creativity::belief::Demand Class Reference

This class represents an author's belief about the per-period demand for books. More...

#include <creativity/belief/Demand.hpp>

Public Member Functions
	Demand ()
	Default constructor: note that unlike a default-constructed BayesianLinear model, this default constructed object is a usable (but noninformative) model. More...
template<typename ... Args>
	Demand (Args &&...args)
	Constructs a demand model with the given prior information. More...
virtual unsigned int	fixedModelSize () const override
	This model always has exactly parameters() parameters.
double	predict (unsigned int draws, double P, double q, unsigned int S, unsigned int nosales, unsigned int age, unsigned int otherBooks, unsigned int lag_marketBooks)
	Given a set of model parameters, this returns an expected value \(Q_b\), the number of sales. More...
std::pair< double, double >	argmaxP (unsigned int draws, double q, unsigned int s, unsigned int z, unsigned int n, unsigned int nosales, unsigned int age, unsigned int otherBooks, unsigned int marketBooks, double c, double max_price)
	Given a set of model parameters (other than \(P_b\)) and a per-unit cost this returns the \(P_b\) value that maximizes expected total profits: More...
virtual std::string	display_name () const override
	Returns "Demand", the name of this model.

Static Public Member Functions
static unsigned int	parameters ()
	Returns the number of parameters of this model (5)
static Eigen::RowVectorXd	row (double P, double q, unsigned S, unsigned nosales, unsigned age, unsigned otherBooks, unsigned lag_marketBooks)
	Given various information about a book, returns an X matrix row of data representing that information. More...
static Eigen::RowVectorXd	bookRow (eris::SharedMember< Book > book, double quality, unsigned int lag_market_books)
	Given a book and perceived quality, this builds an X matrix row of data representing that book. More...

Detailed Description

This class represents an author's belief about the per-period demand for books.

The model is of the form:

\(Q_b = \beta_0 + \beta_1 P_b + \beta_2 q_b + \beta_3 prevSales + \beta_4 nosales + u\) where:

• \(Q_b\) is the quantity (i.e. copies) demanded
• \(P_b\) is the price of a copy (which must be non-negative)
• \(q_b\) is the quality of the work, determined when the work is created, which must be non-negative. Readers observe either a draw based on this, or a predicted quality.
• \(prevSales\) is the number of copies sold in previous periods
• \(nosales\) is the number of periods the book has gone without sales
• \(marketBooks\) is the number of books on the market in the previous period

The following restrictions are imposed on beliefs:

• \(\beta_1 \leq -0.05\) (demand curve is downward sloping (at least for sufficiently small p))
• \(\beta_2 \geq 0\) (higher quality means more sales (at least for low quality))
• \(\beta_4 \leq -1\) (each no sales period decreases quantity demanded by at least 1)

These constraints are combined with a natural conjugate prior for the purposes of updating the beliefs via Bayesian econometrics.

Additionally, when predicting, readers know that Q can never be larger than the number of readers in the world minus the copies already sold, and incorporate this limitation into predictions.

Constructor & Destructor Documentation

Demand() [1/2]

creativity::belief::Demand::Demand ( )

inline

Default constructor: note that unlike a default-constructed BayesianLinear model, this default constructed object is a usable (but noninformative) model.

See also

eris::learning::BayesianLinear::BayesianLinear()

Demand() [2/2]

template<typename ... Args>

creativity::belief::Demand::Demand ( Args &&...  args )

inlineexplicit

Constructs a demand model with the given prior information.

This forwards arguments to the BayesianLinearRestricted constructor, but also adds the model's restrictions.

Parameters

args	prior arguments to forward to the base BayesianLinearRestricted constructor.

