A revised version (April 2004) of this document is located at http://web.mit.edu/paltsev/www/current.html
PDF version of this document is located at http://web.mit.edu/paltsev/www/docs/move.pdf
GAMS-MPSGE codes of the static and dynamic models are located here
Department of Economics, University of Colorado
2
Draft:3
``This is not a paper. These are notes for your own understanding'' - Thomas Rutherford, the author of MPSGE, put these words on the first draft of this document. ``I already have a paper on a Ramsey model. Read it.'' - he added4. At that time I was (and still am) a poor graduate student in desperate need for a research assistantship money. So I haven't told him that I have read his paper several times. Even after that, my understanding of what MPSGE is and how to build a dynamic model was not much better than my understanding of how to build a bridge in Nepal. Therefore, I put these notes together as a reference for myself. Then I thought that if these notes were helpful for me, they might be useful for other beginners. Tom's resolution sent the paper to the bottom drawer of my desk for a long time. It was sitting there until the moment when some people suddenly decided that I know something and asked me the question: ``How do you build a dynamic model in MPSGE?''. Obviously, my advice for them was to read the Rutherford's paper. For some reason, which is still unknown to me, almost all of them returned and asked for something more clear and detailed. My subsequent explanations in the mix of English and Russian left them even more confused. At that moment, I decided that the idea that these are ``notes for understanding'' may not be that bad after all. In short, if you are smart as Tom Rutherford, these notes are not for you. However, if you have overrated yourself a little bit and still have some questions about modelling dynamics in MPSGE, you may try to find the answers here. Tom does not like to write papers at the beginner's level. I hope I am stupid enough to try it. Ready? Then let's go through the basics together.
In the next section, I will try to share my understanding of what is MPSGE and what kind of problems can be solved with its help. Section 3 describes how a simple static model can be represented in the MPSGE format. If you are already familiar with MPSGE, you may skip that part and turn directly to section 4, which shows the transition of a static model into a dynamic model5. This is done for the case where an economy is initially on a steady-state growth path. Section 5 presents adjustments in the model, which are necessary for the off steady-state case. Program listings are provided in the Appendices. I have also included Appendices on MPSGE-MCP conversion and representation of different functional forms in MPSGE format. They are not directly relevant to dynamic models. However, they are important for better understanding the process of economic modelling with MPSGE.
Finally, I am always amazed by those IMF and The World Bank-like disclaimers. Now I also have a chance to provide one. Here it is: ``The views expressed in this document are those of the author and do not necessarily reflect the views of the MPSGE creator.'' In simple English it means that these notes are just notes for my own understanding. If you have any questions about MPSGE, then ``ask Tom''.
MPSGE (Mathematical Programming System for General Equilibrium Analysis) is a programming language designed by Thomas Rutherford [1999] in late 80s for solving Arrow-Debreu economic equilibrium models. The MPSGE is very powerful but badly documented language. Due to that fact, the usual recipe for mastering in MPSGE is ``learning by doing''.
MPSGE uses the programming language GAMS (General Algebraic Modeling System) as an interface6. As a result, if one wants to use MPSGE, he needs to learn basic GAMS syntax7 (Brook, Kendrick, Meeraus [1992]) as well. As it follows from the name, MPSGE solves computable general equilibrium (CGE) economic models. These models consist of economic agents who interact among each other. The word ``general'' means that all economic flows are accounted for, i.e there is a ``sink'' for every ``source''. Finding an equilibrium for a CGE model involves finding equilibrium prices, quantities, and incomes.
As an illustration of a simple economic model, consider the task of finding an equilibrium in the economy (for the future reference I will call it Wonderland), which consists of two economic agents: consumers and producers. Consumers have an initial endowment of labor L and capital K. For simplicity, there is a single representative consumer8 CONS in Wonderland. The consumer derives his income I from the sales of his endowments. Then he purchases his preferred choice of goods. There are two goods, X and Y, in the economy. The consumer obtains utility from consumption of the goods. Producers are the firms that take initial endowments of the consumer as inputs of production and convert them into outputs. Both production sectors, X and Y, are characterized by the available technology F and G, respectively.
We want to determine the prices and quantities which maximize producers' profits and consumer's utility9. The solution to this problem (which is a simple Arrow-Debreu [1954] problem) can be found using a non-linear formulation. It can be represented as an optimization problem of the consumer subject to income, technology, and feasibility constraints.
| maxW(X,Y) | s.t | I = px X+py Y |
| X = F(Kx, Lx) | ||
| Y = G(Ky, Ly) | ||
| L = Lx + Ly | ||
| K = Kx + Ky |
There are some cases (such as a presence of several consumers, taxes, or other distortions) where it is not possible to solve the problem of finding a market equilibrium as an optimization problem. Then the problem could be approached in a different way. It can be turned into a Mixed Complimentarity Problem (MCP) and solved as a system of non-linear equations. NLP problems are a subset of MCP and MPSGE finds an equilibrium as a solution to MCP.
Here is the first test to check if you should be reading these notes or go directly to Rutherford's [1997] paper. I had never heard about MCP formulation before. When I asked about it, a knowledgeable person told me that it is very easy, something like:
Given: f: Rn ® Rn
Find: z Î Rn
s.t. f(z) ³ 0, z ³ 0, zT f(z) = 0.
If everything is clear for you in the statement above, you are wasting your time with these notes. Close them and go to do real work!
