Marginal Propensities to Consume (MPCs)

Updated 25 October 2025

Marginal Propensities to Consume (MPCs) are defined as the derivative of the consumption function with respect to income, measuring how consumer spending responds to income changes.
Theoretical frameworks use nonlinear models, DSGE systems, and behavioral dynamics to capture MPC variability across different wealth and income states.
Empirical and computational studies reveal that heterogeneous MPCs, influenced by wealth effects, social networks, and policy feedbacks, are key to understanding fiscal multiplier effects.

Marginal propensities to consume (MPCs) quantify the responsiveness of consumer expenditure to changes in income or liquid resources. In microeconomic and macroeconomic analysis, the MPC is a central measure for evaluating the transmission of fiscal policy interventions, the heterogeneity of household behavior, and the dynamics of aggregate demand. The MPC is formally defined as the derivative $dc/db$ of the consumption function $c(\cdot)$ with respect to the available resources or income $b$ . Empirical and theoretical advances have demonstrated that the MPC is rarely constant across agents, time, or states of the economy, reflecting structural features, behavioral feedbacks, and the influence of network effects.

1. Analytical Frameworks and Fundamental Models

The classic approach models the time evolution of a consumer’s disposable budget $b(t)$ and links it to consumption via a functional relationship. For example, in "Propensity to spending of an average consumer over a brief period" (Luca et al., 2017), the evolution equation is

$\frac{db}{dt} = y_0 - c(b)$

with fixed income $y_0$ and nonlinear consumption function:

$c(b) = a b^2 + c_0 \quad (b \geq 0)$

where $a>0$ and $c_0$ is the subsistence level. The MPC is directly given by

$MPC = \frac{dc}{db} = 2a b$

illustrating its strict dependence on wealth: as budget $b$ increases, the MPC rises in proportion, implying dynamically amplifying consumption responses among wealthier consumers.

In multi-group or loop-based macroeconomic models, such as (Bar-Yam et al., 2017), MPC heterogeneity naturally arises from the differentiation between wage recipients (workers, high MPC) and capital owners (investors, low MPC). The MPC determines the strength of the fiscal multiplier, which in turn conditions macroeconomic stability and recession risk.

2. State-Dependence, Nonlinearity, and Feedback Effects

Models incorporating behavioral dynamics, such as confidence and learning, endogenize the MPC as a function of both micro and macro state variables. DSGE approaches with reflexive feedback (Morelli et al., 2019, Morelli et al., 2021) typically introduce a "confidence variable" $f_t$ or $G_t$ controlling the fraction of income consumed:

$c_t = G_t Y_t$

with $G_t$ adjusted via nonlinear mappings (e.g., logistic functions) of past consumption and thresholds:

$G_t = \frac{1}{2} [G_{\min} + G_{\max} + (G_{\max} - G_{\min}) \cdot \tanh(\theta_c \mathcal{C}_t)]$

where the confidence index $\mathcal{C}_t$ depends on deviations from subsistence or trend levels. Thus, in periods following negative shocks or policy errors, confidence collapses, $G_t$ (and hence the MPC) declines sharply, triggering demand crises and amplifying the effects of downturns.

Empirically, MPCs respond acutely to business-cycle phase, crisis intensity, and policy narratives, with output volatility and recession probabilities highly sensitive to the underlying MPC through these endogenous feedbacks.

3. Heterogeneity, Wealth Effects, and Asymptotics

General income fluctuation and savings models (Ma et al., 2020, Ma et al., 2020) rigorously characterize the asymptotic MPC as a function of wealth, risk aversion, and stochastic returns. Under weak regularity assumptions—for instance, regularly varying marginal utility $u'(c) = c^{-\gamma}\ell(c)$ —the optimal consumption function $c(a,z)$ in high-wealth regimes becomes asymptotically linear:

$\lim_{a \to \infty} \frac{c(a,z)}{a} = \bar{c}(z)$

with $\bar{c}(z)$ the asymptotic MPC characterized by a fixed-point equation involving the dynamics of discount factors and returns. Notably, necessary and sufficient conditions for $\bar{c}(z)$ to be zero (i.e., vanishing MPC for the rich) are tied to the spectral radius $r(K(1-\gamma)) \geq 1$ , signifying situations where high wealth agents save nearly all additional income. This provides a microfoundation for empirically observed high saving rates among the rich and contributes to the emergence of Pareto wealth tails.

