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Basic rules of differential calculus are useful for understanding economic discourse, such as key discussions in the business or financial media that are nebulous or misleading.

If u(*t*) and v(*t*) are differentiable functions and α is a constant, then we have the following calculus rules:

*Rule* 1: *d **α* = 0

where *d* is the differential operator, such as *d* u(*t*) = *d* u(*t*) / *d t*.

Differential calculus has two independent concepts: the differential and the derivative. The differential is a very small quantity that is negligible except when compared to other differentials. A derivative is the ratio of two differentials as they become smaller and smaller.

*Rule* 2: *d* (α u) = *α* (*d* u)

*Rule* 3: *d* (u ± v) = *d* u ± *d* v

*Rule* 4: *d* (u v) = (*d* u) v + u (*d* v)

*Rule* 5: *d* (u / v) = [v (*d* u) - u (*d* v)] / v^{2}, where v ≠ 0.

Rules 1 to 5 are valid for differentials or derivatives.

The rules above are applied in understanding several fiscal issues using GDP (gross domestic product) and its structural components:

(1) Y = C + I + G + (X - M), so it follows by applying Rule 3:

(2) * d* Y = *d* C + *d* I + *d* G + *d* (X - M)

The variable Y is GDP, C is personal consumption expenditure, I is gross private domestic investment (GPDI), G is government expenditure, X is exports of “goods and services,” and M is imports of “goods and services.”

Goods are a misnomer because destructive objects like weapons or harmful chemicals cannot be regarded as goods. GPDI has several components, which include (*i*) fixed investment and (*ii*) change in private inventories. Fixed investment is divided into *nonresidential* (business structures, equipment, and intellectual property “products”) and *residential*. This exposition is simplified, considering the elementary components of GDP.

According to accounting equation (2), which is valid for every economy, including separate federal or state jurisdictions, GDP can increase between two adjacent periods by the increase of any singular component (C, I, G, X) or by the increase of two or more components.

Assume that *d* C = 0, *d* I = 0, and *d* (X - M) = 0, which means that these structural components are stable between two adjacent periods. It folows that the increase in GDP is driven by the increase in government expenditure:

(3) *d* Y = *d* G

Now, consider a government expenditure behavioral equation to obtain certain fiscal policy implications:

(4) G = T + γ *d* D

In equation (4), T represents taxes (net of transfers) and *d* D is the derivative of the public debt function D(*t*). Greek gamma represents the average (constant between two adjancent periods) interest rate paid on the government debt.

Behavioral equation (4) means that government expenditure must be financed by T (taxes) and/or by a change in D (public debt). Assume the interest rate over the government debt is constant and apply Rule 4 to investigate equation (4):

(5) * d* G = *d* T + γ *d* ^{2} D

where *d* ^{2} = *d ×* *d* represents the acceleration of the government debt.

Next, assume that *d* T = 0, no change in T (taxes) between two adjacent periods, which (despite rhetorical noise) is agreed upon by conservatives and democrats. Then, equation (5) implies a public policy of increased government expenditure by debt acceleration:

(6) * d* G = γ *d* ^{2} D

To get closer to U.S. reality, assume *d* T = 0 and let the interest rate become a variable in equation (4). It is apparent from Rule 4 that the government debt accelerates with increased government expenditure. Thus, a variable interest rate adds complexity [(*d* γ) *d* D] to government deficit financing:

(7) * d* G = (*d* γ) *d* D + γ (*d* ^{2} D)

If *d* C = 0, *d* I = 0, and *d* (X - M) = 0, then the two-period increase in GDP is driven by the acceleration of the government debt, with or without increased taxes. The acceleration of government debt bodes terrible future consequences for the aging population, children, and grandchildren.

*Grosso modo*, the farcical success expressed in equation (7) represents the U.S. economic policy since President Reagan, except that the U.S. economy now suffers the double trouble from the leakage effect that the trade deficit is non-decreasing:

(8) * d* (X - M) < 0.

The government deficit [(G − T) > 0] adds to the increase of GDP, and the trade deficit [(X - M) < 0] subtracts from GDP.

The U.S. economy suffers from twin deficits. Government deficit is regarded as good, but it has become pernicious by debt acceleration beyond reasonable limits. Trade deficits are bad (vide the colonial search for *El Dorado* and the NATO obsession with controlling world petroleum), and can lead to belligerant foreign policy and the demise of diplomany.

As a takeaway, several important fiscal policy effects using the NIPA (national income and product accounts) can be explored from the accounting equation (1), gaining further understanding by introducing a government expenditure behavior equation (4). Recent U.S. economic success measured by the annual increases in GDP financed by debt acceleration is furtive (or dubious), and policy equation (7) shows that the U.S. government (including Congress and the executive branches) is irresponsible, acting as if there is no tomorrow.

The U.S. government’s fiscal profiligacy (or fast debt accumulation) must have limits. The largesse of public policies such as the Chips Act of 2022 (signed by President Biden on August 9, 2022) and the U.S. instigation or “unconditional” support of foreign wars are financed by further public debt acceleration, which are obtuse and reprehensible actions.

Rules 1 to 5 can be read in Samuel Goldberg, *Introduction to Difference Equations*, Wiley, 1958, p. 47 (Table 1.3).

Suppose that a particle moves in a straight line to a certain distance (s) depending on time. The distance as a function of time is written s = ƒ(*t*). Denote the speed (velocity, *vitesse* in French) of movement by v(*t*). Then *v(t)* = *d* s / *d* *t*. The rate of change of the speed is called acceleration, written as: *d* v / *d* *t* = a(*t*) = *d* ^{2} s / *d* *t* ^{2}. See Serge Lange, *A First Course in Calculus* (5th edition), Springer, 1986, pp. 106-107 (rate of change). For this tutorial, GDP and its individual components are like moving particles.

NIPA annual and quarterly GDP tables are available at:

https://apps.bea.gov/iTable/?reqid=19&step=2&isuri=1&categories=survey

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