If you are still here, then let's try to convert the mathematical symbols into simple English. I should tell that it won't help very much to a beginner because it sounds like: given the function f between two n-dimensional sets of real numbers (function f assigns to each member of the first set exactly one member of the second set), find z which belongs to an n-dimensional set of real numbers, such that function f (z) is greater or equal to zero, z is greater or equal to zero, and associated complimentary slackness condition10 is satisfied. The complimentary slackness condition requires that either z equals zero (i.e. the dual multiplier vanishes), or the inequality constraint is satisfied with strict equality, or both.
However, these words is just a usual scientific way of hiding something simple. Do you recall from your 6th grade math the solution to an equation like: x(5-x) = 0? Yes, it is x = 0 and x = 5. To make it more clear for a comparison, the equation above can be rewritten as xf(x) = 0, where f(x) = 5-x. It leads to the condition that either x or f(x) has to be equal zero. This is the main idea behind MCP.
In the case when we have not just one x, but a vector of [`x] = (x1,x2,...,xn), then there is a system of n equations like [`x] f([`x]) = 0, which forms an MCP problem. The word ``mixed'' in MCP reflects the fact that the solution is a mix of equalities f(x) = 0 and inequalities f(x) > 0.
Mathiesen [1985] has shown that the Arrow-Debreu economic equilibrium model can be formulated as MCP, where three inequalities should be satisfied: zero profit condition, market clearance condition, and income balance condition. A set of three non-negative variables is involved in solving MCP problem: prices, quantities (they are called as activity levels in MPSGE), and income levels.
Zero profit condition requires that any activity operated at a positive intensity must earn zero profit (i.e. value of inputs must be equal or greater than value of outputs). Activity levels y for constant returns to scale production sectors are the associated variables with this condition. It means that either y > 0 (a positive amount of y is produced) and profit is zero, or profit is negative and y = 0 (no production activity takes place). In terms of MCP, the following condition should be satisfied for every sector in an economy:
|
Market clearance condition requires that any good with positive price must have a balance between supply and demand and any good in excess supply must have a zero price. Price vector p (which includes prices of final goods, intermediate goods and factors of production) is associated variable. In terms of MCP, the following condition should be satisfied for every good and every factor of production:
|
Income balance condition requires that for each agent (including any government entities) value of income must equal the value of factor endowments. Is it possible to formulate it in terms of MCP:
|
To illustrate the equilibrium conditions, consider the Wonderland example. For the reasons which will be clear later, additional production sector W is introduced to represent utility derived from the consumption of X and Y. Accordingly, Pw is a price of the output of the production sector W. First, consider zero profit conditions. In the Wonderland example, an activity level y is a vector with the following components y = (X, Y, W). Profit in a particular sector is defined as a difference between total revenue and total cost. We assume that activity levels X, Y, and W are positive (i.e., the condition y ³ 0 is satisfied with the strict inequality as y > 0). Hence, zero profit conditions ((-profit) = 0) can be written as
| (1) |
| (2) |
| (3) |
Market clearance condition requires that if prices are positive then supply should be equal to demand. It is represented by the following equations
| (4) |
| (5) |
| (6) |
| (7) |
| (8) |
Income balance condition states that the value of endowments equals consumer's income:
| (9) |
MPSGE would solve nine equations (1-9) for nine unknowns: activity levels, X, Y, W; prices, Px, Py, Pw, w, r; and income I. It should be noted that equilibrium conditions are hard to formulate in an explicit form for certain problems. The good news for a MPSGE user is that one does not need to spend any time deriving the equilibrium conditions. MPSGE builds them for you automatically. However, it is always helpful to understand what the algebraic representation of a particular MPSGE model is12. For a comparison, alternative formulations of a simple model of taxation and international trade (Shoven and Whalley [1984]) in MPSGE and its algebraic form (MCP formulation) are presented in Appendix 2. An MCP formulation of a dynamic Ramsey problem can be found in the Rutherford's paper. In order to run MPSGE program, MCP solvers should be used. GAMS has two MCP solvers: MILES and PATH. In Tom's words, MILES is his abandoned child, which leaves PATH as the first choice.
In the previous section we have briefly discussed how MPSGE finds an equilibrium. It is nice to know that MPSGE will solve it not as an optimization problem, but instead by finding an equilibrium in a system of inequalities. However, in order to create a simple MPSGE program a user does not need to know this. What is much more important for a beginner is the syntax of the MPSGE language13. A big advantage of MPSGE is that it is based on nested constant elasticity of substitution (CES) utility and production functions14. Use of nested functions can provide a flexible representation of how inputs are related. To introduce the MPSGE syntax15, we will consider again the Wonderland example described in Section 2. The GAMS-MPSGE program of the Wonderland static model (move1_1.gms) is listed in Appendix 3.
Economic models are based on data which in turn are usually organized into a Social Accounting Matrix (SAM)16. We assume the following data represented in Table 1 where a SAM has three production sectors X, Y, and W (they are declared in the $SECTORS block of the MPSGE program), one consumer CONS (declared in the $CONSUMERS block), and five markets X, Y, W, L, and K (declared in the $COMMODITIES block). Because the variables which are associated with commodities are their prices, they usually begin with the letter P.
| Production Sectors | Consumers | |||
| Markets | X | Y | W | CONS |
| PX | 100 | -100 | ||
| PY | 100 | -100 | ||
| PW | 200 | -200 | ||
| PL | -40 | -60 | 100 | |
| PK | -60 | -40 | 100 | |
Table 1. A Static Model SAM (the Wonderland example)
A SAM describes a data set, which is usually referred to as a benchmark dataset. Numbers represent the values (price times quantity) of economic transactions in a given period of time. A positive entry means a value of output (or sales). A negative entry reflects a value of input (or purchases). Zero profit, market clearance and income balance conditions imply that row sums and column sums are equal to zero.
