋I(Y)＋I(－) 微分により増加分　　dY＝C’(Y)dY＋I’(Y)dY＋dI(－) C’(Y)は限界消費性向、I’(Y)は限界投資性向（marginal propensity to invest） 　dY＝（1／（1－C’(Y)－I’(Y)））dI(－) 乗数（1／（1－C’(Y)－I’(Y)））を複合乗数（complex multiplier） C’(Y)=0.6、I’(Y)＝0.2とすると、複合乗数は１／0.2＝5 1－C’(Y)＝S’(Y)とおき、限界貯蓄性向で表すと、dY＝（1／（S’(Y)－I’(Y)））dI(－) 複合乗数がプラスの値になるためには、　S’(Y)＞I’(Y)

19 10．Fiscal Multiplier of Lump-sum Tax and Balanced Budget Multiplier 一括税の財政乗数と均衡予算乗数
Government expenditure G is perfectly covered by a direct tax T imposed on income. This is the principle of balanced budget G＝T Lump sum tax T⇒Income after tax is Y－T = disposable income Consumption function C＝C(Y－T)＝a＋c(Y－T) Equilibrium condition formula in the product market Y＝C(Y－T)＋I＋G Regarding the investment I and direct tax T unchanged, taking an increment of income Y and fiscal expenditure G, dY＝C’(Y－T)dY＋dG　　∴　dY＝（1／（1－c））dG Increase in fiscal expenditures in the case of lump-sum direct taxes, as well as an increase in independent investment, has a multiplier effect with a multiplier of 1 / (1 - c) = a fiscal multiplier of lump-sum tax 財政支出（government expenditure）Gが、所得に対して課される直接税（direct tax）Tによって完全に賄われ、G＝Tという均衡予算（balanced budget）の原則 一括税（lump-sum tax）T⇒税引き後の所得はY－T＝可処分所得（disposable income） 　消費関数　C＝C(Y－T)＝a＋c(Y－T) 　生産物市場の均衡条件式　　Y＝C(Y－T)＋I＋G 投資Iと直接税Tは不変のまま、所得Yと財政支出Gの増分をとると、 　　dY＝C’(Y－T)dY＋dG　　∴　dY＝（1／（1－c））dG 一括直接税の場合の財政支出の増加は、独立投資の増加と同様に、1／（1－c）という乗数の乗数効果＝一括税の場合の財政乗数

20 10B．Fiscal Multiplier of Lump-sum Tax and Balanced Budget Multiplier 一括税の財政乗数と均衡予算乗数
Assuming investment I and fiscal expenditure G remain unchanged, taking a increment of income Y and direct tax T, dY＝C’(Y－T)dY－C’(Y－T)dT　　　∴　dY＝（－c／（1－c））dT The tax increase (or tax reduction) in the case of a lump sum direct tax has a multiplier effect of the multiplier of -c / (1 - c) = a multiplier of tax increase ( or decrease), ⇒because 0 <c <1, is smaller than the multiplier of fiscal expenditure Turning all tax increases on increase in fiscal expenditure and maintaining balanced budget G＝T , ⇒　　1／（1－c）－c／（1－c）＝1 The multiplier effect is 1, the theorem of balanced budget multiplier 投資Iと財政支出Gは不変のまま、所得Yと直接税Tの増分をとると 　　dY＝C’(Y－T)dY－C’(Y－T)dT　　　∴　dY＝（－c／（1－c））dT 一括直接税の場合の増税（ないし減税）は、－c／（1－c）という乗数の乗数効果＝増減税の乗数、⇒0＜c＜1ゆえ、財政支出の乗数よりは小 増税をした分をすべて財政支出の増加に回し、均衡予算G＝Tを維持する場合 　　1／（1－c）－c／（1－c）＝1 乗数効果が1、均衡予算乗数の定理（theorem of balanced budget multiplier）

