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
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Sunday, August 28, 2011
Lecture 9. FINDING THE VALUE OF THE MULTIPLIER
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