Extended Samuelson Model (ESM)

• ESM is a memory-dependent extension of the Samuelson multiplier-accelerator model that incorporates two-lag consumption and third-order dynamics for richer economic analysis.
• It employs delayed difference equations and a state-space framework to capture oscillatory business cycle behavior and address equilibrium uniqueness through regularization.
• In equity-market applications, the model leverages normalized excess demand and multi-timeframe analysis to predict market turning points with high accuracy.

The term Extended Samuelson Model (ESM) is used in arXiv literature for distinct extensions of Samuelsonian dynamics. In macroeconomics, it denotes a reformulated multiplier-accelerator model in which consumption depends on two lagged values of national income, private investment depends on consumption changes, and national income is described by a third-order linear delayed difference equation with possible non-unique equilibrium and an associated notion of optimal equilibrium (Ortega et al., 2017). In equity-market research, the same term denotes a framework in which the rate of price change is driven by normalized excess demand and market microstructure effects, with peaks, troughs, turning points, market states, and directional signals analyzed across multiple timescales (Han, 2 Oct 2025).

1. Samuelsonian antecedent and the reformulated macroeconomic model

The reformulated ESM starts from the classical Samuelson multiplier-accelerator structure for national economy. In the original formulation, national income satisfies

$T_k = C_k + I_k + G_k,$

with consumption

$C_k = a T_{k-1}, \quad 0<a<1,$

investment

$I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$

and constant government spending

$G_k=\overline{G}.$

This yields the second-order linear difference equation

$T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$

The reformulated ESM extends this structure by incorporating memory into the system and by assuming that total consumption and private investment depend upon national income values (Ortega et al., 2017).

In the extended formulation, national income is written as

$T_k=C_k+I_k,$

consumption is specified by

$C_k=c_1T_{k-1}+c_2T_{k-2}+P,$

with $c_1,c_2>0$ and $0<c_1+c_2<1$, and $P$ is a constant that also absorbs governmental spending. Private investment remains accelerator-based,

$C_k = a T_{k-1}, \quad 0<a<1,$0

but, after substitution, becomes a function of three lagged income values. The resulting reformulation replaces the original second-order dynamics by a third-order delayed system.

2. Algebraic structure and memory effects

The central equation of the macroeconomic ESM is

$C_k = a T_{k-1}, \quad 0<a<1,$1

This is a third-order linear delayed difference equation for national income (Ortega et al., 2017). Its delay structure is generated by the consumption function with two lags and the induced investment term

$C_k = a T_{k-1}, \quad 0<a<1,$2

The model can be written as a first-order system,

$C_k = a T_{k-1}, \quad 0<a<1,$3

where

$C_k = a T_{k-1}, \quad 0<a<1,$4

Component	Equation	Role
National income	$C_k = a T_{k-1}, \quad 0<a<1,$5	Accounting identity
Consumption	$C_k = a T_{k-1}, \quad 0<a<1,$6	Two-lag memory
Investment	$C_k = a T_{k-1}, \quad 0<a<1,$7	Accelerator term
Main dynamic equation	$C_k = a T_{k-1}, \quad 0<a<1,$8	Third-order delayed dynamics
Matrix form	$C_k = a T_{k-1}, \quad 0<a<1,$9	State-space representation

The explicit dependence on three past values of $I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$0 is the model’s formal memory effect. In the paper’s interpretation, this produces richer dynamic behavior than the original Samuelson model and allows the respective solutions of the third-order polynomial to correspond to the typical observed business cycles of real economy.

3. Equilibrium, singularity, and optimal equilibrium

Equilibrium is defined by a fixed point $I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$1 satisfying

$I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$2

or equivalently

$I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$3

Let $I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$4. The determinant is

$I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$5

Hence a unique equilibrium occurs if $I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$6, that is, when $I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$7 is full rank (Ortega et al., 2017).

In the unique case, the equilibrium is

$I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$8

and because all equilibrium components must coincide, $I_k = b(C_k-C_{k-1}) = ab(T_{k-1}-T_{k-2}),$9, one obtains

$G_k=\overline{G}.$0

If $G_k=\overline{G}.$1, then $G_k=\overline{G}.$2 and $G_k=\overline{G}.$3 is singular. In this case, the system either has no solution or infinitely many equilibrium solutions. The paper therefore introduces an optimal equilibrium, defined as the solution that minimizes the distance in Euclidean norm between $G_k=\overline{G}.$4 and the image $G_k=\overline{G}.$5, regularized to ensure invertibility; it is described as the least-squares, minimum norm solution. With a small parameter $G_k=\overline{G}.$6 and matrix $G_k=\overline{G}.$7 $G_k=\overline{G}.$8, the regularized equilibrium is

$G_k=\overline{G}.$9

The paper identifies this construction with standard Tikhonov regularization and links it to minimum norm or Moore-Penrose pseudoinverse solutions in linear algebra.

4. Dynamic interpretation and business-cycle content

The reformulated ESM is presented as a memory-dependent extension of Samuelson’s multiplier-accelerator model in which delayed effects are intended to better capture economic behavior. The paper states that decisions on consumption and investment in a real economy may depend on more than just the immediate past, and the two-lag consumption rule formalizes that dependence (Ortega et al., 2017).