To illustrate how SAM is balanced, consider the following row and column of Table 1. A production sector X purchases 40 units17 of labor and 60 units of capital, and sales 100 units of good X . The sum over X column is equal to zero. It shows that there is no excess profit (zero profit) in the sector X. A row PX shows that 100 units of X which are produced and sold (positive entry of 100) are consumed by sector W (negative entry of 100). The sum over PX row is equal to zero. It shows that supply equals to demand (market clearance) for a good X.
For simplicity, sometimes a Harberger convention is adopted in a benchmark SAM. It consists in normalizing all prices to 1. Then quantities in a SAM represent expenditures, or how much of a good or factor one can buy for $1. It should be noted that an Arrow-Debreu economy only depend upon relative prices. Doubling all prices double money profits and income, which is resulted in the same solution for quantities (or activity levels). The absolute price level has no impact on the equilibrium outcome.
As it has already been noted, MPSGE does not require a user to formulate an algebraic representation of production and utility functions. The user only needs to provide reference quantities, reference prices, and elasticities. Based on provided information, MPSGE constructs underlying production and utility functions. At the solution, MPSGE returns equilibrium values for the variables, described in $SECTORS, $COMMODITIES, and $CONSUMERS.
In the Wonderland example, there are two products (X and Y), two factors (L and K), and one consumer (CONS). An extra row and column (PW and W) are introduced in order to explicitly represent utility derived from an aggregate consumption. As we will discuss later, it could have been done directly in the $DEMAND block18.
The program starts with declaration of the
title, which will be printed in a solution listing. Usually,
GAMS-formatted declarations of scalars, parameters, and sets then
follow. We do not need them in our simple static model.
The next portion is in MPSGE format.
$ONTEXT
$MODEL:NAME
These two blocks shifts control from GAMS to MPSGE compiler. Model
name is chosen by a user. For example, our model has the name
MOVE1_1. The model name has to be a legitimate filename, because a
file NAME.GEN (MOVE1_1.GEN in our case) is going to be generated.
$SECTORS:
In this block, production sectors are declared. They determine how
inputs are converted into outputs. The variables here are activity levels
and they are associated with zero profit conditions (i.e. they make
sure that a production sector earn zero profit). In the Wonderland
model, we have three sectors: X, Y, and W. Comments can be made in the same
line after ! mark.
$COMMODITIES:
Commodities are declared in this block. The variables here are prices.
They are associated with market clearance conditions (i.e. they make
sure that supply is equal to demand). We have five commodities: two
output prices PX, PY, two factor prices PK, PL, and price index PW for
welfare (i.e. utility derived from an aggregate consumption).
$CONSUMERS:
Consumers who supply factors and receive utility are described. The
variable here is income from all sources. It is associated with income
balance condition (i.e. it makes sure that total income is equal to
total consumption (or total demand)). In our case, we have one representative
consumer CONS.
$PROD:
Production block constructs the underlying production function (i.e., the
rule how inputs are converted into output). Sectors and commodities
used in a production block should be declared in respective blocks.
Consider the first production block for X (Production blocks for Y
and W have a similar structure):
$PROD:X s:1
O:PX Q:100
I:PL Q:40
I:PK Q:60
This block describes a Cobb-Douglas production function (we can see
it from s:1, which means that the elasticity of substitution
between inputs is equal to one). Inputs here are PL and PK (we can
see it from I:PL and I:PK). Output is PX (
O:PX). This production sector converts 40 units of PL and 60 units
of PK into 100 units of PX. Mathematically, the production
function for this technology can be written as X = fL0.4 K0.6
(See Appendix 1 for conversion of production and utility function
into the MPSGE format). Figure 1 illustrates the calibration of a
production function to benchmark prices and quantities.
MPSGE specifies the production function with a single reference point. In this example we have explicitly specified reference quantities only. Reference prices are set to 1 by default. Relative price of inputs determines the slope of isocost (which in turn equals to the slope of isoquant at the reference quantities). It means that our production block is identical to:
$PROD:X s:1
O:PX Q:100 P:1
I:PL Q:40 P:1
I:PK Q:60 P:1
The curvature of isoquant is determined by s, the elasticity of
substitution between inputs (s:1 corresponds to a Cobb-Douglas
production function). Default value for elasticity is zero. The
reference quantities and prices are used only for calibration.
They are not used by the solver as starting values of the
variables. Production block Y is similar to X.
Production block for W (Welfare) serves as a tool for conversion of goods X and Y consumption into utility derived from an aggregate consumption. In this simple model we have introduced this production block for clarity purposes only. We could have eliminated the production block W and changed the demand block into19
$DEMAND:CONS
D:PX Q:100
D:PY Q:100
E:PL Q:100
E:PK Q:100
However, in more complicated models the representation of welfare
as a production block may be very useful (for example, in the
analysis of a consumption tax).
$DEMAND:CONS
D:PW Q:200
E:PL Q:100
E:PK Q:100
A consumer receives income from its L and K endowments ( E:PK
and E:PL) and demands good PW ( D:PW). Reference
quantities (Q fields) are used for calibration of the utility function in
the same way that the production block calibrates production function.
$INCLUDE NAME.GEN
SOLVE NAME USING MCP;
These two blocks are used for running the model.
The solution of the static model (reported in a listing
(move1_1.lst) file) is:
LOWER LEVEL UPPER MARGINAL
VAR X . 1.0000 +INF .
VAR Y . 1.0000 +INF .
