And ask your friends to do the same.Finding Total Consumer Spending
In the previous lecture we developed the folowing formulas: (1) TCS1 = CS1 / (1 − Pp) = STIM × Pc × Cp / (1 − Pp) = STIM × 0.47 × 0.63 / (1 − 0.5) = STIM × 0.2961 / 0.5 = STIM × 0.5922 = $1,000 × 0.5922 = $592.20 This is the total consumer spending received by the first wave of retail producers as a result of being stimulated by STIM. This money remains to be spent as a further stimulus to the economy. Dividing equation (1) by STIM yields: (2) TCS1/STIM = (Pc × Cp) / (1 − Pp) = 0.5922 This is the ratio between the output and input of our “producer / consumer" model. In equations (1) and (2), the value 0.5922 is dependent only on Assumptions (1) and (2) of the previous lecture and is independent of the value of STIM, Jane‘s stimulus check.
. While we display the cells in Column 3 as distinct groups of retail producers, each cell simply represents a generic group of retail producers. No mathematical rigor is lost by combining all the cells in Column 3 into one cell so that all the retail producers in Column 3 become a new “Jane” with a stimulus check for $592.20 (TCS1) in hand, ready to be spent just as STIM was spent.
Column 1 2 3 Row Pp=0.5 Pc=0.47 Cp=0.63 1 $1,000.00 $470.00 $296.10 2 500.00 235.00 148.05 3 250.00 117.50 74.025 - ------- ------- ------- _ _______ _______ _______ Total $2,000.00 $940.00 $592.20 Justification
Before we follow that procedure, let us note that the Congressional Budget Office (CBO) limits the multiplier value to 2.5 for purchases of goods and services by the federal government and transfers to state and local governments for infrastructure. Other types of transfers or payments or tax cuts are restricted to multipliers of lower value. Thus, according to the CBO, unemployment benefits would have a lower multiplier. Let us be clear about the new Jane. Whereas the old Jane was a contractor for infrastructure to Uncle Sam, the new Jane is a retail producer. The CBO holds that infrastructure investment has a higher multiplier than unemployment benefits, which stimulate retail sales. This would imply that creating a new Jane as a retail producer would result in a lower value of TCS1/STIM than we calculated for the old Jane, the government contractor. Such a limitation would invalidate our above plan for the new Jane. Nevertheless, we shall assume that Assumptions (1) and (2) from the previous lecture apply as well to retail producers. Our justification for this assumption? 1. Economics is complicated. A large economy is akin to a 9-dimensional chess game. And there are no Bobby Fischers to give us assurance, not even in the CBO. Multiplier values used by the spectrum of very reputable economists range widely. We believe that the multiplier value should be derived from the model used. In our model, Assumption 1 described our average consumer. We believe the numbers used for that consumer are not far from reality and will apply to any model. Assumption 2 described our average government contractor for infrastructure. From his gross income, we allowed small percentages for taxes and retained earnings and then split the rest almost equally between other producers and individuals. We see no reason to believe that an average retail producer has a profile that is significantly different than that of the contractor for infrastructure. Nor do we believe that the average producer who supplies the retail producer with goods and services has a profile that is significantly different than the profile of a producer who supplies the contractor for infrastructure. We welcome comment from those who disagree. 2. We want to develope a formula for the multiplier that will be based entirely upon the numbers used in Assumptions 1 and 2. Once given the formula, any value for the percentages used can be easily changed and the results can be examined. We believe that the the model is robust and that the value of the multiplier will not be very sensitive to a change in the numbers we used. 3. We are not seeking the exact value of a multiplier. We are trying to demonstrate how government spending stimulates the economy and what varialbles determine the value of the multiplier. We believe that our use of Assumptions 1 and 2 are useful for that purpose and that there is no deception in doing so. With that justification, we will assume that the new Jane with a check for $592.20 dollars will have an effect upon the economy that will be proportional to the effect that the old Jane had with a check for $1,000.