See also

eris::learning::BayesianLinear::BayesianLinear

Member Function Documentation

argmaxP()

std::pair<double, double> creativity::belief::Demand::argmaxP	(	unsigned int	draws,
		double	q,
		unsigned int	s,
		unsigned int	z,
		unsigned int	n,
		unsigned int	nosales,
		unsigned int	age,
		unsigned int	otherBooks,
		unsigned int	marketBooks,
		double	c,
		double	max_price
	)

Given a set of model parameters (other than \(P_b\)) and a per-unit cost this returns the \(P_b\) value that maximizes expected total profits:

\[ P_b Q_b(P_b, \ldots) - c_b Q_b(P_b, \ldots) \]

which, due to the linearity of P_b in Q_b, has a maximum at:

\[ P_b^* = \frac{X\beta_{-1}}{-2\beta_1} + \frac{c_b}{2} \]

where \(X\) is all of the model terms that do not depend on \(P_b\) and \(\beta_{-1}\) are the associated coefficients.

Note, however, that there is a limit to Q: it can never be larger than the number of readers in the world who haven't read the book, in other words, the population minus the copies previously sold minus the copies pirated minus 1 (an author doesn't buy his own book).

Parameters

draws	the number of draws to use for prediction
q	the quality of the book
s	prior book sales
z	book pirated copies
n	simulation readers size (including the author)
nosales	the number of periods the book has gone without sales (age is used if this is > age).
age	the age of the book, in simulation periods
otherBooks	the number of other books created by this book's author. This parameter also determines the onlyBook dummy (= 1 iff otherBooks == 0).
marketBooks	the number of books on the market in the last pre-prediction period (including this book, if this book was on the market)
c	the per-unit cost of copies.
max_price	the maximum price to set (if optimum is predicted above this, reduce to this)

Returns

a std::pair of double values where .first is the maximizing price and .second is the quantity predicted at that price. If optimal price is below c (i.e. demand is 0 or negative at a price of c), 0 is returned for both fields.

Exceptions

std::domain_error

if c < 0 or q < 0

bookRow()

static Eigen::RowVectorXd creativity::belief::Demand::bookRow ( eris::SharedMember< Book >  book, double  quality, unsigned int  lag_market_books  )

static

Given a book and perceived quality, this builds an X matrix row of data representing that book.

This method may only be called in the inter-period optimization stage, after t has been incremented but before new books/markets have been created. The book must still be on the market: its current market price is used to build the row.

Note that this method typically should only be used beginning in t=3: before that, lag_market_books would be 0.

predict()

double creativity::belief::Demand::predict	(	unsigned int	draws,
		double	P,
		double	q,
		unsigned int	S,
		unsigned int	nosales,
		unsigned int	age,
		unsigned int	otherBooks,
		unsigned int	lag_marketBooks
	)

Given a set of model parameters, this returns an expected value \(Q_b\), the number of sales.

Parameters

draws	the number of draws to use for prediction
P	the price of a copy of the book
q	the quality of the book
S	prior book sales
nosales	the number of time periods since the book last had a sale; if greater than age, age is used (so that sim->t() - book->lastSale() can be passed)
age	of the book
otherBooks	the number of other books created by this book's author. This parameter also determines the onlyBook dummy (= 1 iff otherBooks == 0).
lag_marketBooks	the number of books on the market in the previous period

Exceptions

std::domain_error

if P < 0 or q < 0.

row()

static Eigen::RowVectorXd creativity::belief::Demand::row ( double  P, double  q, unsigned  S, unsigned  nosales, unsigned  age, unsigned  otherBooks, unsigned  lag_marketBooks  )

static

Given various information about a book, returns an X matrix row of data representing that information.

Parameters

P	the price of a copy of the book
q	the quality of the book
S	prior book sales
nosales	the number of time periods since the book last had a sale; if greater than age, age is used (so that sim->t() - book->lastSale() can be passed)
age	of the book
otherBooks	the number of other books created by this book's author. This parameter also determines the onlyBook dummy (= 1 iff otherBooks == 0).
lag_marketBooks	the number of books on the market in the previous period

Inheritance diagram for creativity::belief::Demand:

Collaboration diagram for creativity::belief::Demand:

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The documentation for this class was generated from the following file:

• creativity/belief/Demand.hpp