§9 Examination of the equilibrium theory method in deriving the investment multiplier effect formula 

DCE Hayashi, Yuichiro 

October 16, 2006

The MPC doesn’t relate to the analysis of the GVA fluctuation problems.

As far as the derivation of Keynes's investment multiplier effect formula is concerned, the  demand function line and the supply function line are plotted on the same 45-degrees line, therefore the equilibrium theory method is not applicable.

The equilibrium theory method is not applicable to a nａｔｉｏｎａｌ production analysis where ｔｈｅ relationship between national incomes and final products are expressed by a break-even type chart.

The profit maximization equation does not exist in economy.

9.1 The conventional derivation method  for the investment multiplier effect formula using the equilibrium theory

   The derivation of Keynes’ multiplier effect formula in  §3 will be reviewed here in Fig.3-3.

   In Fig.3-3, the horizontal axis represents gross national income Y₁ and the vertical axis represents aggregate demand Y_2D or aggregate supply Y_2S. The aggregate demand function Y_2D is given by Eq. (3-2), and it is shown as the points on the sloped line, which goes through point K* in the figure. In the multiplier effect model, G₂ is a component of Y₂ and is a kind of independent constant (fｉｘｅｄ cost) variable to the other component variables. Eq.(3-7) shows ｔhｅ relationship between Y_2S and gross income Y₁ 

From Assumption 1 in §3, the consumption demand is represented in Eq.(3-3).

   The aggregate supply and demand reaching equilibrium will be explained in Fig.9-1. The sloped line, given by Eq.(3-3), going through point K* shows  'intended or planned aggregate demand' by demanders( firms, households and government). The 45° line represents actual expenditure i.e. 'realized aggregate supply' including inventories by firms. Suppose firms' supply is at the level Y₁ which is greater than the equilibrium level of point K*, then unplanned inventories [Y₁−E₁] will increase. In this case, the firms will try to reduce the excess inventories. 

  This results in a wage cut or the layoff of workers i.e. decrease of income. If the firms' supply is at the level Y₂, the reverse will take place. In this way, actual production immediately converges at the cross point of the aggregate demand line with the 45° line. The reason that Keynes's principle of effective is recognized as a dominant economic theory equally with Walras's general equilibrium theory is that the logic of this equilibrium process is considered mathematically or in accounting reasonable.  

   In addition, in a short run case where the price of goods is sticky, the convergence to the equilibrium point is realized through the adjustment of product quantities. Since the equilibrium point between supply and demand in products is not necessarily the same between labor supply and demand, involuntary unemployment may exist. In order to improve the unemployment condition, a government have only to increase public investment. This is the principle of effective demand.  

Let's return to Fig.3-3. Suppose supply Y_2S  and demand Y_2D is in equilibrium at point K*, then equilibrium gross national income Y*₁ is given as Eq. (3-9) which is the solution between  Eq. (3-7) and Eq. (3-2) substituted Eq.(3-3). The following should be noted: the constants a _B and MPC surely exist statistically; we have assumed that I*₂ and NX*₂ are constants; G*₂ is an independent variable which doesn't have a linear relationship to Y*₁. 

ΔY 1 = Δ G2 / (1 -MPC)

(3-13-1)

   Suppose now that only the variable of government demand G*₂ has been increased by ΔG_2D to become G**₂( = G*₂ +ΔG_2D) by means of  government bonds, and as a result, income Y_1S* reaches Y **_1S( = Y *_1S+ΔY₁), where K**  is a new equilibrium point. Then we easily obtain Eq.(3-13-1) from Eq.(3-9). If we go through Eq.(3-10) in §7, we have the same result. Eq. (3-13-1) is obtained as the solution between the two linear equations or as the cross point of the two lines. This  inevitably results from the conventional equilibrium theory method. 

   It should be noted that Fig.3-3 is the break-even type chart shown below in 5.2, §5 and explained  in [Question-8], §4. The reason is that if the chart is the time-series type output chart shown in Fig.4.2 in §4 and explained in 5.2, §5, the sum of parts C(Y) and I(Y), at any intended time, must be equal to the whole Y which is the 45° line.