The New Jane
Therefore, we could regenerate the same table with TCS1 ($592.20) as the input displayed in the Row 1, Column 1 cell instead of STIM ($1,000.00) The new table would yield TCS2, the reduced fraction of TCS1, as the output of the second wave of spending. The next table using TCS2 as input would yield a further reduced TCS3 as output. And so forth, until TCSn would approach zero. The following equations display that process: TCS1 = STIM × 0.5922 = $1,000 × 0.5922 = $592.20 TCS2 = TCS1 × 0.5922 = $592.20 × 0.5922 = $350.70 TCS3 = TCS2 × 0.5922 = $350.70 × 0.5922 = $207.69 Obviously, we have another geometric series with the same formula: in which Sum Total = TCS = A / (1 - R) where A = the first term in the series = TCS1 = STIM × Pc × Cp / (1 − Pp) (from equation (1) above) = STIM × 0.5922 (from equation (1) above) and R = the ratio between terms in the series = (Pc × Cp) / (1 − Pp) (from equation (2) above) = 0.5922 Noting that A = STIM × R, Therefore, (3) TCS = A / (1 − R) = STIM × R / (1 - R) = $1,000 × 0.5922 / (1 - 0.5922) = $1,000 × 1.45218 = $1,452.18 This is the total amount of consumer spending stimulated by Jane's $1,000 stimulus check.
The Multiplier Formula
We can now develop the formula for the multiplier M based upon Assumptions (1) and (2) in the previous lecture. Dividing equation (3) by STIM, we have: (5) TCS/STIM = R / (1 - R) = 1.45218 (from equation (3)) Since (from Lecture 4, equation (5)), (6) M = 1 + (Consumer Spending / Stimulus) = 1 + (TCS / STIM) = 1 + 1.45218 = 2.45218 The rounded multiplier is M = 2.45. Thus, with Assumptions (1) and (2), the value of the multiplier is 2.45 and a $1T stimulus will generate $1.45T of consumer spending and $2.45T of GDP growth. To derive the formula for M, we can substitute equation (5) into equation (6) yielding: (7) M = 1 + [R / (1 - R)] = 1 + [Pc × Cp / (1 − Pp)] / {1 − [(Pc × Cp) / (1 − Pp)]} Noting that the denominator of equation (7) can be expressed as: [(1 − Pp) - (Pc × Cp)] / (1 − Pp), which has the same denominator as the numerator of equation (7), we can eliminate both denominators, yielding: (8) M = 1 + {(Pc × Cp) / [[(1 − Pp) - (Pc × Cp)]}
Input - Output View of the Multiplier
Subtracting 1 from each side of equation (6) yields: (Consumer Spending / Stimulus) = M − 1 which is equivalent to: TCS / STIM = M − 1 and, multiplying by the denominator: (9) TCS = STIM × (M − 1) In effect, stimulus is a process in which input = STIM and output = STIM + TCS = STIM + [STIM × (M − 1)] (from (9)) = STIM × [1 + (M − 1)] = STIM × M This conforms to our Lecture 4 equation (4): GDP growth = M × Stimulus
Remarks on the Multiplier Formula
The derived value for the multiplier depends entirely upon the values chosen for the variables in Assumptions (1) and (2). It will rise or fall as consumer and producer variables allow more or less of consumers' gross income to be spent rather than being taxed or saved. Hence, the need for both a middle class tax rebate and increased government benefits and payments to individuals during a recession. The Marginal Propensity to Save (MPC) does not appear in the formula but affects the value of Ct. The variables Pt (percentage of producer's gross revenue paid to governments at all levels for taxes) and Pe (percentage of producer's gross revenue retained as owner's equity) do not appear in the formula but they do affect the sum of Pp and Pc. We also note here that Obama's economists limit the multipler value to 1.6 for estimation of stimulus effect. We encourage readers to supply their own constants in the formula or to develope a different model and report their findings.
Timing is Everything
In the analysis above, we have ignored the effects of time. In fact, the time between Jane’s first purchase and the spending of TCSn’s last penny could take many months. This time effect is important, as we will discuss in the next section. In general, the less time to exhaustion, the better result for the economy. The time effect is related to factors MPC and Pe. Hesitation to spend or invest affects the economy in important ways. It is interesting to note that all the factors in Assumptions (1) and (2) except MPC and Pe are set either by government or by the nature of industry. The MPC is the only factor that is related to the spending behavior of consumers. The Pe is related to the producers’ view of his circumstances. Since Cp is computed (in Assumption 1) in an equation that includes MPC and Ct, increasing MPC increases the value of the multiplier. We will examine this in the next lecture.