VAR W . 1.0000 +INF .
VAR PX . 1.0000 +INF .
VAR PY . 1.0000 +INF .
VAR PL . 1.0000 +INF .
VAR PK . 1.0000 +INF .
VAR PW . 1.0000 +INF .
VAR CONS . 200.0000 +INF .
In the solution, LOWER and UPPER columns show the bounds of respective variables. Zero is represented by ``.''. INF means infinity. The LEVEL column reports the solution value. The MARGINAL column shows the value of the complimentary slackness (shadow) variable. We need to pay close attention to LEVEL and MARGINAL columns. Complimentary slackness implies that in equilibrium, either the level of a variable will be positive or the marginal value will be positive, but not both. If both of them are positive, we need to check our model.
The solution to the static model tells us that the benchmark data (represented in the SAM) are consistent with the equilibrium of the model. We can see it because all marginals are equal to zero after benchmark replication (i.e., after solving the model with ITERLIM=0). Equilibrium activity levels X, Y, and Z are equal to 1 (and not to 100, 100, and 200 as in SAM !). Equilibrium prices are equal to 1. Equilibrium income level of a representative consumer is equal to 200.
Somewhat confusing for beginners is that activity levels in the solution are all equal to one. This is just due to rescaling and it is done for the purpose of future analysis of counterfactual experiments, where a user can easily see the percentage change of a value deviated from 1. If we want to see unscaled activity levels, we need to multiply the levels from the solution by Q: fields in the production blocks. For example, for the activity level X it equals to 1*100=100.
There are other ways to get actual activity levels in the solution listing. One possible way is to rescale production blocks and assign initial values for activities as it is done in the program move1_3.gms. The solution for this case directly replicates the benchmark data:
LOWER LEVEL UPPER MARGINAL
VAR X . 100.0000 +INF .
VAR Y . 100.0000 +INF .
VAR W . 200.0000 +INF .
VAR PX . 1.0000 +INF .
VAR PY . 1.0000 +INF .
VAR PL . 1.0000 +INF .
VAR PK . 1.0000 +INF .
VAR PW . 1.0000 +INF .
VAR CONS . 200.0000 +INF .
However, for numerical reasons, it is advisable to scale equilibrium
values close to unity. That is why it is better to use a REPORT block
as it is shown in the program move1_4.gms, where the
initial SAM is recreated using the reported values.
All initial values for activity levels and prices are equal to 1 by default. A variable CONS, which represents a consumer income level, is somewhat different. It is determined by endowments. Remember from income balance condition that income is equal to total value of endowments, and in turn equal to total demand. In the case of static model it is equal to 200 because endowment of capital (E:PK) is 100 and endowment of labor (E:PL) is 100, and prices PK and PL are equal to 1 by default.
As we have noted, the model determines only relative prices. So if we assign all initial prices as being equal to 2, then variable CONS will be calculated to be equal to 400. We can check it by modifying the static model. Initial price levels can be assigned after $SYSINCLUDE mpsgeset NAME, but before $INCLUDE NAME.GEN command in the following way:
PX.L = 2; PY.L = 2; PW.L = 2; PK.L = 2; PL.L = 2;Then the solution listing shows an increase in all prices and the income level.
LOWER LEVEL UPPER MARGINAL
VAR X . 1.0000 +INF .
VAR Y . 1.0000 +INF .
VAR W . 1.0000 +INF .
VAR PX . 2.0000 +INF .
VAR PY . 2.0000 +INF .
VAR PL . 2.0000 +INF .
VAR PK . 2.0000 +INF .
VAR PW . 2.0000 +INF .
VAR CONS . 400.0000 +INF .
Due to determination of relative prices only, usually one price is fixed by
a user. If no price is fixed initially (i.e. no good is chosen as a
numeraire), then MPSGE will do a default price normalization. In the
case of our static model, a consumer's income level CONS is fixed. It can
be seen in the solution listing file (.lst) by looking at the
statement Default price normalization using income for CONS.
The model can also be specified in a different format. Instead of using numbers in reference quantity and price field, we could have introduced parameters at the beginning of the program and used them for calibration in the fields of production and demand blocks. An example of the vector syntax is presented in the program move1_5.gms.
After benchmark calibration, counterfactual experiments (for example, an introduction of taxes or changes in labor endowment) are performed. It is important to remember to change the iteration limit before running counterfactuals, which is done with the GAMS statement like NAME.ITERLIM=2000;. The solver will stop as soon as this limit is reached if the solution is not found before a user defined number of iterations. In this paper, we will not describe any counterfactuals with our static model20. Our interest here is different and consists of the conversion of the static model into a dynamic one.
When I had started to learn economic modeling, I immediately wanted to build a dynamic model. I wanted to predict the future, so what kind of predictions one can make with a static model? Later on, I realized that my attempts of dynamic modeling were like a desire to build a spaceship with a screwdriver only. Besides, I wanted too much out of dynamic models. Dynamic models are nowhere near the goal of predicting the future. A model can more or less tell you what is going to happen if there are no shocks and structural changes in an economy. In addition, one need to make assumptions about the rate of economic growth for several decades into the future, the rate of time preference, the growth rate of population, inflation, depreciation, etc. All these necessary assumptions bring us very far from the reality. But policy makers still need to make their decisions and economists need to provide answers about the future. Therefore, dynamic general equilibrium models are important tools for economic policy evaluation. But now I know that it is much more informative to develop a good static model than a poor dynamic one (and so far, I have never ever seen a good dynamic model).