   Based upon the aforementioned knowledge, correct derivation of the multiplier effect formula is introduced. Suppose that the assumptions I₂=const. and NX ₂=const. are preserved. Eq. (3-2) is rewritten in the following:

Derivation1

   If we intend to increase only G₂ in Eq.(9-1) into G₂+ΔG₂, the incremental equation for Eq.(9-1) is given as follows:  

From Eq.(3-7),  Eq.(3-2) and Eq.(9-1), we obtain:

ΔY1 = Δ G2

(9-2）

Derivation2

   If we want to increase C₂ into C₂+ΔC₂, and G₂ into G₂+ ΔG₂, the incremental equation for Eq.(3-2) is yielded in the following:

Y2D+ΔY2D = C2 + ΔC2 + I 2(const.) + NX2 (const.)+ G2 + ΔG2

(9-3）

From Eq.(3-7),  Eq.(3-2),  Eq.(3-3) and Eq.(9-3), we obtain:

ΔY1 = ΔG2 / (1 - MPC)

(3-13-1)

  If we want to increase only G₂ in Eq.(3-2) into G₂+ΔG₂, it is natural that, Eq.(7-3) is correct. Although Keynes wanted to obtain the income through Eq.(3-13-1), as a matter of course, his intention cannot be realized.

   Fig.9-2 illustrates the processes of the Derivation1 and Derivation2.This chart shows where the equilibrium theory method has caused the erroneous result. 　

   At a glance of Fig.9-2, we might recognize that the system in the interval of the origin〜K* was a one-degree of freedom system with only one independent variable ΔC, and that in K*〜K** interval was a two-degrees of freedom system with independent variables of both ΔG and ΔC. However, the fact is the reverse. In the former interval, the system is a two-degrees of freedom system with independent variables of both ΔG and ΔC( =0), and In the latter interval, it is a one-degree of freedom system with only the independent variable of a given ΔG or a given ΔC (26/Sept./2006 ).

   Now, which is correct; that this system has two-degrees of freedom shown in the origin〜K* interval( more properly, at only the point K* in place of the interval ) or that this has a one-degree of freedom shown in the K*〜K** interval in this incremental problem of ΔG? If the financial resource of ΔG is due to public funds, the funds have been financed by a part of savings in households. That is to say, the resource which would have been expended in consumption has flown to the public funds, so private consumption ΔC （= ΔG）is not generated. Consequently, as this system has two-degrees of freedom from beginning to end, it is correct that the change ΔG is preserved throughout. Now we have obtained a new principle: 'the MPC principle cannot be applied to government additional expenditure due to public bonds'. We replaced the notations as: a_K (t) =a_C (t).

   Although it is considered that only the relationship between ΔC and ΔY has a marginal propensity to expend, this is not valid. As shown in Fig.4-7, the other expenditures I, G and NX have their own marginal propensity to expend statistically as well. This shows that Eq.(3-R7) plays a big role in economic fluctuation analyses.　

(1) none of the economic fluctuations which don't satisfy Eq.(3-R7) exist;

(2) we must not set up such an assumption that it doesn't satisfy Eq.(3-R7);  

(3) if we set up, for example, such an assumption a_I  (t)＝0 for analyzing an economic fluctuation problem, we must take the following means; we set up the assumption that investment I doesn't exist from the beginning in Y, or we take an incremental fixed cost variable into consideration in incremental variables.

   Why has the planned aggregate demand line, which passed through K**, appeared in Fig.3-3? This is because we made the assumption I=const. and NX=const. With this assumption we have a_I  (t) ＝  a_NX (t) ＝0, so we obtain, from Eq. (3-R7), the following:

Multiplying Y into the both sides gives:

if both the equations Y=Y* and Y＝Y*+ΔY are satisfied in Eq.(9-5),  the two substituted equations yield:

   Depending on this analytical view, we know that only Derivation1 remains. Consequently, the actual aggregate demand line inevitably always equals the aggregate supply line in shape, i.e. 45° line ( really, a line with any slope) in Ｆｉｇ.3-3.The correct chart in place of Fig.3-3 is presented in Fig.9-3. Fig.9-3 shows that a national products chart must be the time series type chart in which the expenditures line goes through the origin, at least for the ΔG₂ change problem.