Proceed to: Lecture 10. STIMULUS ON STEROIDS
Thanks for your interest. Marvin Sussman, retired engineer If you don’t like the world as it is, change it this way!:
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Sunday, August 28, 2011
Lecture 9. FINDING THE VALUE OF THE MULTIPLIER
Lecture 10. STIMULUS ON STEROIDS
The MPC Effect
In the assumptions we made in the previous lecture, we did not explain the significance of the word “shortly”. In economic theory, time is of the essence. If Jane and all her friends were to reduce movie attendance from once a week to once a month, Hollywood would suffer. The MPC explains much of the difference between prosperity and depression. Indeed, Keynes considered “hoarding” of cash to be the cause of recessions. The very rich may not change their spending habits much during a recession, but the rest of us begin to save at an unusually high rate. That has an immediate effect on industries dependent upon discretionary spending: stocks and bonds, real estate, autos, electronics, home furnishings, clothing, restaurants, sports, entertainment, and even food on the table. And that starts the downward spiral. The key to increasing the value of the multiplier and the efficacy of a stimulus and the growth of the GDP is to increase the MPC. In the previous lecture, in Assumption (2), we set the value of MPC at 90%. The resulting value of the multiplier was 2.45. If we give MPC the value 100% = 1,0, recalculation of the multiplier would (using data from Assumption 1, Lecture 8) proceed as follows: Cp = MPC × (1 - Ct) = 1.0 × (1 - 0.3) = 1.0 × 0.7 = 0.7 Pc × Cp = 0.47 × 0.7 = 0.329 (1)) M = 1 + {(Pc × Cp) / [(1 − Pp) - (Pc × Cp)]} = 1 + {0.329 / [(1 − 0.5) - 0.329]} = 1 + (0.329 / 0.171) = 1 + 1.924 = 2.92 An 11% MPC increase produced a 19% increase in the multiplier value. This lends credence to the importance of the MPC. To study the question further, let's do some accounting to show the effect of the MPC increase.
An MPC Experiment
Using mid-year 2011 approximate results from usdebtclock.org: Pre-stimulus ND: $14.5T Pre-stimulus GDP: $14.8T Pre-stimulus DR: 98.0% Stimulus: $ 1.0T Tax burden* / GDP = 0.3 *for all levels of government
Then for MPC = 90%:
Multiplier = 2.45 GDP growth, ΔGDP = M x S / 5 = 2.45 x $1T / 5 = $0.490T Tax revenue gain, ΔTR = ΔGDP x TB × 5 = $0.490T x 0.3 × 5 = $0.735T Infrastructure purchase discount, IPD = ΔTR / S = $0.735T / $1T = 73.5% New ND = ND + S - ΔTR = $14.5T + $1.0T - $0.735T = $14.765T New GDP = GDP + ΔGDP = $14.8T + $0.490T = $15.290T New DR = New ND / New GDP = $14.765T / $15.290T = 96.56% DR change, ΔDR = New DR - DR = 96.56% - 98.0% = -1.44%
Now, let’s raise MPC to 100%
Multiplier = 2.92 GDP growth, ΔGDP = M x S / 5 = 2.92 x $1T / 5 = $0.584T Tax revenue gain, ΔTR = ΔGDP x TB × 5 = $0.584T x 0.3 × 5 = $0.876T Infrastructure purchase discount, IPD = ΔTR / S = $0.876T / $1T = 87.6% New ND = ND + S - ΔTR = $14.5T + $1.0T - $0.876T = $14.624T New GDP = GDP + ΔGDP = $14.8T + $0.584T = $15.384T New DR = New ND / New GDP = $14.624T / $15.384T = 95.0% DR change, = ΔDR = New DR - DR = 95.0% - 98.0% = -3.0%
To contrast the change induced by raising MPC:
| MPC | MULTIPLIER | DISCOUNT | DR DROP |
|---|---|---|---|
| 90% | 2.45 | 73.5% | 1.44% |
| 100% | 2.92 | 87.6% | 3.0% |
Maximize the Marginal Propensity to Consume
The effect is striking. If we accept the previous lecture's model and assumptions and as long as there are idle resources, we can reduce the DR by building infrastructure! In any case, the larger the multiplier, the larger the purchase discount. But can we realistically set the effective value of MPC equal to 100%? People do indeed save money, sometimes more 10% of their disposable income. And the recession guarantees a high saving rate. Q: Where do people save their money? A: In banks. Q: What do banks do with their assets? A: To increase income, they lend as much of it as possible as soon as possible. Q: What do borrowers do with their loans? A: To reduce interest cost and begin reaping gains from their venture, they invest it as soon as possible. Q: Can the effective MPC conceivably approach 100%? A: Yes, if enough entrepreneurs have profitable ventures in mind and banks are able to grant the loans judiciously. Q: Can that happen in a recession? A: Yes, if enough of the entrepreneurs have federal government contracts in hand when applying for the loan. Q: How soon could that happen? A: As soon as Congress passes and the President signs a stimulus bill large enough to give overtime