Neural Demand Estimation with Habit Formation and Rationality Constraints

Marta Grzeskiewicz

econ.GNMar 2, 2026

Neural Demand Estimation with Habit Formation and Rationality Constraints

Authors: Marta Grzeskiewicz

Organizations: University of Cambridge

Abstract

We develop a flexible neural demand system for continuous budget allocation that estimates budget shares on the simplex by minimizing KL divergence. Shares are produced via a softmax of a state-dependent preference scorer and disciplined with regularity penalties (monotonicity, Slutsky symmetry) to support coherent comparative statics and welfare without imposing a parametric utility form. State dependence enters through a habit stock defined as an exponentially weighted moving average of past consumption. Simulations recover elasticities and welfare accurately and show sizable gains when habit formation is present. In our empirical application using Dominick's analgesics data, adding habit reduces out-of-sample error by c.33%, reshapes substitution patterns, and increases CV losses from a 10% ibuprofen price rise by about 15-16% relative to a static model. The code is available at https://github.com/martagrz/neural_demand_habit .

Figures & tables

Model	CES
LA-AIDS	$0.00979\pm 0.00010$
BLP (IV)	$0.02207\pm 0.00025$
QUAIDS	$0.00978\pm 0.00012$
Series Estm.	$0.00302\pm 0.00006$
LD (Shared)	$0.15411\pm 0.00030$
LD (GoodSpec)	$0.15532\pm 0.00040$

Table 1: Post-Shock RMSE: CES DGP (mean

\pm

SE, 5 runs)

Figure 1: Observed vs. predicted budget share for Good 0 under the CES DGP. Left : Neural Demand (static). Right : LA-AIDS. The neural demand system achieves near-exact prediction across the full support; LA-AIDS exhibits a systematic fan pattern reflecting misspecification of the CES Marshallian demand curvature.

Panel of Figure 2 — Figure 2: Predicted budget-share demand curves under the CES DGP ( $\pm$ 1 SE across runs). Budget shares plotted as a function of Good-1 (Fuel) price $p_{1}$ ; all other prices and income held at simulation means.

Figure 3: Full $3\times 3$ price elasticity matrices under the CES DGP for Ground Truth, LA-AIDS, QUAIDS, and Neural Demand (static). Diagonal: own-price quantity elasticities. Off-diagonal: cross-price quantity elasticities. Neural Demand recovers both diagonal and off-diagonal elements with near-exact accuracy; LA-AIDS and QUAIDS produce small biases on the Good-2 own-price elasticity and the Food-Other cross-price terms.

Model	$\hat{\varepsilon}_{00}$	$\hat{\varepsilon}_{11}$	$\hat{\varepsilon}_{22}$
LA-AIDS	$-1.422\pm 0.001$	$-1.487\pm 0.001$	$-1.697\pm 0.003$
BLP (IV)	$-1.485\pm 0.004$	$-1.662\pm 0.005$	$-1.787\pm 0.004$
QUAIDS	$-1.421\pm 0.001$	$-1.487\pm 0.001$	$-1.699\pm 0.003$
Series Estm.	$-1.424\pm 0.001$	$-1.478\pm 0.001$	$-1.713\pm 0.001$
Linear Demand (Shared)	$-1.522\pm 0.008$	$-1.613\pm 0.012$	$-1.519\pm 0.013$
Linear Demand (GoodSpec)	$-1.560\pm 0.007$	$-1.703\pm 0.011$	$-1.555\pm 0.014$

Table 2: Own-Price Quantity Elasticities — Simulation (CES DGP)

Model	CV	Error vs Truth (%)
LA-AIDS	$-122.0925\pm 0.2934$	0.4%
BLP (IV)	$-120.4891\pm 0.3883$	1.7%
QUAIDS	$-122.1117\pm 0.3041$	0.4%
Series Estm.	$-122.8321\pm 0.2975$	0.2%
Linear Demand (Shared)	$-92.5375\pm 0.2241$	24.5%
Linear Demand (GoodSpec)	$-92.0899\pm 0.2394$	24.9%

Table 3: Compensating Variation — Simulation (CES DGP, 20% shock)

