Introduction

This vignette demonstrates how to fit individual (fixed-effect) demand curves using fit_demand_fixed(). This function fits separate nonlinear least squares (NLS) models for each subject, producing per-subject estimates of demand parameters like $Q_{0}$ (intensity) and $\alpha$ (elasticity).

When to use fixed-effect models: Fixed-effect models are appropriate when you want independent curve fits for each participant. They make no assumptions about the distribution of parameters across subjects and are the simplest approach to demand curve analysis. For hierarchical models that share information across subjects, see vignette("mixed-demand"). For guidance on choosing between approaches, see vignette("model-selection").

Data format: All modeling functions expect long-format data with columns id (subject identifier), x (price), and y (consumption). See vignette("beezdemand") for details on data preparation and conversion from wide format.

Fitting with Different Equations

fit_demand_fixed() supports several equation forms. We demonstrate the three most common below using the apt (Alcohol Purchase Task) dataset.

Hursh & Silberberg (“hs”)

The exponential model of demand (Hursh & Silberberg, 2008):

\[\log_{10}(Q) = \log_{10}(Q_0) + k \cdot (e^{-\alpha \cdot Q_0 \cdot x} - 1)\]

fit_hs <- fit_demand_fixed(apt, equation = "hs", k = 2)
fit_hs
#> 
#> Fixed-Effect Demand Model
#> ==========================
#> 
#> Call:
#> fit_demand_fixed(data = apt, equation = "hs", k = 2)
#> 
#> Equation: hs 
#> k: fixed (2) 
#> Subjects: 10 ( 10 converged, 0 failed)
#> 
#> Use summary() for parameter summaries, tidy() for tidy output.

Koffarnus (“koff”)

The exponentiated model (Koffarnus et al., 2015):

\[Q = Q_0 \cdot 10^{k \cdot (e^{-\alpha \cdot Q_0 \cdot x} - 1)}\]

fit_koff <- fit_demand_fixed(apt, equation = "koff", k = 2)
fit_koff
#> 
#> Fixed-Effect Demand Model
#> ==========================
#> 
#> Call:
#> fit_demand_fixed(data = apt, equation = "koff", k = 2)
#> 
#> Equation: koff 
#> k: fixed (2) 
#> Subjects: 10 ( 10 converged, 0 failed)
#> 
#> Use summary() for parameter summaries, tidy() for tidy output.

Simplified (“simplified”)

The simplified exponential model (Rzeszutek et al., 2025) does not require a scaling constant $k$:

\[Q = Q_0 \cdot e^{-\alpha \cdot Q_0 \cdot x}\]

fit_simplified <- fit_demand_fixed(apt, equation = "simplified")
fit_simplified
#> 
#> Fixed-Effect Demand Model
#> ==========================
#> 
#> Call:
#> fit_demand_fixed(data = apt, equation = "simplified")
#> 
#> Equation: simplified 
#> k: none (simplified equation) 
#> Subjects: 10 ( 10 converged, 0 failed)
#> 
#> Use summary() for parameter summaries, tidy() for tidy output.

The k Parameter

The scaling constant $k$ controls the range of the demand function. For the "hs" and "koff" equations, k can be specified in several ways:

`k` value	Behavior
Numeric (e.g., `2`)	Fixed constant for all subjects (default)
`"ind"`	Individual $k$ per subject, computed from each subject’s data range
`"share"`	Single shared $k$ estimated across all subjects via global regression
`"fit"`	$k$ is a free parameter estimated jointly with $Q_0$ and $\alpha$

## Fixed k (default)
fit_demand_fixed(apt, equation = "hs", k = 2)

## Individual k per subject
fit_demand_fixed(apt, equation = "hs", k = "ind")

## Shared k across subjects
fit_demand_fixed(apt, equation = "hs", k = "share")

## Fitted k as free parameter
fit_demand_fixed(apt, equation = "hs", k = "fit")

The param_space argument controls whether optimization is performed on the natural scale ("natural", the default) or log10 scale ("log10"). The log10 scale can improve convergence for some datasets:

fit_demand_fixed(apt, equation = "hs", k = 2, param_space = "log10")

Inspecting Fits

All beezdemand_fixed objects support the standard tidy(), glance(), augment(), and confint() methods for programmatic access to results.