work to all of our unemployed and under-empoyed. And that happened during WW II, as the historical evidence in a future lecture will show. Q: How much can banks lend. A: With our fractional reserve system, banks can lend up to a multiple of their reserves. The multiple has a value at least 10 and often 20, depending upon the policies of management. Q: What is the effect of a stimulus upon bank reserves and bank funds available for loans? A: With Marginal Propensity to Save (MPS) = 10% of disposable income and Marginal Propensity to Consume (MPC) = 90% of disposable income and Total Consumer Spending / Stimulus = 1.45, (from the previous lecture) hence: Total Consumer Saving / Stimulus = 10% / 90% = 0.16 and $1T of stimulus could add $0.16T to reserves. With a minimum multiple of 10, $1T of stimulus could add $1.6T of funds available for loans and creditworthy loan applications may be funded. Q: What is the effect of stimulus on loan applications? A: A large stimulus could increase consumer confidence and improve business sentiment toward investment. If a stimulus increased private investment by more than 10% of the value of the stimulus, the multiplier would effectively increase by more than 10%. A multiplier approaching 3.0 (as shown above) would approach 3.3. Q: With a multiplier approaching 3.3, what is the effect of infrastructure cost upon the federal deficit? A: With Infrastructure Purchase Discount (IPD) = Multiplier x Tax Burden = 3.3 x 30% = 99%, the effect of infrastructure cost upon the federal budget deficit In any case, the larger the multiplier, the larger the purchase discount. But can we realistically set the effective value of MPC equal to 100%? People do indeed save money, sometimes more 10% of their disposable income. And the recession guarantees a high saving rate. Q: Where do people save their money? A: In banks. Q: What do banks do with their assets? A: To increase income, they lend as much of it as possible as soon as possible. Q: What do borrowers do with their loans? A: To reduce interest cost and begin reaping gains from their venture, they invest it as soon as possible. Q: Can the effective MPC conceivably approach 100%? A: Yes, if enough entrepreneurs have profitable ventures in mind and banks are able to grant the loans judiciously. Q: Can that happen in a recession? A: Yes, if enough of the entrepreneurs have federal government contracts in hand when applying for the loan. Q: How soon could that happen? A: As soon as Congress passes and the President signs a stimulus bill large enough to give overtime work to all of our unemployed and under-empoyed. And that happened during WW II, as the historical evidence in a future lecture will show. Q: How much can banks lend. A: With our fractional reserve system, banks can lend up to a multiple of their reserves. The multiple has a value at least 10 and often 20, depending upon the policies of management. Q: What is the effect of a stimulus upon bank reserves and bank funds available for loans? A: With Marginal Propensity to Save (MPS) = 10% of disposable income and Marginal Propensity to Consume (MPC) = 90% of disposable income and Total Consumer Spending / Stimulus = 1.45, (from the previous lecture) hence: Total Consumer Saving / Stimulus = 10% / 90% = 0.16 and $1T of stimulus could add $0.16T to reserves. With a minimum multiple of 10, $1T of stimulus could add $1.6T of funds available for loans and creditworthy loan applications may be funded. Q: What is the effect of stimulus on loan applications? A: A large stimulus could increase consumer confidence and improve business sentiment toward investment. If a stimulus increased private investment by more than 10% of the value of the stimulus, the multiplier would effectively increase by more than 10%. A multiplier approaching 3.0 (as shown above) would approach 3.3. Q: With a multiplier approaching 3.3, what is the effect of infrastructure cost upon the federal budget? A: With Infrastructure Purchase Discount (IPD) = Multiplier x Tax Burden = 3.3 x 30% = 99%, the net addition of infrastructure appropriations to the federal budget approaches zero! During a recession, with IPD approaching 100%, a sufficiently generous stimulus would bring full employment; increase tax revenue; lower the DR; and repair, rebuild, and renew our infrastructure at almost zero net deficit.
Proceed to: Lecture 11. INDEBTEDNESS: THE STIMULUS EFFECT
Thanks for your interest.
Marvin Sussman, retired engineer
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