Model	Quasilinear	Leontief	Stone–Geary	Endog. CES
LA-AIDS	$0.00003\pm 0.00000$	$0.01884\pm 0.00019$	$0.00581\pm 0.00004$	$0.08610\pm 0.00178$
BLP (IV)	$0.00577\pm 0.00000$	$0.02762\pm 0.00030$	$0.00478\pm 0.00003$	$0.02902\pm 0.00189$
QUAIDS	$0.00003\pm 0.00000$	$0.01879\pm 0.00022$	$0.00576\pm 0.00005$	$0.08721\pm 0.00182$
Series Estm.	$0.00003\pm 0.00000$	$0.00319\pm 0.00002$	$0.00112\pm 0.00001$	$0.08696\pm 0.00248$
LD (Shared)	$0.47159\pm 0.00079$	$0.09059\pm 0.00052$	$0.12343\pm 0.00012$	$0.16474\pm 0.00038$
LD (GoodSpec)	$0.47185\pm 0.00076$	$0.09465\pm 0.00060$	$0.12352\pm 0.00013$	$0.16524\pm 0.00043$

Table 4: Post-Shock RMSE: Other DGPs (mean

\pm

SE, 5 runs)

Figure 4: Post-shock RMSE across five DGPs (

\pm

SE, 5 runs). LDS (Shared) and LDS (GoodSpec) dominate the Quasilinear column at RMSE

\approx 0.47

; Var. Mixture dominates Leontief at

0.17

. Neural demand variants are essentially invisible on static DGPs. BLP (IV) is included for reference only; it is not an appropriate structural comparator for the continuous-allocation DGPs (Section 2.8.4 ).

Figure 5: RMSE heatmap: key model comparators across six DGPs (mean RMSE, SE in parentheses). Red = high error; green = low error.

Panel of Figure 6 — Figure 6: Predicted budget-share demand curves under the Habit Formation DGP ( $\theta=0.3$ , $\delta_{\text{DGP}}=0.7$ ; habit stocks held at training means). The true Fuel demand (Good 1, centre) exhibits a non-monotone shape driven by habit persistence that all static models miss. Neural Demand (habit) qualitatively captures the pattern. BLP is included for reference only; it is not an appropriate structural comparator for this DGP.

Model	RMSE	KL Div.	RMSE Red. (%)
LA-AIDS	$0.04625\pm 0.00116$	$0.00887\pm 0.00039$	0.0%
BLP (IV)	$0.04771\pm 0.00118$	$0.01959\pm 0.00041$	-3.2%
QUAIDS	$0.04619\pm 0.00117$	$0.00885\pm 0.00040$	0.1%
Series Estm.	$0.04553\pm 0.00113$	$0.00843\pm 0.00037$	1.6%
Linear Demand (Shared)	$0.14856\pm 0.00048$	$0.11231\pm 0.00094$	-222.0%
Linear Demand (GoodSpec)	$0.14935\pm 0.00052$	$0.11339\pm 0.00099$	-223.7%

Table 5: Habit-State Augmentation Advantage — Simulation (5 runs; mean

\pm

SE)

Figure 7: Habit-state augmentation advantage on the Habit Formation DGP. Left : RMSE. Right : KL divergence. All static models cluster at RMSE

\approx 0.047

–

0.050

(equivalent to LA-AIDS). Neural Demand (habit) reduces RMSE by 80% and KL by a factor of more than 20.

Figure 8: Profile KL criterion for

\delta

identification (

\pm

1 SE, 5 seeds). The minimum coincides with the true value

\delta_{\text{DGP}}=0.7

(red dashed line). The flat region spanning approximately

[0.54,0.77]

under the 2-SE rule constitutes the identified set: values below 0.5 and above 0.85 are sharply rejected; intermediate values are observationally near-equivalent.

Quantity	Value
True $\delta$	0.7
$\hat{\delta}$ (mean ± SE)	0.680 ± 0.014
IS lo (mean)	0.487
IS hi (mean)	0.762
IS width (mean±SE)	0.276 ± 0.122
Coverage (%)	100

Table 6: Partial Identification of

\delta

(Habit-Decay Parameter) — 5 seeds

Train: 28,644 obs Test: 4,287 obs Stores: 93 Weeks: 1–399
Good	Mean Price	Std Price	Mean Share	Std Share
	($/100 tablets)
Aspirin	7.311	1.557	0.2107	0.0519
Acetaminophen	10.554	1.427	0.5026	0.0785
Ibuprofen	10.506	1.428	0.2867	0.0819

Table 7: Descriptive Statistics: Dominick’s Analgesics Scanner Panel

Model	RMSE	MAE
LA-AIDS	$0.07069$	$0.05593$
BLP (IV)	$0.07766$	$0.05991$
QUAIDS	$0.06956$	$0.05519$
Series Est.	$0.06514$	$0.05077$
Linear Demand (Shared)	$0.09201$	$0.07361$
Linear Demand (GoodSpec)	$0.09197$	$0.07356$

Table 8: Out-of-Sample Predictive Accuracy — Dominick’s Analgesics (5 independent re-estimations; mean

\pm

std)

Figure 9: Out-of-sample RMSE — Dominick’s Analgesics. The habit-augmented models achieve RMSE

\approx 0.048

, approximately 33% below the LA-AIDS baseline.