21 Disposable income is (1-τ)Y, consumption function is C＝C(Y)＝a＋(1－τ)cY
11．Fiscal Multiplier of Proportional Tax and Built-in Stabilizer 比例税の財政乗数とビルトイン・スタビライザー Direct tax as a proportional tax, tax rate is τ (0 <τ <1), the amount of the tax is τY . Disposable income is (1-τ)Y, consumption function is C＝C(Y)＝a＋(1－τ)cY Equilibrium condition in the product market, Y＝C(Y)＋I＋G＝a＋(1－τ)cY＋I＋G With investment I and tax rate τ unchanged, taking increments of income Y and government spending G, dY＝C’(Y)dY＋dG＝(1－τ)cdY＋dG　　dY＝（1／（1－(1－τ)c））dG Increase in government expenditure in case of the proportional tax ⇒ multiplier = 1／（1－(1－τ)c） It is a smaller multiplier dY＝（1／（1－(1－τ)c））dI than that of government expenditure in case of lump-sum direct tax. 比例税（proportional tax）の直接税、税率をτ（0＜τ＜1）、税額はτY 　可処分所得は(1－τ)Y、　消費関数は　C＝C(Y)＝a＋(1－τ)cY 生産物市場の均衡条件式、　　Y＝C(Y)＋I＋G＝a＋(1－τ)cY＋I＋G 投資Iと税率τは不変のまま、所得Yと財政支出Gの増分、 　dY＝C’(Y)dY＋dG＝(1－τ)cdY＋dG　　dY＝（1／（1－(1－τ)c））dG 比例税の場合の財政支出の増加⇒乗数＝1／（1－(1－τ)c） 一括直接税の場合の財政支出の増加と比べて小さな乗数　dY＝（1／（1－(1－τ)c））dI

22 Progressive tax rate τ goes up as income Y increases,
11B．Fiscal Multiplier of Proportional Tax and Built-in Stabilizer 比例税の財政乗数とビルトイン・スタビライザー In comparison with lump-sum direct tax T, because the amount of a tax τY grows greater as an increase in income Y in the case of the proportional tax , an increase in disposable income is suppressed, and an increase in income (multiplier effect) is smaller when independent expenditure increases. Because the amount of a tax τY becomes smaller with a decrease in income Y, a decrease in disposable income is suppressed, and a decrease in income is smaller when independent expenditure decreases, and multiplier effect is smaller. ⇒ a function to suppress and stabilize economic fluctuations = automatic stable function (built-in stabilizer) Progressive tax rate τ goes up as income Y increases, ⇒ Because it is τ=τ(Y), τ' (Y)>0, and an increase (decrease) in the amount of a tax τY becomes bigger with increase (decrease) in income Y, the multiplier effect is small, and the automatic stability function of the economy is bigger. 一括直接税Tに比べて比例税では、所得Yの増加と共に税額τYが大きくなるので、可処分所得の増加が抑えられ、独立支出が増えても所得の増加（乗数効果）は小さい。 所得Yの減少と共に税額τYが小さくなるので、可処分所得の減少が抑えられ、独立支出が減っても所得の減少は少なく、乗数効果が小さい。 ⇒経済変動を抑制し安定化させる機能＝自動安定機能（built-in stabilizer: ビルトイン・スタビライザー） 所得Yが増えるにつれて税率τが上がる累進税率（progressive tax rate） ⇒τ＝τ（Y)、τ’（Y)＞0であり、所得Yの増加（減少）と共に税額τYの増加（減少）は更に大きくなるので、乗数効果は小さくて、景気の自動安定機能は更に大きい

23 12．Trade Multiplier 貿易乗数
Export X is the demand from foreign countries. Import M is the demand to foreign countries and an increasing function of national income Y. M＝M(Y)　 M’(Y)＞0 The equilibrium condition of supply and demand in the product market is as follows. Y＝C(Y)＋I＋G＋X－M(Y) Assuming investment I and government expenditure G unchanged, taking increments of income Y and export X , dY＝C’(Y)dY＋dX－M’(Y)dY It is rewritten as follows, with setting the marginal propensity to import being m＝M’(Y). dY＝（1／（1－c＋m））dX An increase in export X induces an increase in national income Y and import M, the trade multiplier （1／（1－c＋m）） is greater than that of independent investment. 輸出Xは海外からの需要、輸入Mは海外への需要、国民所得Yの増加関数 　　M＝M(Y)　 M’(Y)＞0 生産物市場の需給均衡条件式は以下のようになる。 　　Y＝C(Y)＋I＋G＋X－M(Y) いま投資Iと財政支出Gは不変のまま、所得Yと輸出Xの増分を取ると、 　　dY＝C’(Y)dY＋dX－M’(Y)dY となる。整理すると、m＝M’(Y)を限界輸入性向として、以下のようになる。 　　dY＝（1／（1－c＋m））dX 輸出Xの増加は、国民所得Yの増加、輸入の増加を誘発、貿易乗数（1／（1－c＋m））は独立投資の乗数よりは大