   I（Y）is uniquely linked to C(Y)  by the principle of a one-degree of freedom system between C(Y) and I（Y）. Consequently, supply C(Y) + I(Y) usually becomes the actual supply by producers corresponding to the foregoing demand C(Y) + I(Y). At this moment, the maximum utility at the  point of producers’ preference of depreciation D, being really a constraint, wages W and profit π are obtained for producing products corresponding to the given demands C(Y) +I(Y). By the way, if consumers want to save money in a growing economy in which MPC exists ( even if the type is Keynes's one) and its government has no debts, then producers must theoretically finance by borrowing the money  in order to produce investment goods I(Y).  

   We have the following cases: a demand line overlaps the supply line; the equilibrium point or the cross point cannot be obtained from the solution of the two linear equations; if we determine a demand (monetary unit) in advance, then the supply is automatically determined as the same value, and this holds conversely. Such cases are caused by the following:

(b) When a demand function for a good is assumed, the statistics observed in the real economy for the function are used;

(c) In the modern economy, supply (monetary unit) can always respond to demand (monetary unit) in quantities, so if demand is given, supply is reproducible, or the present economy is in a stable state;

(d) The statistics of goods in the stable, real economy express equilibrium points between supply and demand.

   Thus, we have the following conclusions: as far as the derivation of Keynes’s investment multiplier effect formula by the equilibrium theory method is concerned, a planned aggregate demand line and an actual aggregate supply line are plotted on the same 45° line, therefore we cannot find a new equilibrium point after a GVA change as the solution of the two equations in both the break-even type chart and the time-series type chart(21/9/2006). 

   However, this doesn't cause any special difficulty in this analytical process. That is to say, the equation, the effective demand = aggregate supply, always and simultaneously holds. This now produces a new problem concerning how the effective demand is determined.  [This part will be reconsidered later.]

  We can confirm that Fig.9-3 is correct by the following proof.

   The graphic production condition doesn't theoretically exist, as shown in Fig.9-1, in which the actual supply line (the 45°line) and an intended demand line crossed with each other and their differences represent an excess demand or supply,  whether in the time series type production chart or in the break-even type one. The reason is as follows:

(a)  If the economic model shown in Fig.9-1 is for the time series type production chart, each economic item such as profit π, consumption of fixed capital D and wages W etc. is a part of the whole Y( final products or national income), and therefore we cannot assume that Δπ/ΔY = 0 or ΔD/ΔY = 0, because e.g. Δπ/ΔY is a proportion of ΔY; refer to Eq.(3-R7). Consequently, both the actual supply line and the planned demand line must go through the origin （Y＝0）. However, the planned line doesn't go through the origin in Fig.9-1, and therefore the economic model shown in Fig.9-1 must be for the break-even type production chart.    

(b) The problem is that whether it is theoretically possible for the graphic condition shown in Fig.9-1 to exist in such a way as explained as follows: the actual supply line (the 45° line) and the intended demand line crosses with each other and their differences represent an excess demand or supply.

    The correct break-even chart under absorption costing was firstly presented by the author as shown in Fig.2.5 in §5, Part 1. When we don't apply the profit maximization equation to the absorption costing, Fig.2.5 is directly applicable. In this chart, η(ε) is the net carryover manufacturing overhead applied in inventories allocated from actual manufacturing overhead C^F(ε). In direct cost D^X(ε), the effect of net inventory is already taken into account. See the details in Part 1.

    If we might apply the profit maximization equation to a break-even chart under absorption costing, both the profit line of P^P₁(ε) in the first chart and the one of P^P₂(ε) in the second chart are parallel to the 45°lines, and therefore the both slopes of the variable cost lines of  D^X(ε) +G^V(ε) (variable selling and general administrative expense) are 45°, respectively. Consequently, in the third chart none of the lines which crosses the 45°line exists. Thus, in any case, such a condition as explained in Fig.9-1 is never produced. Regarding this argumentation, refer to " The cause of Solomons's error in his break-even chart for absorption costing" presented in Part 1.  

