Complete Cross-Price Analysis Workflow
This section demonstrates a complete analysis workflow using data
that contains multiple experimental conditions: target consumption when
the alternative is absent (alone), target consumption when
the alternative is present (own), and alternative
consumption as a function of target price (alt).
Loading the cp Dataset
# Load the cross-price example dataset
data("cp", package = "beezdemand")
# Examine structure
glimpse(cp)
#> Rows: 48
#> Columns: 5
#> $ id <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
#> $ x <dbl> 0.00, 0.01, 0.03, 0.06, 0.13, 0.25, 0.50, 1.00, 2.00, 4.00, 8.0…
#> $ y <dbl> 24.3430657, 22.8832117, 22.6058394, 21.0291971, 19.7810219, 15.…
#> $ target <chr> "alone", "alone", "alone", "alone", "alone", "alone", "alone", …
#> $ group <chr> "cigarettes", "cigarettes", "cigarettes", "cigarettes", "cigare…
# View conditions
table(cp$target)
#>
#> alone alt own
#> 16 16 16The cp dataset contains:
- alone: Target commodity (cigarettes) consumption when alternative is not available
- own: Target commodity consumption when alternative is available
- alt: Alternative commodity (e-cigarettes) consumption as a function of target price
Note that x represents the target commodity
price throughout all conditions.
Step 1: Fit Target Demand (Alone Condition)
First, we fit a standard demand curve to the target commodity when the alternative is absent:
# Filter to alone condition
alone_data <- cp |>
dplyr::filter(target == "alone")
# Fit demand curve (modern interface)
fit_alone <- fit_demand_fixed(
data = alone_data,
equation = "koff",
k = 2
)
# View results
fit_alone
#>
#> Fixed-Effect Demand Model
#> ==========================
#>
#> Call:
#> fit_demand_fixed(data = alone_data, equation = "koff", k = 2)
#>
#> Equation: koff
#> k: fixed (2)
#> Subjects: 1 ( 1 converged, 0 failed)
#>
#> Use summary() for parameter summaries, tidy() for tidy output.Step 2: Fit Target Demand (Own Condition)
Next, fit the same demand model to target consumption when the alternative is present:
# Filter to own condition
own_data <- cp |>
dplyr::filter(target == "own")
# Fit demand curve
fit_own <- fit_demand_fixed(
data = own_data,
equation = "koff",
k = 2
)
# View results
fit_own
#>
#> Fixed-Effect Demand Model
#> ==========================
#>
#> Call:
#> fit_demand_fixed(data = own_data, equation = "koff", k = 2)
#>
#> Equation: koff
#> k: fixed (2)
#> Subjects: 1 ( 1 converged, 0 failed)
#>
#> Use summary() for parameter summaries, tidy() for tidy output.Step 3: Fit Cross-Price Model (Alt Condition)
Finally, fit the cross-price model to alternative consumption as a function of target price:
# Filter to alt condition
alt_data <- cp |>
dplyr::filter(target == "alt")
# Fit cross-price model
fit_alt <- fit_cp_nls(
data = alt_data,
equation = "exponentiated",
return_all = TRUE
)
# View results
summary(fit_alt)
#> Cross-Price Demand Model Summary
#> ================================
#>
#> Model Specification:
#> Equation type: exponentiated
#> Functional form: y ~ (10^log10_qalone) * 10^(I * exp(-(10^log10_beta) * x))
#> Fitting method: nls_multstart
#> Method details: Multiple starting values optimization with nls.multstart
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> log10_qalone 1.1720228 0.0025378 461.8250 < 2.2e-16 ***
#> I -0.3712609 0.0066765 -55.6075 < 2.2e-16 ***
#> log10_beta -0.1271127 0.0217331 -5.8488 5.696e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Confidence Intervals:
#> 2.5 % 97.5 %
#> log10_qalone 1.1665 1.17751
#> I -0.3857 -0.35684
#> log10_beta -0.1741 -0.08016
#>
#> Fit Statistics:
#> R-squared: 0.9973
#> AIC: 1.3
#> BIC: 4.39
#>
#> Parameter Interpretation (natural scale):
#> qalone (Q_alone): 14.86 - consumption at zero alternative price
#> I: -0.3713 - interaction parameter (substitution direction)
#> beta: 0.7463 - sensitivity parameter (sensitivity of relation to price)
#>
#> Optimizer parameters (log10 scale):
#> log10_qalone: 1.172
#> log10_beta: -0.1271fit_cp_nls() uses a log10-parameterized optimizer
internally (for numerical stability), but predict() returns
y_pred on the natural y scale. For the
"exponential" form, predictions may also include
y_pred_log10.
Comparing Results Across Conditions
# Extract key parameters for each condition
coef_alone <- coef(fit_alone)
Q0_alone <- coef_alone$estimate[coef_alone$term == "q0"]
Alpha_alone <- coef_alone$estimate[coef_alone$term == "alpha"]
coef_own <- coef(fit_own)
Q0_own <- coef_own$estimate[coef_own$term == "q0"]
Alpha_own <- coef_own$estimate[coef_own$term == "alpha"]
comparison <- data.frame(
Condition = c("Alone (Target)", "Own (Target)", "Alt (Cross-Price)"),
Q0_or_Qalone = c(
Q0_alone,
Q0_own,
coef(fit_alt)["qalone"]
),
Alpha_or_I = c(
Alpha_alone,
Alpha_own,
coef(fit_alt)["I"]
)
)
comparison
#> Condition Q0_or_Qalone Alpha_or_I
#> 1 Alone (Target) 23.54818 0.01406589
#> 2 Own (Target) 21.19336 0.01562876
#> 3 Alt (Cross-Price) NA -0.37126092Interpretation:
- Comparing
Q0between alone and own conditions shows how the presence of an alternative affects baseline consumption of the target commodity - The
Iparameter from the cross-price model indicates whether the products are substitutes (I < 0) or complements (I > 0)
Combined Visualization
# Create prediction data
x_seq <- seq(0.01, max(cp$x), length.out = 100)
# Get demand predictions for each condition
pred_alone <- predict(fit_alone, newdata = data.frame(x = x_seq))$.fitted
pred_own <- predict(fit_own, newdata = data.frame(x = x_seq))$.fitted
# Cross-price model predictions (always on the natural y scale)
pred_alt <- predict(fit_alt, newdata = data.frame(x = x_seq))$y_pred
# Combine into plot data
plot_data <- data.frame(
x = rep(x_seq, 3),
y = c(pred_alone, pred_own, pred_alt),
Condition = rep(c("Target (Alone)", "Target (Own)", "Alternative (Cross-Price)"), each = length(x_seq))
)
# Plot
ggplot() +
geom_point(data = cp, aes(x = x, y = y, color = target), alpha = 0.6) +
geom_line(data = plot_data, aes(x = x, y = y, color = Condition), linewidth = 1) +
scale_x_log10() +
scale_y_log10() +
labs(
x = "Target Price",
y = "Consumption",
title = "Cross-Price Analysis: All Conditions",
color = "Condition"
) +
theme_bw()