Model	$\epsilon_{00}$ (ASP)	$\epsilon_{11}$ (ACET)	$\epsilon_{22}$ (IBU)
LA-AIDS	$-0.849$	$-1.420$	$-1.140$
BLP (IV)	$-0.998$	$-1.769$	$-0.587$
QUAIDS	$-0.882$	$-1.423$	$-1.129$
Series Est.	$-0.708$	$-1.629$	$-1.224$
Linear Demand (Shared)	$-0.163$	$-0.215$	$-0.190$
Linear Demand (GoodSpec)	$-0.164$	$-0.205$	$-0.181$

Table 9: Own-Price Quantity Elasticities — Dominick’s Analgesics (5 runs; mean

\pm

std)

Figure 10: Cross-price elasticity matrices for seven key specifications — Dominick’s Analgesics. Evaluated at mean test prices; habit stock fixed at cross-sectional mean (MDP, sorting removed). Diagonal: own-price quantity elasticities. Off-diagonal: cross-price quantity elasticities. The aspirin–ibuprofen cross-price effects (row 1 column 3; row 3 column 1) converge toward zero under all neural demand specifications and are essentially zero under the habit and habit, FE models. The acetaminophen–ibuprofen cross (row 2 column 3; row 3 column 2) is robustly positive across all specifications. The full eleven-specification matrix is reported in Appendix Figure 15 .

Figure 11: Neural demand decomposition: structural price effect vs. state–segment correlation — Dominick’s Analgesics. Blue : Neural Demand (static), plotting the model-implied conditional mean aspirin budget share as aspirin price varies. Dashed green : Neural Demand (habit) evaluated at a fixed state (habit stock set to the cross-sectional mean

\bar{x}

), isolating the structural price effect holding the state constant. Solid teal : Neural Demand (habit) evaluated at the price-conditional state distribution (allowing

\bar{x}

to vary with price as in the data), combining the structural effect with state selection. Shaded : contribution of state–segment correlation (gap between teal and dashed green).

Figure 12: Market segmentation and state selection — Dominick’s Analgesics, test set. Left: Aspirin budget share $w_{0}$ vs. ibuprofen budget share $w_{2}$ (Spearman $\rho=-0.361$ ), consistent with persistent segmentation across stores. Right: Aspirin habit stock $\bar{x}_{0}$ vs. ibuprofen price (Spearman $\rho=-0.033$ ; binned mean $\pm 1$ SE), indicating little cross-good state selection with respect to ibuprofen price.

Figure 13: Cross-price demand matrix — Dominick’s Analgesics (

\pm 1

SD,

n=5

). Row = price that varies; column = budget share response. All other prices and income are held at evaluation-sample means. Neural Demand (habit, fixed

\bar{x}

) evaluates the habit model at

\bar{x}=\mathbb{E}[\bar{x}]

to remove state selection. The aspirin–ibuprofen cross-price slopes largely vanish under the fixed-state habit specification, while acetaminophen–ibuprofen substitution remains robustly positive.

Model	CV Loss ($)	vs Neural Demand (static)
LA-AIDS	$-34.8710$	+1.1%
BLP (IV)	$-38.1498$	-8.2%
QUAIDS	$-34.9710$	+0.8%
Series Est.	$-36.1016$	-2.4%
Linear Demand (Shared)	$-42.1538$	-19.6%
Linear Demand (GoodSpec)	$-42.1721$	-19.6%

Table 10: Welfare Loss from 10% Ibuprofen Price Increase — Dominickś Analgesics (5 run(s); mean

\pm

std)

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 14: Training KL divergence by DGP across epochs (

\pm

1 SE, 5 seeds). Each row corresponds to a specific Data Generating Process (DGP), and each column represents a distinct ND architecture: Static , Habit-persistent , Control Function (CF) , and the integrated Habit-CF model. The y-axis displays the Training Kullback-Leibler (KL) divergence over 10,000 epochs, illustrating the optimization trajectory for each estimator. Solid and dashed lines represent the mean training loss across 5 independent seeds, while the shaded regions denote

\pm 1

standard error. Vertical dotted lines and markers indicate the checkpoint corresponding to the minimum training loss.