  For the reasons mentioned above, the cause of the incorrect derivation of the multiplier effect formula by the conventional equilibrium theory method has been demonstrated clearly.

9.3 The significance of Keynes's principle of effective demand

(1) Keynes' investment multiplier effect for national income doesn't exist, because its formula is mathematically wrong( incorrect/ flawed). 

(2) The MPC doesn't relate to economic fluctuations regardless of a stable or unstable economic condition.

(3) For a demand in an economic product, as far as the statistic demand value is used, its planned demand (monetary unit) function always equals the supply function (monetary unit)). If Keynes’s investment multiplier effect formula is correctly derived using the equilibrium theory method, the solution obtained becomes undetermined except for apparent solutions e.g. a given additional demand = the additional supply corresponding to it.

Except for the incorrect part of the investment multiplier effect formula for national income, essential parts in Keynes' principle of effective demand are as follows (the author agrees with these points):

(5) Financing public funds into public works, in order to remedy unemployment, surely benefits its purpose in such an amount as wages included in the works. However, the amount of the public funds issuance increases. The reason is that the public investment by public bonds(= ΔG) covers the drop of the private investment. Since consumption runs closely with investment, ΔY (= ΔG) remains constant if the drop (monetary unit) equals the government expenditure. At this moment, a part of the past accumulative savings by consumers changes into consumption, because the depreciation of the government investment goods cannot be recovered by sales by its nature. However, the wages included in the goods relieves unemployment. In this manner, government investment by borrowing is necessary in an emergency, but such a policy must not be executed for long periods of time.

(7) The author conjectures that the demand of labor( monetary unit) by enterprise is determined by conflict between their maximum profit π and the minimum wages W which ensures both production corresponding to goods demand and stable employment . The expression of this relationship is the break-even chart. When the GDP is decreasing in a recession, the power of keeping minimum profits by enterprise is stronger than the power of desiring to leave jobs due to low wages on the side of labor. At this moment, although laborers take the wage level and don't want to leave their jobs,  unemployment is produced.

  Apart from the indeterminate conclusions, depending on the author's analytical results being valid, what now is the significance of Keynes' principle of effective demand? Presently, everybody knows the break-even chart in business management. At this moment we know that Keynes' investment multiplier effect formula is wrong( incorrect/ false), it merely means as follows: when depreciation is constant and prices of products( including labor price) are sticky, enterprise’s sales determine products quantities( including labor quantity) and profits; the transformation from private investment to government investment at the reinvestment of accumulative private savings under a recession relieves unemployment as much as the wages included in the government expenditure. This is surely right.

  The author considers finally that it was a creative theory that could produce Say's law, a question which had been dominant for more than 100 years until the emergence of Keynes' theory. It was surely evolutional in the history of economics.

  Keynes’s principle of effective demand is inseparably linked to the equilibrium theory method. The reason is as follows: we use a constant for investment I(Y) and Keynes’ type consumption function for consumption C(Y) as a demand function of goods; the supply function is the 45° line; the investment multiplier effect formula is obtained as the solution of the two equations or the cross point between both lines. Keynes's theory denies only the part of free changes of goods prices, and its theoretical ground of the derivation of the investment multiplier effect formula is the equilibrium theory.

   By the way, no economist doubts nor criticizes Keynes's assumption that the demand function, for companies, I(Y) = a constant. Therefore, the author infers that  both demand and production functions can take any mathematical expressions which simulate real  human thought and behavior. Based on this realization, he now considers Walras’s general equilibrium theory. The reason is as follows: the equilibrium theory uses marginal utility as a theoretical basis for personal preferences i.e. human senses to goods using symbolic variables; therefore, when we try to apply the theory to real economies, the validity of the analytical result by the theory will ultimately depend on the assumptions for demand and production functions which reflect real economies however sophisticated the equilibrium theory may be. 

(a) was the erroneous investment multiplier effect formula produced from the flaw in  the equilibrium theory method itself?; 

(b) was it produced from the improper measures of assuming that incremental investment is a constant, or was the error due to adopting Keynes's type consumption function as a demand function, even though the equilibrium theory method is correct? ;

(c) cannot we apply the equilibrium theory method to only the derivation of Keynes's formula for some reason, even though the equilibrium theory method is correct?