With such an encouragement, we will attempt to develop a dynamic model based on the Wonderland example. There are different approaches to dynamic economic modeling (see, Ginsburgh and Keyzer [1997] for an overview). We limit our discussion to a simple Ramsey model. The model behaves differently if it is on a so called steady-state than it is not. A steady-state is defined as a situation in which the various quantities (such as capital, output, consumption, etc.) grow at constant rates. We start our consideration with the situation when benchmark data describe an economy on a steady-state in a base period.
In a dynamic version of the Wonderland example, a representative consumer maximizes the present value of his lifetime utility
| (10) |
| (11) |
| (12) |
| (13) |
| (14) |
| (15) |
| (16) |
as C(RKt, Wt) and demand function
as D(Pt, M), where M is consumer's income. Then MCP can be formulated as follows.
Zero profit conditions:
| (17) |
| (18) |
| (19) |
Market clearance conditions:
| (20) |
| (21) |
| (22) |
Income balance condition:
| (23) |
As noted before, a modeller does not need to program these equilibrium conditions explicitly. MPSGE constructs them automatically. However, we need to describe a change in capital over time. This requires modification of the SAM, presented in Table 1. We add one more production sector for investment. This, in turn, will change the composition of the welfare block W, because now output can be consumed or invested, rather than just consumed as in the static case.
For the Wonderland example, a steady-state requires that investment is equal to 70. How did we come to this number? In order to quantify the value for investment on a steady-state growth path, we need to describe the evolution of capital and labor over time. It requires assumptions on the growth rate of labor, g, capital depreciation rate, d, and interest rate, r. Denoting the initial labor force as L0, labor at time t is
| (24) |
| (25) |
| (26) |
In addition, we assume a constant interest rate r, so all future prices (including price of labor and capital) in terms of present value are:
| (27) |
Capital can be bought or rented. Therefore, implementation of dynamics involves two prices for capital: purchase price, PK, and rental price, RK. It should be stressed that in a static SAM (Table 1), the total value of capital endowment, VK (Check the intersection of PK and CONS in Table 1 to see that VK =100), equals to capital earnings (and not to capital stock!). The relationship between VK, rental price RK and a stock of capital K is as follows:
| (28) |
Now we need to consider the first-order conditions for capital and investment. Let's rewrite them as
| (29) |
| (30) |
| (31) |
| (32) |
| (33) |
| (34) |
| (35) |
| Production Sectors | Consumers | ||||
| Markets | X | Y | W | inv | CONS |
| PX | 100 | -65 | -35 | ||
| PY | 100 | -65 | -35 | ||
| PW | 200 | -200 | |||
| PL | -40 | -60 | 100 | ||
| PK | -60 | -40 | 100 | ||
| sav | -70 | 70 | |||
Table 2. Dynamic Model SAM
An important characteristic of the dynamic problem is a treatment of capital in the last period of modeling. We cannot solve numerically for an infinite number of periods, hence, some adjustments are needed for approximation of finite horizon model to the infinite horizon choices. Special procedures should be introduced for the terminal capital, otherwise, all capital would be consumed in the last period and nothing would be invested. We have followed the Rutherford's paper and introduced the level of post-terminal capital as a variable and added a constraint on the growth rate of investment in the terminal period:
| (36) |
A program listing of the dynamic model is represented in Appendix 3 (file move2_1.gms). Below we list and then discuss the changes which are necessary to make in the code of the static model.
Introduction of time sets.
Introduction of time sets requires three sets T, TFIRST, and
TLAST, GAMS code for which is
SET T /1*10/, TFIRST(T), TLAST(T); TFIRST(T) = YES$(ORD(T) EQ 1); TLAST(T) = YES$(ORD(T) EQ CARD(T));In the first line we declare the sets. We have 10 periods. Because of the importance of the base year period TFIRST and the terminal period TLAST, they are declared as separate sets (and subsets of T). The next two lines assign the values for the first and last periods. It is done in a way which may seem a bit odd for a novice in GAMS. The purpose of this is convenience; because it allows you to change the time dimension of the model in a single line. For example, extension of the model to 100 periods requires only one change to T /1*100/ .
Declaration of the assumed parameters.
We assume g = 0.02, d = 0.05, and r = 0.05, which is
declared in the following GAMS statement.
SCALAR DELTA DISCOUNT FACTOR /0.05/
R INTEREST RATE /0.05/
G GROWTH RATE /0.02/;
Declaration of initial parameters for capital and
investment.
We directly assign a value for initial capital earnings, VK, as
SCALAR VK INITIAL CAPITAL EARNINGS /100.0/;Also, from equations (33), (28), and (34) the values for RKO, KO and IO are
RK0 = DELTA+R; K0 = VK/RK0; I0 = (DELTA + G) * K0;
Declaration of growth rates for prices and quantities.
This step implements equations (26) and (27) for the growth rates in quantities and prices, which are
QREF(T) = (1 + G) ** (ORD(T) - 1); PREF(T) = (1/(1 + R)) ** (ORD(T) - 1);We have ORD(T)-1 as an exponent to represent the fact that in the base year QREF and PREF are equal to 1 and grow thereafter.
Production blocks for capital and investment.