Third-order difference equations can naturally produce persistent oscillatory behavior. In the paper’s interpretation, the roots of the characteristic third-order polynomial correspond to business cycle phenomena observed empirically, including cycles in GDP or national income. By suitably choosing the parameters $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$0, $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$1, and $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$2, the model can generate cycles with different amplitudes, periods, and damping characteristics. The incorporation of memory therefore makes it possible for the model to generate both stable trajectories and cyclical, including damped or sustained, fluctuations.

The equilibrium-selection problem also has an economic interpretation. In the non-unique case, the optimal equilibrium is presented as a systematic method for selecting a meaningful reference-point equilibrium. The paper further states that, if multiple static steady-state outcomes are possible, the optimal equilibrium provides a criterion for policy targets or expectations.

5. Equity-market usage of the same name

A later paper applies the name Extended Samuelson Model to equity-market price dynamics and traces the framework to Samuelson’s 1941 insight that the rate of price change in an asset is driven by excess demand. In that formulation,

$T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$3

where $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$4 is price, $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$5 and $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$6 are demand and supply at price $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$7, $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$8 is an increasing function of normalized excess demand with $T_{k+2} - a(1+b)T_{k+1} + abT_k = \overline{G}.$9 and $T_k=C_k+I_k,$0, and $T_k=C_k+I_k,$1 denotes the effect of price-makers and market microstructure (Han, 2 Oct 2025).

The core observable is normalized excess demand,

$T_k=C_k+I_k,$2

which is described as reflecting the aggregate bias of price-takers. The framework is presented as causal and dynamic rather than statistical and correlational: micro-level inputs such as order flow and normalized excess demand are aggregated to macro-outcomes such as trends, crashes, and rebounds, while liquidity takers and liquidity providers are modeled as interacting forces.

Within this market-oriented ESM, peaks occur when normalized excess demand reaches local maxima and troughs when it reaches minima. Turning points are derived by monitoring normalized excess demand transitions and T2/T4 thresholds, computed from coarser timeframes and projected into finer intervals. The paper also defines temporal compatibility: all broader timeframe signals are present in finer timeframes, while finer intervals provide additional, sharper prediction. The hierarchy is monthly $T_k=C_k+I_k,$3 weekly $T_k=C_k+I_k,$4 daily $T_k=C_k+I_k,$5 hourly $T_k=C_k+I_k,$6 5min $T_k=C_k+I_k,$7 1min.

The framework codifies eight market states based on normalized excess demand at three nested timeframes—daily, weekly, and monthly—from maximum pessimism, State 1, to euphoria, State 8. It also defines six directional signals: Uptrend Signal, Downtrend Signal, Reverse to UP, Reverse to DOWN, Selling by Informed Takers, and Buying by Informed Takers. These are used for entry and exit, scaling and filtering, multi-timeframe fusion, and stop-loss or take-profit placement around T2/T4 turning points and prior high/low reversals.

The paper reports reversal prediction accuracy of $T_k=C_k+I_k,$8 with 15-min data and $T_k=C_k+I_k,$9 with 5-min data in real-time trading tests, and states that intraday normalized excess demand accounts for $C_k=c_1T_{k-1}+c_2T_{k-2}+P,$0 of daily S&P 500 return variation over ten years. It further claims that, even during calm periods, the framework can capture 10-point swings in the S&P 500, equivalent to $C_k=c_1T_{k-1}+c_2T_{k-2}+P,$1 per E-mini futures contract. The same paper compares its state-based approach to state-based approaches attributed to Renaissance Technologies’ Medallion Fund.

6. Terminology, scope, and acronym ambiguity

The macroeconomic reformulation in "Optimal equilibrium for a reformulated Samuelson economical model" (Ortega et al., 2017) and the equity-market framework in "Equity Market Price Changes Are Predictable: A Natural Science Approach" (Han, 2 Oct 2025) are not the same model. The former is a third-order delayed difference-equation model for national income with memory, equilibrium analysis, and optimal equilibrium under singularity. The latter is a differential, excess-demand and microstructure framework for market states, directional signals, and real-time reversal prediction. The shared name reflects a common appeal to Samuelsonian dynamic adjustment, but the state variables, governing equations, and empirical domains differ.

A further source of ambiguity is the acronym ESM itself. In unrelated arXiv literature, ESM denotes the Extended Space Model, a proposed generalization of special relativity from $C_k=c_1T_{k-1}+c_2T_{k-2}+P,$2 to $C_k=c_1T_{k-1}+c_2T_{k-2}+P,$3 in which the interval $C_k=c_1T_{k-1}+c_2T_{k-2}+P,$4 becomes a fifth coordinate and scalar gravity is unified with electromagnetism (Andreev et al., 2013). This suggests that references to “ESM” require disambiguation by field, equation set, and claimed physical or economic mechanism.

For research usage, the term Extended Samuelson Model is therefore best treated as a context-dependent label rather than a uniquely standardized model class. In the macroeconomic usage, its distinguishing feature is delayed national-income dynamics with non-unique equilibrium handled by regularized least squares. In the market usage, its distinguishing feature is normalized excess demand, temporal compatibility across timeframes, and state/signal-based identification of turning points.