tidy(): Per-Subject Parameter Estimates

tidy(fit_hs)
#> # A tibble: 40 × 10
#>    id    term  estimate std.error statistic p.value component estimate_scale
#>    <chr> <chr>    <dbl>     <dbl>     <dbl>   <dbl> <chr>     <chr>         
#>  1 19    Q0       10.2      0.269        NA      NA fixed     natural       
#>  2 30    Q0        2.81     0.226        NA      NA fixed     natural       
#>  3 38    Q0        4.50     0.215        NA      NA fixed     natural       
#>  4 60    Q0        9.92     0.459        NA      NA fixed     natural       
#>  5 68    Q0       10.4      0.329        NA      NA fixed     natural       
#>  6 106   Q0        5.68     0.300        NA      NA fixed     natural       
#>  7 113   Q0        6.20     0.174        NA      NA fixed     natural       
#>  8 142   Q0        6.17     0.641        NA      NA fixed     natural       
#>  9 156   Q0        8.35     0.411        NA      NA fixed     natural       
#> 10 188   Q0        6.30     0.564        NA      NA fixed     natural       
#> # ℹ 30 more rows
#> # ℹ 2 more variables: term_display <chr>, estimate_internal <dbl>

glance(): Model-Level Summary

glance(fit_hs)
#> # A tibble: 1 × 12
#>   model_class      backend equation k_spec     nobs n_subjects n_success n_fail
#>   <chr>            <chr>   <chr>    <chr>     <int>      <int>     <int>  <int>
#> 1 beezdemand_fixed legacy  hs       fixed (2)   146         10        10      0
#> # ℹ 4 more variables: converged <lgl>, logLik <dbl>, AIC <dbl>, BIC <dbl>

augment(): Fitted Values and Residuals

augment(fit_hs)
#> # A tibble: 146 × 6
#>    id        x     y     k .fitted  .resid
#>    <chr> <dbl> <dbl> <dbl>   <dbl>   <dbl>
#>  1 19      0      10     2   10.2  -0.159 
#>  2 19      0.5    10     2    9.69  0.314 
#>  3 19      1      10     2    9.24  0.760 
#>  4 19      1.5     8     2    8.82 -0.819 
#>  5 19      2       8     2    8.42 -0.421 
#>  6 19      2.5     8     2    8.04 -0.0444
#>  7 19      3       7     2    7.69 -0.689 
#>  8 19      4       7     2    7.03 -0.0334
#>  9 19      5       7     2    6.45  0.554 
#> 10 19      6       6     2    5.92  0.0820
#> # ℹ 136 more rows

confint(): Confidence Intervals

confint(fit_hs)
#> # A tibble: 40 × 6
#>    id    term  estimate conf.low conf.high level
#>    <chr> <chr>    <dbl>    <dbl>     <dbl> <dbl>
#>  1 19    Q0       10.2      9.63     10.7   0.95
#>  2 30    Q0        2.81     2.36      3.25  0.95
#>  3 38    Q0        4.50     4.08      4.92  0.95
#>  4 60    Q0        9.92     9.02     10.8   0.95
#>  5 68    Q0       10.4      9.75     11.0   0.95
#>  6 106   Q0        5.68     5.10      6.27  0.95
#>  7 113   Q0        6.20     5.85      6.54  0.95
#>  8 142   Q0        6.17     4.92      7.43  0.95
#>  9 156   Q0        8.35     7.54      9.15  0.95
#> 10 188   Q0        6.30     5.20      7.41  0.95
#> # ℹ 30 more rows

summary(): Formatted Summary

The summary() method provides a formatted overview including parameter distributions across subjects:

summary(fit_hs)
#> 
#> Fixed-Effect Demand Model Summary
#> ================================================== 
#> 
#> Equation: hs 
#> k: fixed (2) 
#> 
#> Fit Summary:
#>   Total subjects: 10 
#>   Converged: 10 
#>   Failed: 0 
#>   Total observations: 146 
#> 
#> Parameter Summary (across subjects):
#>   Q0:
#>     Median: 6.2498 
#>     Range: [ 2.8074 , 10.3904 ]
#>   alpha:
#>     Median: 0.004251 
#>     Range: [ 0.001987 , 0.00785 ]
#> 
#> Per-subject coefficients:
#> -------------------------
#> # A tibble: 40 × 10
#>    id    term      estimate std.error statistic p.value component estimate_scale
#>    <chr> <chr>        <dbl>     <dbl>     <dbl>   <dbl> <chr>     <chr>         
#>  1 106   Q0         5.68     0.300           NA      NA fixed     natural       
#>  2 106   alpha      0.00628  0.000432        NA      NA fixed     natural       
#>  3 106   alpha_st…  0.0257   0.00176         NA      NA fixed     natural       
#>  4 106   k          2       NA               NA      NA fixed     natural       
#>  5 113   Q0         6.20     0.174           NA      NA fixed     natural       
#>  6 113   alpha      0.00199  0.000109        NA      NA fixed     natural       
#>  7 113   alpha_st…  0.00812  0.000447        NA      NA fixed     natural       
#>  8 113   k          2       NA               NA      NA fixed     natural       
#>  9 142   Q0         6.17     0.641           NA      NA fixed     natural       
#> 10 142   alpha      0.00237  0.000400        NA      NA fixed     natural       
#> # ℹ 30 more rows
#> # ℹ 2 more variables: term_display <chr>, estimate_internal <dbl>

Normalized Alpha ($\alpha^*$)

When $k$ varies across participants or studies (e.g., k = "ind" or k = "fit"), raw $\alpha$ values are not directly comparable. The alpha_star column in tidy() output provides a normalized version (Strategy B; Rzeszutek et al., 2025) that adjusts for the scaling constant:

\[\alpha^* = \frac{-\alpha}{\ln\!\left(1 - \frac{1}{k \cdot \ln(b)}\right)}\]

where $b$ is the logarithmic base (10 for HS/Koff equations). Standard errors are computed via the delta method. alpha_star requires $k \cdot \ln(b) > 1$; otherwise NA is returned.

tidy(fit_hs) |>
  filter(term == "alpha_star") |>
  select(id, term, estimate, std.error)
#> # A tibble: 10 × 4
#>    id    term       estimate std.error
#>    <chr> <chr>         <dbl>     <dbl>
#>  1 19    alpha_star  0.00836  0.000249
#>  2 30    alpha_star  0.0240   0.00276 
#>  3 38    alpha_star  0.0172   0.00146 
#>  4 60    alpha_star  0.0176   0.000592
#>  5 68    alpha_star  0.0113   0.000394
#>  6 106   alpha_star  0.0257   0.00176 
#>  7 113   alpha_star  0.00812  0.000447
#>  8 142   alpha_star  0.00969  0.00163 
#>  9 156   alpha_star  0.0193   0.000668
#> 10 188   alpha_star  0.0321   0.00184

Plotting

Basic Demand Curves

The plot() method displays fitted demand curves with observed data points. The x-axis defaults to a log10 scale:

plot(fit_hs)

Faceted by Subject

Use facet = TRUE to show each subject in a separate panel:

plot(fit_hs, facet = TRUE)

Axis Transformations

Control the x- and y-axis transformations with x_trans and y_trans:

plot(fit_hs, x_trans = "pseudo_log", y_trans = "pseudo_log")

Diagnostics

Model Checks

check_demand_model() summarizes convergence, residual properties, and potential issues:

check_demand_model(fit_hs)
#> 
#> Model Diagnostics
#> ================================================== 
#> Model class: beezdemand_fixed 
#> 
#> Convergence:
#>   Status: Converged
#> 
#> Residuals:
#>   Mean: 0.0284
#>   SD: 0.5306
#>   Range: [-1.458, 2.228]
#>   Outliers: 3 observations
#> 
#> --------------------------------------------------
#> Issues Detected (1):
#>   1. Detected 3 potential outliers across subjects

Residual Plots

plot_residuals() produces diagnostic plots. Use $fitted for a residuals-vs-fitted plot:

plot_residuals(fit_hs)$fitted

Predictions

Default Predictions

predict() with no arguments returns fitted values at the observed prices:

predict(fit_hs)
#> # A tibble: 160 × 3
#>        x id    .fitted
#>    <dbl> <chr>   <dbl>
#>  1     0 19      10.2 
#>  2     0 30       2.81
#>  3     0 38       4.50
#>  4     0 60       9.92
#>  5     0 68      10.4 
#>  6     0 106      5.68
#>  7     0 113      6.20
#>  8     0 142      6.17
#>  9     0 156      8.35
#> 10     0 188      6.30
#> # ℹ 150 more rows