  The problem (b) will be examined. We have assumed that I(Y) = a constant. Is this assumption incorrect? Yes, it is a little incorrect but not absolutely. When we use Keynes's type consumption function as a consumption function C(Y), assuming any expression other than I(Y) = [ Ｙ - Keynes's type consumption function C(Y)] aｓ a consumption function I(Y) will give all incorrect results. The reason is that the expression of the aggregate supply function, i.e. the 45° line is correct but the expression of the aggregate demand function is incorrect from the beginning because its line is not the 45° line. 

  We shall examine the item (a). in the process of deriving Keynes's investment multiplier effect formula,  I(Y)= a constant was assumed. Was this assumption erroneous? This is a little erroneous, but this is not necessarily wrong( false). 

   See both the contradictory and the correct expressions below for (a) in Fig.9-3 . Errors, in the equilibrium theory method logic, which haven't been found through Eq.(3-13-1), is shown clearly.   

Contradictory expression: 5=Y=3（being proportional to 5)+2（not proportional to 5）; variable cost ratio ν= 3/5. 3=5·(3/5); 5=5·(3/5)+2; 5·(1 - 3/5)=2;　5=Y*=2/(1 - 3/5)( being correct); G= a variable which doesn't relate to the volume of Y; 5+ΔG = Y** = (2+ΔG) / (1 - 3/5); ΔG = ΔG/(1 - 3/5); ΔG=2.5ΔG (being incorrect); Note that the origin of measuring the variable cost ratio ν is Y=0. 

Correct expression: 5 =Y* = 2/(1 - 3/5)( being correct); G= a variable which doesn't relate to the volume of Y; 5+ΔG = 3+( 2+ΔG) =Y**; ΔG=ΔG( being correct); 3 = (5+ΔG)·(3/(5+ΔG)); variable cost ratio ν= 3/(5+ΔG); 5+ΔG = 3+( 2+ΔG) = (5+ΔG)·(3/(5+ΔG)) +( 2+ΔG); 5+ΔG= Y** = ( 2+ΔG)/（1-3/（5+ΔG））; ΔG=ΔG( being correct); Note that the origin of measuring the variable cost ratio ν is Y=0 which is also the origin of the previous equilibrium. readers will imagine the break-even equilibrium, if they increase ΔG as ΔG = 0,1,２, · · and substitute them into the break-even sales formula. We should always have input( monetary unit) = output( monetary unit). That means if an incremental ΔG is given only by an increase of F component, we have final tanθ=3/（5+ΔG） after the increase.

  The variable cost ratio ν at the first equilibrium point is ν=3/5. When the government demand G (fixed cost type variable) increases as much as ΔG, the new variable cost ratio ν changes into ν=3/(5+ΔG). That is to say, the variable cost ratio ν is a nonlinear function of G in the process of the change ΔG. Therefore, the demand function is a nonlinear function. Consequently, the supply and demand equilibrium identity (simultaneous equations) which determines the new equilibrium point and are expressed with price p multiplied by quantity n, become nonlinear simultaneous equations, whether we assume any side as a constant (parameter) of price p or quantity n in a term p · n. 

   Why did ΔC enter into the equations? The reason is that the MPC actually exists in the real economy. Then, is it possible that the MPC has influence on an incremental goods fluctuation analysis? In the economy, the MPC has the roleof connecting C(Y) and I(Y), and restrains free activities with each other. At that moment,  economic motions are expressed by one motion of ΔY ( C(Y) or I(Y)), and at the same time, the MPC disappears from the economic fluctuation analysis. 

   The way of all goods production is the break-even chart type, which consists of parallel and diagonal lines. In the break-even chart, there exists production costs consisting of fixed costs and variable ones. Here the profit maximization principle doesn’t exist irrespective of a short or long period. According to Keynes’s logic, a labor quantity is determined at an equilibrium or cross point, at which demand leads supply, between supply and demand of goods regardless of labor factor, and so this causes involuntary unemployment. This author conjectures that his logic is almost surely correct regardless of price changes or price rigidity. However, his investment multiplier effect formula is incorrect.