New production block for K(T) implements capital
accumulation as it represented by equation (29). The MPSGE format of
the production block is (we skip terminal capital for a moment):
$PROD:K(T)
O:PK(T+1) Q:(K0*(1-DELTA))
O:RK(T) Q:(K0*(R+DELTA))
I:PK(T) Q:K0
Those who learn the MPSGE representation of the Ramsey dynamic model are mostly confused by the reference quantity of RK(T). The difference with equation (29) is that in the production block the reference quantity for capital rental price 24 is K0*(R+DELTA), whereas in equation (29) (r+d) does not appear as a coefficient of RKt. The dynamic model would not calibrate without this reference quantity. The reason for this is the fact that RK is used in several production blocks: X, Y, and K and we need to to reflect the relationship between RKt and Pt given by equation (33)
Alternatively, we could have used K0 as a reference price for O:RK(T) and assign the rental price growth as RK.L(T) = (DELTA+R)*PREF(T);, but in this case the input fields for production of X and Y blocks should also be changed as represented in the program move2_2.gms of Appendix 3.
Production block I introduces equation (30). We assume that investment is split equally between two production sectors.
$PROD:I(T) S:0
O:PK(T+1) Q:I0
I:PY(T) Q:(0.5*I0)
I:PX(T) Q:(0.5*I0)
Note that S:0 corresponds to Leontief production function,
i.e., elasticity of substitution between inputs is equal to zero. If
no elasticity is specified in a production block, MPSGE uses zero
elasticity as a default value. Therefore, S:0 can be skipped
from the code of this block.
Change in the Welfare block.
We need to make changes in the production of W, because in the
dynamic version of the model not all output is consumed but some
portion of it is invested. From Table 2 we know the new values for output
(Q:130) and inputs (Q:65).
Change in the Demand block.
For the same reason, in the DEMAND block we need to modify a
reference quantity for PW(T) from 200 to 130. Also, growth in
consumption and labor over time is represented by QREF(T).
We use the reference price PREF(T) as the value of the discount on the future consumption given by equation (10). Introduction of individual time-preference as a separate parameter is not needed because in the Ramsey model, the steady-state leads to a constant pattern of per capita consumption and to the condition that r = r.
Another change in the DEMAND block involves the usage of initial capital stock K0 as a reference quantity for PK(TFIRST), instead of the value of capital VK as in the static case. This is determined by income balance condition of the dynamic model.
One more modification in the DEMAND block is the introduction of
intertemporal elasticity of substitution s:1. Note that we
have assumed a Cobb-Douglas case in our simple dynamic
model. In general, we could have introduced a scalar SIGMA
and use s:SIGMA as an elasticity parameter (see Appendix 1
for conversion utility functions into the MPSGE format).
Terminal capital constraint.
In order to introduce a constraint in MPSGE, we need to use two types
of blocks, which we have not discussed yet. A constraint is declared
in $AUXILIARY block and then formulated in
$CONSTRAINT block. For our model, we need to introduce a
constraint on terminal capital given by equation (36). Therefore, a
constraint TK is declared. Also, in order to induce investment in
the last period of modeling, a price PKT is introduced. It
represents the market for a post-terminal capital.
The post-terminal capital is output of the production blocks K and I. The condition $TLAST(T) shows that the post-terminal capital is going to be produced only in the last period. In CGE models, everything which is produced has to be consumed, therefore, an endowment of terminal capital in the $DEMAND block is introduced as
E:PKT Q:(-1) R:TK
Negative value of the endowment represent the fact that terminal
capital is going to be consumed. The field R:TK implies that
the following condition has to
be satisfied:
SUM(T$TLAST(T), I(T)/I(T-1) - Y(T)/Y(T-1)) =G= 0;
Summation sign, SUM, is used due to the GAMS syntax which
does not directly allow the assignment of a set member to a variable
with a different domain.
Initial values.
The assignment is done after the $SYSINCLUDE line but
before the $INCLUDE line of the program. The initial values in our
model are assigned to reflect present values (prices are adjusted for
PREF(T) which is given by equation (27)) and growth in
quantities (quantities are adjusted for QREF(T) as
represented by equation (26)). The only one exception is
PK.L(T)=(1+R)*PREF(T), which reflects equation (31).
The initial values for the level and price of a terminal capital are determined by the same logic as for other quantities and prices as25:
TK.L = K0 * (1 + G) ** CARD(T);
PKT.L = SUM(TLAST, PK.L(TLAST)/(1+R));
The solution listing (run the program move2_1.gms to obtain the listing) shows that all marginals are equal to zero, activity levels are rising over time and prices are falling over time. Total consumption has the value of 1145. The price of capital differs from all other prices by (1+r). Table 3 summarizes the results26. After benchmark replication, one can run counterfactual experiments, such as introduction of taxes, change in growth rates, elasticities, etc.
| Quantities | Prices | X | Y | W | K | I | |
| 1 | 1.0000 | 1.0000 | 100.00 | 100.00 | 130.00 | 1000.00 | 70.00 |
| 2 | 1.0200 | 0.9524 | 102.00 | 102.00 | 132.60 | 1020.00 | 71.40 |
| 3 | 1.0404 | 0.9070 | 104.04 | 104.04 | 135.25 | 1040.40 | 72.83 |
| 4 | 1.0612 | 0.8638 | 106.12 | 106.12 | 137.96 | 1061.21 | 74.29 |
| 5 | 1.0824 | 0.8227 | 108.24 | 108.24 | 140.72 | 1082.43 | 75.77 |
| 6 | 1.1041 | 0.7835 | 110.41 | 110.41 | 143.53 | 1104.08 | 77.29 |
| 7 | 1.1262 | 0.7462 | 112.62 | 112.62 | 146.40 | 1126.16 | 78.83 |
| 8 | 1.1487 | 0.7107 | 114.87 | 114.87 | 149.33 | 1148.69 | 80.41 |
| 9 | 1.1717 | 0.6768 | 117.17 | 117.17 | 152.32 | 1171.66 | 82.02 |
| 10 | 1.1951 | 0.6446 | 119.51 | 119.51 | 155.36 | 1195.09 | 83.66 |
Table 3. Steady-state prices and quantities
Data for a base period may be inconsistent with a steady-state growth path. Table 4 gives an example of the situation where investment is ``too big'', i.e., from equation (35) the value of investment is 80 > (0.05+0.02) ·100/(0.05+0.05) = 70.