Custom Price Grid

Supply newdata to predict at specific prices:

new_prices <- data.frame(x = c(0, 0.5, 1, 2, 5, 10, 20))
predict(fit_hs, newdata = new_prices)
#> # A tibble: 70 × 3
#>        x id    .fitted
#>    <dbl> <chr>   <dbl>
#>  1     0 19      10.2 
#>  2     0 30       2.81
#>  3     0 38       4.50
#>  4     0 60       9.92
#>  5     0 68      10.4 
#>  6     0 106      5.68
#>  7     0 113      6.20
#>  8     0 142      6.17
#>  9     0 156      8.35
#> 10     0 188      6.30
#> # ℹ 60 more rows

Aggregated Models

Instead of fitting each subject individually, you can fit a single curve to aggregated data.

Mean Aggregation

agg = "Mean" computes the mean consumption at each price across subjects, then fits a single curve to those means:

fit_mean <- fit_demand_fixed(apt, equation = "hs", k = 2, agg = "Mean")
fit_mean
#> 
#> Fixed-Effect Demand Model
#> ==========================
#> 
#> Call:
#> fit_demand_fixed(data = apt, equation = "hs", k = 2, agg = "Mean")
#> 
#> Equation: hs 
#> k: fixed (2) 
#> Aggregation: Mean 
#> Subjects: 1 ( 1 converged, 0 failed)
#> 
#> Use summary() for parameter summaries, tidy() for tidy output.

plot(fit_mean)

Pooled Aggregation

agg = "Pooled" fits a single curve to all data points pooled together, retaining error around each observation but ignoring within-subject clustering:

fit_pooled <- fit_demand_fixed(apt, equation = "hs", k = 2, agg = "Pooled")
fit_pooled
#> 
#> Fixed-Effect Demand Model
#> ==========================
#> 
#> Call:
#> fit_demand_fixed(data = apt, equation = "hs", k = 2, agg = "Pooled")
#> 
#> Equation: hs 
#> k: fixed (2) 
#> Aggregation: Pooled 
#> Subjects: 1 ( 1 converged, 0 failed)
#> 
#> Use summary() for parameter summaries, tidy() for tidy output.

plot(fit_pooled)

Conclusion

The fit_demand_fixed() interface provides a modern, consistent API for individual demand curve fitting with full support for the tidy() / glance() / augment() workflow. For datasets where borrowing strength across subjects is desirable, consider the mixed-effects or hurdle model approaches.

Fixed-Effect Demand Modeling with `beezdemand`

Brent Kaplan

2026-03-02

Introduction

Fitting with Different Equations

Hursh & Silberberg (“hs”)

Koffarnus (“koff”)

Simplified (“simplified”)

The k Parameter

Inspecting Fits

tidy(): Per-Subject Parameter Estimates

glance(): Model-Level Summary

augment(): Fitted Values and Residuals

confint(): Confidence Intervals

summary(): Formatted Summary

Normalized Alpha (\(\alpha^*\))

Plotting

Basic Demand Curves

Faceted by Subject

Axis Transformations

Diagnostics

Model Checks

Residual Plots

Predictions

Default Predictions

Custom Price Grid

Aggregated Models

Mean Aggregation

Pooled Aggregation

Conclusion

See Also

Fixed-Effect Demand Modeling with beezdemand

Brent Kaplan

2026-03-02

Introduction

Fitting with Different Equations

Hursh & Silberberg (“hs”)

Koffarnus (“koff”)

Simplified (“simplified”)

The k Parameter

Inspecting Fits

tidy(): Per-Subject Parameter Estimates

glance(): Model-Level Summary

augment(): Fitted Values and Residuals

confint(): Confidence Intervals

summary(): Formatted Summary

Normalized Alpha (\(\alpha^*\))

Plotting

Basic Demand Curves

Faceted by Subject

Axis Transformations

Diagnostics

Model Checks

Residual Plots

Predictions

Default Predictions

Custom Price Grid

Aggregated Models

Mean Aggregation

Pooled Aggregation

Conclusion

See Also

Fixed-Effect Demand Modeling with `beezdemand`