Calculates the maximum expenditure (Omax) and the price at maximum expenditure (Pmax) for the exponential demand model used in the two-part hurdle model.
Arguments
- Q0
Intensity parameter (consumption at price 0).
- k
Scaling parameter for the exponential decay.
- alpha
Elasticity parameter (rate of decay).
- price_range
Numeric vector of length 2 specifying the price range to search for Pmax. Default is
NULL, which uses an adaptive range based on alpha (approximately 0 to 10/alpha).
Value
A named list with:
- Pmax
Price at maximum expenditure
- Omax
Maximum expenditure (price * quantity)
- Qmax
Quantity at Pmax
Details
For the demand function: $$Q(p) = Q_0 \cdot \exp(k \cdot (\exp(-\alpha \cdot p) - 1))$$
Expenditure is E(p) = p * Q(p). Omax is the maximum of E(p) and Pmax is the price at which this maximum occurs. These are found numerically.
The search range is automatically adjusted based on alpha to ensure the maximum is found. For small alpha values, Pmax can be quite large.
Examples
# Calculate for group-level parameters
calc_omax_pmax(Q0 = 10, k = 2, alpha = 0.5)
#> Warning: `calc_omax_pmax()` was deprecated in beezdemand 0.2.0.
#> ℹ Please use `beezdemand_calc_pmax_omax()` instead.
#> Warning: Note: k (2.000) < e (~2.718); the expenditure function has no interior maximum. Returning the maximum over a bounded search interval via numerical optimization.
#> $Pmax
#> [1] 19.99993
#>
#> $Omax
#> [1] 27.06941
#>
#> $Qmax
#> [1] 1.353476
#>
#> $note
#> [1] "k < e: bounded-range maximum over [0.001, 20.000] via numerical optimization"
#>
# With k >= e (~2.718), a local maximum exists
calc_omax_pmax(Q0 = 10, k = 3, alpha = 0.5)
#> $Pmax
#> [1] 1.238123
#>
#> $Omax
#> [1] 3.100397
#>
#> $Qmax
#> [1] 2.504112
#>