| Production Sectors | Consumers | ||||
| Markets | X | Y | W | Inv | CONS |
| PX | 100 | -60 | -40 | ||
| PY | 100 | -60 | -40 | ||
| PW | 200 | -200 | |||
| PL | -40 | -60 | 100 | ||
| PK | -60 | -40 | 100 | ||
| sav | -80 | 80 | |||
Table 4. A Dynamic Model SAM off a Steady-State
One possible solution for model calibration in the case of the
off-steady-state data is to assume that g
is the labor growth rate, but the capital growth rate is different
initially and converges to g over time27. We use the
calibration which is based on the secant algorithm. It finds the
solution to an equation f(x) = 0 as:
1. Initial guess on x(n), where n is the number of iteration.
2. Calculate x(n+1) as28 :
| (37) |
To reiterate our current dynamic problem, we know the initial values given in the SAM. The SAM is not consistent with a steady-state growth path. Also, we know (assumed) depreciation rate and long-run interest rate but the rental price in a base period is unknown. Our goal is calibrating a dynamic model to the data represented in the SAM. In order to do that we use iterations based on the secant algorithm described above. The calibration involves the following iteration process: guessing initial rental rate of capital, RK0, calculating a model of the transition path, observing investment, I0, in the current equilibrium, and adjusting RK0.
The program listing for the off-steady-state dynamic model is provided in Appendix 3 (file move3_1.gms). At first, we guess about the rental price, RKG, in the first period. In the on-steady-state case the rental price (which is also equal to return to capital) is d+ r. We make our guess using this knowledge. Note that if our guess is ``too far'', the algorithm does not work. Also, if investment is ``too far'' from the steady-state value, the algorithm does not work as well.
The off-steady-state dynamic program has the same structure as before, except for the fact that we have used an alternative formulation similar to the program described in Appendix 3 (file move2_2.gms). We apply equation (37) for finding f(x), which is in our case the difference between the value of investment calculated in the program and the one given in the SAM. Accordingly, x(n) is the guess on rental price at the first period. During the first iteration we simply add 0.1 to our guess value for RK0 in order to produce a basis for further iterations. During all other iterations we reproduce equation (37) as
RKG = RKG + ERROR(ITER)/(ERROR(ITER-1)-ERROR(ITER))
* (RK0VAL(ITER)-RK0VAL(ITER-1));
where ERROR represents29 f(x) and RK0VAL represents x(n).
The last portion of the program includes graphing the results for investment, consumption, growth rate and interest rate. We have used GNUPLOT utility to produce graphs. It can be seen from the graphs (run the model move3_1.gms to obtain the graphs) that the values eventually converge to a steady-state growth path. The speed of convergence to a steady-state is determined by production and preference parameters (Barro and Sala-i-Martin [1995]).
Adjustments similar to the described above can be applied to a static model of the GTAPinGAMS package (Rutherford [1998b]). The listing of the dynamic model is presented in Appendix 4.
Learn MPSGE!!!
MPSGE represents production and preferences as nested CES functions.
Production:
A production technology is identically represented by either the
production function or corresponding cost function. A general form of
a nested CES production function is X(K,L) = (aKr+ (1-a)Lr)[1/( r)] where r is the substitution
parameter (-¥ < r < 1), which is related to the elasticity
of substitution s as s = [1/( 1-r)]. When r = -¥ (or s = 0), CES production function converts into
Leontief (or fixed proportions) production function X1(K,L) = min(aK1,(1-a)L1). When r = 1 (or s = ¥), then
inputs are perfect substitutes in production X2 = aK2 + (1-a)L2.
When r = 0 (or s = 1), then CES production function
converts into a Cobb-Douglas case X3(K,L) = K3a L31-a.
Let's consider MPSGE representation of a particular example of a
Cobb-Douglas production function: X(K,L) = K0.6L0.4. The
following production blocks represent the same technology:
a)
$PROD:X s:1
O:PX Q:1
I:PK Q:0.6
I:PL Q:0.4
b)
$PROD:X s:1
O:PX Q:100
I:PK Q:60
I:PL Q:40
c)
$PROD:X s:1
O:PX Q:100
I:PK Q:60 P:2
I:PL Q:40 P:2
d)
$PROD:X s:1
O:PX Q:100 P:2
I:PK Q:60
I:PL Q:40
e)
$PROD:X s:1
O:PX Q:100 P:2
I:PK Q:60 P:2
I:PL Q:40 P:2
Remember that reference prices and quantities are used for
calibration only and not transferred to a solver as initial values!
Another important issue is that only relative prices matter. As
such, use of P:1 or P:2 for both inputs gives
the same marginal rate of substitution. Price of output does not
matter in this example because we have one output only. In the case
of two outputs relative prices would matter as they determine
marginal rate of transformation.
How about this production block?
$PROD:X s:1
O:PX Q:1
I:PK Q:1 P:0.6
I:PL Q:1 P:0.4
It still represents the same production technology (?) as all above
MPSGE production blocks.
Now consider a production block:
$PROD:X s:1
O:PX Q:1
I:PK Q:0.6 P:2
I:PL Q:0.4
This production block is described by the function
X(K,L) = K0.75L0.25, because value shares are:
K = [0.6/( 1.2+0.4)] = 0.75 and L = [0.4/( 1.2+0.4)] = 0.25.
