# Structural Dynamic Model (alias Structural Growth Model) This function computes the general equilibrium and simulates the economic dynamics. The key part of this function is an exchange function (see `F_Z`), which is expounded in Li (2010, 2019). ```r sdm( A, B = diag(nrow(A)), n = nrow(B), m = ncol(B), S0Exg = matrix(NA, n, m), p0 = matrix(1, nrow = n, ncol = 1), z0 = matrix(100, nrow = m, ncol = 1), GRExg = NA, moneyOwnerIndex = NULL, moneyIndex = NULL, pExg = NULL, tolCond = 1e-5, maxIteration = 200, numberOfPeriods = 300, depreciationCoef = 0.8, thresholdForPriceAdjustment = 0.99, priceAdjustmentMethod = "variable", priceAdjustmentVelocity = 0.15, trace = TRUE, ts = FALSE, policy = NULL, exchangeFunction = F_Z ) ``` ## Arguments - `A`: a demand coefficient n-by-m matrix (alias demand structure matrix) or a function A(state) which returns an n-by-m matrix. - `B`: a supply coefficient n-by-m matrix (alias supply structure matrix) or a function which returns an n-by-m matrix. If (i,j)-th element of S0Exg is not NA, the value of the (i,j)-th element of B will be useless and ignored. - `n`: the number of commodities. - `m`: the number of economic agents (or sectors). - `S0Exg`: an initial exogenous supply n-by-m matrix. This matrix may contain NA, but not zero. - `p0`: an initial price n-vector. - `z0`: an m-vector consisting of the initial exchange levels (i.e. activity levels, production levels or utility levels). - `GRExg`: an exogenous growth rate of the exogenous supplies in S0Exg. If GRExg is NA and some commodities have exogenous supply, then GRExg will be set to 0. - `moneyOwnerIndex`: a vector consisting of the indices of agents supplying money. - `moneyIndex`: a vector consisting of the commodity indices of all types of money. - `pExg`: an n-vector indicating the exogenous prices (if any). - `tolCond`: the tolerance condition. - `maxIteration`: the maximum iteration count. If the main purpose of running this function is to do simulation instead of calculating equilibrium, then maxIteration should be set to 1. - `numberOfPeriods`: the period number in each iteration. - `depreciationCoef`: the depreciation coefficient (i.e. 1 minus the depreciation rate) of the unsold products. - `thresholdForPriceAdjustment`: the threshold for the fixed percentage price adjustment method. - `priceAdjustmentMethod`: the price adjustment method. Normally it should be set to "variable". If it is set to "fixed", a fixed percentage price adjustment method will be used. - `priceAdjustmentVelocity`: the price adjustment velocity. - `trace`: if TRUE, information is printed during the running of sdm. - `ts`: if TRUE, the time series of the last iteration are returned. - `policy`: a policy function. - `exchangeFunction`: the exchange function. ## Returns sdm returns a list containing the following components: - **tolerance**: the tolerance of the results. - **p**: equilibrium prices. - **z**: equilibrium exchange levels (i.e. activity levels, output levels or utility levels). - **S**: the equilibrium supply matrix at the initial period. - **e**: equilibrium foreign exchange rates in a multi-money economy. - **growthRate**: the endogenous equilibrium growth rate in a pure production economy. - **A**: the equilibrium demand coefficient matrix. - **B**: If B is a function, the equilibrium supply coefficient matrix is returned. - **ts.p**: the time series of prices in the last iteration. - **ts.z**: the time series of exchange levels (i.e. activity levels, production levels or utility levels) in the last iteration. - **ts.S**: the time series of supply matrix in the last iteration. - **ts.q**: the time series of sales rates in the last iteration. - **ts.e**: the time series of foreign exchange rates in the last iteration. - **policy.data**: the policy data. ## Details The parameters A may be a function A(state) wherein state is a list consisting of p (the price vector), z (the output and utility vector), w (the wealth vector), t (the time) and e (the foreign exchange rate vector if any). state indicates the states at time t. The parameters B also may be a function B(state) wherein state is a list consisting of p (the price vector), z (the output and utility vector) and t (the time). ## References LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese) LI Wu (2010) A Structural Growth Model and its Applications to Sraffa's System. http://www.iioa.org/conferences/18th/papers/files/104_20100729011_AStructuralGrowthModelanditsApplicationstoSraffasSstem.pdf Torres, Jose L. (2016, ISBN: 9781622730452) Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press. Varian, Hal R. (1992, ISBN: 0393957357) Microeconomic Analysis. W. W. Norton & Company. ## Author(s) LI Wu ## See Also `iep`; `Example2.2`; `Example2.3`; `Example.Section.3.1.2.corn`; `Example3.1`; `Example3.2`; `Example3.4`; `Example3.8`; `Example3.9`; `Example3.10`; `Example3.12`; `Example3.14`; `Example4.2`; `Example4.8`; `Example4.9`; `Example4.10`; `Example4.11.1`; `Example4.11.2`; `Example4.12`; `Example4.13`; `Example4.15`; `Example4.16`; `Example5.1`; `Example5.2`; `Example5.3.2`; `Example5.4`; `Example5.5`; `Example5.6`; `Example5.10`; `Example5.11.1`; `Example5.11.2`; `Example6.2.1`; `Example6.2.2`; `Example6.3`; `Example6.4`; `Example6.5`; `Example6.6.1`; `Example6.6.2`; `Example6.6.3`; `Example6.7`; `Example6.9`; `Example6.10`; `Example6.11`; `Example7.1`; `Example7.2`; `Example7.3`; `Example7.4`; `Example7.5.1`; `Example7.5.2`; `Example7.6`; `Example7.7`; `Example7.8`; `Example7.9X`; `Example7.10`; `Example7.10.2`; `Example7.11`; `Example7.12`; `Example7.13`; `Example7.14`; `Example7.15`; `Example8.1`; `Example8.2`; `Example8.7`; `Example8.8`; `Example8.9`; `Example9.3`; `Example9.4`; `Example9.5`; `Example9.6`; `Example9.7`; `Example9.10`; ## Examples ```r ## the example on page 352 in Varian (1992) ge <- sdm( A = function(state) { a <- 0.5 alpha <- rep(1, 3) Beta <- matrix(c(0, a, a, 0.5, 0, 0, 0.5, 1 - a, 1 - a), 3, 3, TRUE) #the demand coefficient matrix. CD_A(alpha, Beta, state$p) }, B = diag(3), S0Exg = matrix(c(NA, NA, NA, NA, 1, NA, NA, NA, 1), 3, 3, TRUE), GRExg = 0, tolCond = 1e-10 ) ge$p/ge$p[1] ## the example (see Table 2.1 and 2.2) of the canonical dynamic ## macroeconomic general equilibrium model in Torres (2016). discount.factor <- 0.97 return.rate <- 1 / discount.factor - 1 depreciation.rate <- 0.06 ge <- sdm( n = 4, m = 3, A = function(state) { a1 <- CD_A(1, rbind(0, 0.65, 0.35, 0), state$p) a2 <- CD_A(1, rbind(0.4, 1 - 0.4, 0, 0), state$p) a3 <- c(1, 0, 0, state$p[1] * return.rate / state$p[4]) cbind(a1, a2, a3) }, B = matrix(c( 1, 0, 1 - depreciation.rate, 0, 1, 0, 0, 0, 1, 0, 1, 0 ), 4, 3, TRUE), S0Exg = { tmp <- matrix(NA, 4, 3) tmp[2, 2] <- 1 tmp[4, 2] <- 1 tmp }, priceAdjustmentVelocity = 0.03, maxIteration = 1, numberOfPeriods = 5000, ts = TRUE ) ge$A %*% diag(ge$z) # the demand matrix ge$p / ge$p[1] plot(ge$ts.z[, 1], type = "l") ```

sdm function