The implication is that the distribution of MPCs itself is highly non-uniform, varying systematically across the wealth and income distributions and affected by both lifecycle factors and stochastic environments. In policy terms, elevated aggregate MPCs are associated with stronger aggregate demand responses and more potent fiscal multipliers, particularly when stimulus is targeted toward high-MPC groups.

Network-based consumption models (Schulz et al., 2022) formalize how agents’ MPCs are modulated by social context and reference group structures:

$C(i) = w Y(i) + (1-w) c [C(j|i) - w Y(i)]$

where $w$ is a baseline MPC and $c$ governs the strength of social comparison. The effective MPC for agent $i$ is recursively determined by the incomes and consumption of peers, with network topology (especially homophily/segregation) strongly affecting the propagation of upward-looking consumption externalities. Weakly segregated networks (low homophily) yield elevated MPCs among low-income agents, whereas strong segregation suppresses these social spillovers.

Crucially, models of this genre can replicate aggregate empirical patterns traditionally attributed to permanent income hypothesis-driven smoothing, but with distinctive heterogeneity and responsiveness features, particularly relevant for understanding debt dynamics and expenditure cascades in times of rising inequality.

5. Measurement, Estimation Techniques, and Empirical Implementation

Contemporary approaches to estimating MPCs out of income or latent shocks employ semiparametric and nonparametric methods (Lee et al., 2022, Dong et al., 2022). In semiparametric GLMs, marginal effects—the econometric analog of MPCs—are efficiently estimated by approximate maximum likelihood using nonparametric spline expansions for unknown density components. These estimators are robust to model misspecification and enable formal inference on derivative-based quantities central to consumption analysis.

For buffer-stock saving models, density-weighted average derivative estimators (Dong et al., 2022) address measurement error using deconvolution kernels and repeated noisy measures:

$\hat{\theta}_1 = -\frac{2}{n^2 b_n^3} \sum_{j=1}^n \sum_{k=1}^n (Y_j - W_j) \int \hat{\mathcal{K}}((x - X_j)/b_n) \hat{\mathcal{K}}'((x - X_k)/b_n) dx$

Applied to panel data, these estimators reject the null of unit MPC out of permanent income, supporting the buffer-stock view that consumers exhibit sub-unit MPCs due to precautionary motives. This is a significant departure from classical complete-insurance models and directly informs the theoretical calibration of policy simulations.

6. Computational and Econophysical Perspectives

Kinetic models of wealth exchange (Cui et al., 2022) operationalize MPCs as complements to agent-specific saving propensities $\lambda_i$ via the simple result $MPC = 1 - \lambda_i$ . Simulation of transaction rules and heterogeneous agent types demonstrates that lower MPCs (higher saving rates) lead to a more concentrated, less unequal wealth distribution, with contour analyses of Gini and Kolkata indices mapping how increases in saving (decreases in MPC) systematically reduce inequality. These results have tractable theoretical implication for the design of fiscal policy and the interpretation of aggregate expenditure data as an ensemble of heterogeneous micro behaviors.

7. Modern Behavioral and Machine Learning Models

Recent machine learning and reinforcement learning-based models (Kuriksha, 2021, Kaplowitz, 23 Oct 2025) embed policy-function learning directly in agent decision frameworks. Neural network agents and Q-learning agents, updating consumption-saving rules via experience-driven learning (e.g., using temporal difference errors), empirically generate higher average MPCs and excess sensitivity. For example, agents with histories of low assets or unemployment exhibit MPCs out of stimulus transfers of 0.50 compared to 0.34 for high-asset households, with persistent "scarring" effects on consumption due to machine learning approximation errors rather than belief updating or ex-ante heterogeneity. Such frameworks offer a micro-foundation for observed patterns in consumption data previously inconsistent with rational expectations models and provide a numerical laboratory for evaluating the distributional and welfare effects of policy interventions.

In summary, the literature documents rich heterogeneity and dynamic behavior in marginal propensities to consume, spanning analytical, empirical, and computational domains. The MPC is shaped by wealth, income, behavioral feedbacks, social networks, policy, and learning. Modern estimation techniques and simulation approaches have enabled precise modeling of these effects, refuting canonical models of constant MPC and grounding fiscal policy recommendations in the true complexity of consumption response. The theoretical and empirical frontier continues to advance as models integrate new mechanisms and datasets, contributing to our evolving understanding of consumption and its role in both microeconomic welfare and macroeconomic stability.