Price equation	First-stage $F$	First-stage $R^{2}$	Partial $R^{2}$
Aspirin	43285.80	0.645	0.603
Acetaminophen	49300.16	0.664	0.633
Ibuprofen	35854.14	0.619	0.557

Appendix

Table 11: First-Stage Diagnostics — Hausman Instruments (Dominick’s Analgesics)

Figure 15: Full $3\times 3$ cross-price elasticity heatmaps for the control function specifications — Dominick’s Analgesics. Rows: Neural Demand (CF) and Neural Demand (habit, CF), evaluated at $\hat{v}_{t}=0$ (causal demand function).

Figure 16: Training KL divergence on the Dominick’s analgesics panel (

\pm

1 SE, 5 seeds) for the neural demand models over 10,000 epochs.

Model	Test KL	$\mathbb{E}[D_{\mathrm{sym}}]$	$\max D_{\mathrm{sym}}$	$\Pr(\lambda_{\max}(\tilde{S})>0)$	$\mathbb{E}[D_{\mathrm{curv}}]$	$\mathbb{E}[D_{\mathrm{hom}}]$
Static (no reg)	0.0192	0.0499	0.5395	0.937	0.0027	0.0576
Static (reg)	0.0201	0.0353	0.4358	0.937	0.0019	0.0487
Habit (no reg)	0.0114	0.0157	0.2463	0.931	0.0004	0.0186
Habit (reg)	0.0113	0.0078	0.1279	0.930	0.0001	0.0109
FE static (reg)	0.0164	0.0438	0.2553	0.937	0.0043	0.0722
FE habit (reg)	0.0114	0.0107	0.1009	0.937	0.0003	0.0175

Appendix

Table 12: Near-integrability diagnostics (“regularity dashboard”).

Quantity	Value
$\hat{\delta}$ mean ± SE)	0.800 ± 0.000
IS lo (mean)	0.640
IS hi (mean)	0.900
IS width (mean±SE)	0.260 ± 0.024

Appendix

Table 13: Profile KL Identification of

\delta

— Dominick’s Analgesics (5 seeds)

Figure 17: Profile KL criterion for

\delta

— Dominick’s Analgesics (

\pm 1

SD, 5 seeds). Minimum at

\hat{\delta}=0.868

(dotted vertical line). The identified set

[0.68,0.90]

(shaded) spans values within the 2-SD threshold. Values below

\delta=0.5

are sharply rejected; intermediate values are observationally near-equivalent.

Model	CV Loss ($)	vs Neural Demand (static)
LA-AIDS	$-23.2182$	+2.3%
BLP (IV)	$-22.1176$	+6.9%
QUAIDS	$-23.3916$	+1.5%
Series Est.	$-23.3300$	+1.8%
Linear Demand (Shared)	$-28.2325$	-18.8%
Linear Demand (GoodSpec)	$-28.1783$	-18.6%

Appendix

Table 14: Welfare Loss from 10% Aspirin Price Increase — Dominickś Analgesics (5 run(s); mean

\pm

std)

Model	CV Loss ($)	vs Neural Demand (static)
LA-AIDS	$-53.2076$	-2.8%
BLP (IV)	$-47.9725$	+7.3%
QUAIDS	$-52.9144$	-2.2%
Series Est.	$-51.2835$	+1.0%
Linear Demand (Shared)	$-46.4907$	+10.2%
Linear Demand (GoodSpec)	$-46.5685$	+10.1%

Appendix

Table 15: Welfare Loss from 10% Acetaminophen Price Increase — Dominickś Analgesics (5 run(s); mean

\pm

std)

Integrable Elasticity via Neural Demand Potentials
May 21, 2026Carlos Heredia, Daniel Roncel
IAMM Research, Department of Applied Artificial Intelligence DAMM, Carrer del Rossell´o 515, 08025 Barcelona, Catalonia, Spain
Learning Consumer Preferences from Bundle Sales Data
Sep 11, 2022Ningyuan Chen, Setareh Farajollahzadeh, Qingwei Jin +2User Preference Modeling
University of Toronto · McGill University · Zhejiang University
Selective Tie Learning: Mitigating Spurious Correlations in Preference Optimization
Oct 26, 2026Christian Moya, Purav Matlia, Elliott Thornley +1Preference Optimization Spurious Correlation Robustness
Purdue University · National University of Singapore

Abstract

Explore similar work

Integrable Elasticity via Neural Demand Potentials

Learning Consumer Preferences from Bundle Sales Data

Selective Tie Learning: Mitigating Spurious Correlations in Preference Optimization