   Walras’s general equilibrium theory is as follows: goods are produced in such a manner that labor and capital behave with maximum efficiency or minimum costs through the production; the produced  goods correspond to the individuals' endowments; each endowment of goods is allocated to each other in such a manner that each individual’s preference to consume for the goods is most optimally satisfied through price changes.

   The marginal utility (plus the profit maximization principle) theory is an everlasting achievement in Walras’s general equilibrium theory. However, Walras’s theory has been criticized for many years because of the comparison of the real economy to the economy realized by Walras’s theory. On the other hand, Walras’s theory has developed and now been ultimately refined.

   We examine the analytical process of  Walras’s general  equilibrium theory. In the theory also, operating profit as a definition of accounting can be defined. We accept both the marginal utility theory for consumption goods and the law of decreasing returns to scale for labor input in production. We assume that business managers always desire maximal gain (corporate growth). The marginal rate of transformation, for two goods, which is provided by the principle of profit  maximization, surely equals the marginal rate of substitution,. for the goods, which is given by the optimum consumption planning

   According to Walras’s law in a general equilibrium, if there is excess supply (monetary amount) in one market( e.g. involuntarily unemployed labor), then there must be a matching excess demand elsewhere ( e.g. for goods). No flaw is found anywhere.

   Why then has his theory been attacked  as it doesn’t express the real economy? That is to say, historical economists have thought that Keynes’s theory seems more correct than Walras’s because although involuntary unemployment actually exists in the real economy, they cannot break down the following completely mathematical logical system: one is Walras’s law  which is backed by Leontief’s input-output table concept; the other is the extremum-seeking method with side conditions which is used to seek the optimum solution for consumption utility and production efficiency. In addition, such problems as a selection between actual labor hours and those of leisure, a semi-self-sufficient economy, the way in which goods quantities  converge and so on are not essential in Walras’s theory.

   Walras’s theory has broken through all attempts to disprove it mathematically as wrong for about 130 years. The disproof of the theory in unemployment problem analysis is the most difficult problem of all. However, if this is not achieved, the author’s conjecture will be regarded as mere imagination. Furthermore,  Keynes’s historical achievement will come to nothing.

   The author thinks that the main purpose in economics is presenting avoidance strategies and solutions for unemployment. There is no economic success under involuntary unemployment. The economic theories including economic growth, financial and fiscal ones exist for only one purpose, the prevention of unemployment.  Both presenting how to win and to help losers in competitive society is possible, though it is difficult to present jobs to people who cannot enter the arena! This is because humans in present society are evolving in the direction of living in a market exchange society by exchanging a value made by one for a value made by another.

   When the author reached the area of logic for the equilibrium theory, he knew at last that Keynes’s investment multiplier effect formula gave the incorrect equality ΔG=2.5ΔG, and therefore it is a nonsensical equation. That means that as he has written this paper regarding that the multiplier has any economic significance, he has now a doubt that this paper may have small erroneous expressions in itself. However, he cannot go back to the beginning and rewrite all the sentences. If he begins to describe this paper from the above conclusion, it will be difficult for readers to understand the incorrectness of Keynes’s investment multiplier effect formula. They must trace the author’s trail of logic to reach the field of this equilibrium theory and now wrestle with the theory which produced such an incorrect conclusion.

   The argument for the equilibrium theory method itself in this part 9.4 is as yet unfinished. This theme was originally created by Leon Walras (1834-1910) and has related to the most elemental theoretical foundation in present microeconomics, and therefore the author cannot easily give conclusive comment without careful consideration. 

   He hopes to create and present a third new theoretical system in which faults found in  Walras's theory will be pointed out and improved, and both Walras's theory and Keynes's theory are made consistent adding a new theoretical structure by the author. For that reason, the author will now take a break from the present theme.

16/Oct./2006　publication

8/May/2007　Reconstruction

23/April/2009　Minor modification

(c) Dec. 2003, Yuichiro Hayashi