# Compute Demand Coefficients of an Agent with a Demand Structural Tree Given a price vector, this function computes the demand coefficients of an agent with a demand structural tree. The class of a demand structural tree is Node defined by the package data.tree. ```r demand_coefficient(node, p, trace = FALSE) ``` ## Arguments - `node`: a demand structural tree. - `p`: a price vector with names of commodities. - `trace`: FALSE (default) or TRUE. If TRUE, calculation intermediate results will be recorded in nodes. ## Returns A vector consisting of demand coefficients. ## Details Demand coefficients often indicate the quantity of various commodities needed by an economic agent in order to obtain a unit of output or utility, and these commodities can include both real commodities and financial instruments such as tax receipts, stocks, bonds and currency. The demand for various commodities by an economic agent can be expressed by a demand structure tree. Each non-leaf node can be regarded as the output of all its child nodes. Each node can be regarded as an input of its parent node. In other words, the commodity represented by each non-leaf node is a composite commodity composed of the commodities represented by its child nodes. Each non-leaf node usually has an attribute named type. This attribute describes the input-output relationship between the child nodes and the parent node. This relationship can sometimes be represented by a production function or a utility function. The type attribute of each non-leaf node can take the following values. * SCES. In this case, this node also has parameters alpha, beta and es (or sigma = 1 - 1 / es). alpha and es are scalars. beta is a vector. These parameters are parameters of a standard CES function (see `SCES` and `SCES_A`). * CES. In this case, this node also has parameters alpha, beta, theta (optional) and es (or sigma = 1 - 1 / es) (see CGE::CES_A). * Leontief. In this case, this node also has the parameter a, which is a vector and is the parameter of a Leontief function. * CD. CD is Cobb-Douglas. In this case, this node also has parameters alpha and beta, which are parameters of a Cobb-Douglas function. * CESAK. In this case, this node also has parameters es, alpha, betaK and alphaK, which are parameters of the CESAK function (see `CESAK_dc`). Moreover, the first child node should represent capital goods. * FIN. That is the financial type. In this case, this node also has the parameter rate or beta. If the parameter beta is not NULL, then the parameter rate will be ignored. The parameter rate applies to all situations, while the parameter beta only applies for some special cases. For FIN nodes, the first child node should represent for a physical commodity or a composite commodity containing a physical commodity, and other child nodes represent for financial instruments. The parameter beta indicates the proportion of each child node's expenditure. The parameter rate indicates the expenditure ratios between financial-instrument-type child nodes and the first child node. The first element of the parameter rate indicates the amount of the first child node needed to get a unit of output. * FUNC. That is the function type. In this case, this node also has an attribute named func. The value of that attribute is a function which calculates the demand coefficient for the child nodes. The argument of that function is a price vector. The length of that price vector is equal to the number of the child nodes. * StickyLinear or SL. That is the sticky linear type. In this case, this node also has an attribute named beta that contains the coefficients of the linear utility or production function. In order to avoid too drastic changes in the demand structure, the adjustment process of the demand structure has a certain stickiness when prices change. ## Examples ```r #### a Leontief-type node dst <- node_new("firm", type = "Leontief", a = c(0.5, 0.1), "wheat", "iron" ) print(dst, "type") node_print(dst) plot(dst) node_plot(dst, TRUE) demand_coefficient(dst, p = c(wheat = 1, iron = 2)) # the same as a = c(0.5, 0.1) #### a CD-type node dst <- node_new("firm", type = "CD", alpha = 1, beta = c(0.5, 0.5), "wheat", "iron" ) demand_coefficient(dst, p = c(wheat = 1, iron = 2)) # the same as the following CD_A(1, c(0.5, 0.5), c(1, 2)) #### a SCES-type node dst <- node_new("firm", type = "SCES", alpha = 2, beta = c(0.8, 0.2), es = 0.5, "wheat", "iron" ) demand_coefficient(dst, p = c(wheat = 1, iron = 2)) # the same as the following SCES_A(alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), es = 0.5) CES_A(sigma = 1 - 1 / 0.5, alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), Theta = c(0.8, 0.2)) #### a FUNC-type node dst <- node_new("firm", type = "FUNC", func = function(p) { CES_A( sigma = -1, alpha = 2, Beta = c(0.8, 0.2), p, Theta = c(0.8, 0.2) ) }, "wheat", "iron" ) demand_coefficient(dst, p = c(wheat = 1, iron = 2)) # the same as the following CES_A(sigma = -1, alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), Theta = c(0.8, 0.2)) #### p <- c(wheat = 1, iron = 3, labor = 2, capital = 4) dst <- node_new("firm 1", type = "SCES", sigma = -1, alpha = 1, beta = c(1, 1), "cc1", "cc2" ) node_set(dst, "cc1", type = "Leontief", a = c(0.6, 0.4), "wheat", "iron" ) node_set(dst, "cc2", type = "SCES", sigma = -1, alpha = 1, beta = c(1, 1), "labor", "capital" ) node_plot(dst) demand_coefficient(dst, p) #### p <- c(product = 1, labor = 1, money = 1) dst <- node_new("firm", type = "FIN", rate = c(0.75, 1 / 3), "cc1", "money" ) # a financial-type node node_set(dst, "cc1", type = "Leontief", a = c(0.8, 0.2), "product", "labor" ) node_plot(dst) demand_coefficient(dst, p) #### the same as above p <- c(product = 1, labor = 1, money = 1) dst <- node_new("firm", type = "Leontief", a = c(0.8, 0.2), "cc1", "cc2" ) node_set(dst, "cc1", type = "FIN", rate = c(0.75, 1 / 3), "product", "money" ) node_set(dst, "cc2", type = "FIN", rate = c(0.75, 1 / 3), "labor", "money" ) node_plot(dst) demand_coefficient(dst, p) #### the same as above p <- c(product = 1, labor = 1, money = 1) dst <- node_new("firm", type = "FIN", rate = c(1, 1 / 3), "cc1", "money" ) # Financial-type Demand Structure node_set(dst, "cc1", type = "Leontief", a = c(0.6, 0.15), "product", "labor" ) node_plot(dst) demand_coefficient(dst, p) ```

demand_coefficient function