Leontief production function can be represented either as
$PROD:X s:0
O:PX Q:100
I:PK Q:60
I:PL Q:40
or as
$PROD:X
O:PX Q:100
I:PK Q:60
I:PL Q:40
because default value for top level elasticity is zero.
Preferences:
General form of a nested CES utility function is U(X,Y) = (aXr + (1-a)Yr)[1/( r)] where r is an elasticity parameter. Elasticity of substitution s is related to elasticity parameter as s = [1/( 1-r)]. $DEMAND block represents utility derived from consumption of goods and consumer's endowments. If quantities demanded and endowments are all equal to 100, then demand block in MPSGE is:
$DEMAND:CONS s:(1/1-RHO)
D:PX Q:100
D:PY Q:100
E:PL Q:100
E:PK Q:100
There is a particular utility function which is widely used in a dynamic analysis. It is a Constant Intertemporal Elasticity of Substitution utility function (CIES), which has a form: U(Ct) = [(Ct1-q-1)/( 1-q)], where s = [1/( q)] is the elasticity of substitution.
As q® 0, U(C) approaches a linear case U(Ct) = Ct - 1 (which represent the same preferences as U(Ct) = Ct). As q®1, then U(Ct) = ln(Ct).
In order to see a relationship between CES and CIES utility functions, we start with CES form: U(Ct) = (aCtr)[1/( r)]. Then substitute r = 1-q because s = [1/( q)] and s = [1/( 1-r)]. It leads to U(C) = (aC1-q)[1/( 1-q)]. Maximization of (aC1-q)[1/( 1-q)] and [(aC1-q)/( 1-q)] produces the same result for optimal value of C (because maxX1/b Û maxXb Û maxX/b). For utility maximization over infinite horizon we sum over instanteneous utilities and adjust for a discount factor, which leads to a traditional form of utility function:
U = å¥t = 0 ([1/( 1+r)])t [(Ct1-q-1)/( 1-q)].
If a consumer has an endowment of labor and capital and elasticity of intertemporal substitution SIGMA between consumption in different periods is defined as SIGMA=1/THETA; to reflect the relationship between s and q, then $DEMAND block for CIES utility function has the following form:
$DEMAND:CONS s:SIGMA
D:PC(T) Q:QREF(T) P:PREF(T)
E:PL(T) Q:QREF(T)
E:PK(TFIRST) Q:K0
Note that if consumer gets utility from two goods (for example, in a Cobb-Douglas form) we can keep the same $DEMAND block as above and add a production block
$PROD:C(T) s:1
O:PC(T) Q:C0
I:PX(T) Q:X0
I:PY(T) Q:Y0
Multilevel nests
Consider a case when T=1,2. Then from production block
a)
$PROD:C s:1
O:PC Q:C0
I:PX(T) Q:X0
I:PY(T) Q:Y0
elasticity between PX1, PX2, PY1, PY2 is 1.
b)
$PROD:C s:1 a:0.5
O:PC Q:C0
I:PX(T) Q:X0 a:
I:PY(T) Q:Y0
elasticity between PX1 and PX2 is 0.5, and elasticity between PY1,
PY2 and PX1:PX2 is 1.
c)
$PROD:C s:1 a:0.5 b:1.5
O:PC Q:C0
I:PX(T) Q:X0 a:
I:PY(T) Q:Y0 b:
elasticity between PX1 and PX2 is 0.5, elasticity between PY1 and PY2
is 1.5, and elasticity between PX1:PX2 and PY1:PY2 is 1.
d)
$PROD:C s:1 a:0.5
O:PC Q:C0
I:PX(T) Q:X0 a:
I:PY(T) Q:Y0 a:
top level elasticity does not matter and the nest elasticity
PX1:PX2:PY1:PY2 is 0.5.
e)
$PROD:C s:1 T.tl:0.5
O:PC Q:C0
I:PX(T) Q:X0 T.tl:
I:PY(T) Q:Y0 T.tl:
elasticity between PX1 and PY1 is 0.5, elasticity between PX2 and PY2
is 0.5, and elasticity between PX1:PY1 and PX2:PY2 is 1.
Appendix 2 contains the MPSGE and algebraic (MCP) versions for the following problem (Shoven and Whalley, Applied General-Equilibrium Models of Taxation and International Trade: An Introduction and Survey, Journal of Economic Literature, Vol. 22, No. 3. (Sep., 1984), pp. 1007-1051). It is provided for illustration purposes. One can compare the degree of difficulty in both versions. If you are interested in an MCP version of a dynamic Ramsey model, see Rutherford's [1997] paper. I have some preliminary notes on conversion of MPSGE into MCP. Check with me if I have finished them yet.
In this model, there are two final commodities (M, N), two factors of production (K, L) and two consumers (R,P). Producers are competitive, minimizing unit cost taking market prices as given. Production functions are of the constant-elasticity-of-substitution (CES) form:
|
| Sector | fi | di | si |
| Manufacturing (M) | 1.5 | 0.6 | 2.0 |
| Non-Manufacturing (N) | 2.0 | 0.7 | 0.5 |
Consumers are endowed with primary factors form which they earn income. This income is used to purchase goods to maximize utility. The CES utility functions are:
|
| Household | K | L | aM | aN | sc |
| Rich | 25 | 0 | 0.5 | 0.5 | 1.5 |
| Poor | 0 | 60 | 0.3 | 0.7 | 0.75 |
After computing the benchmark equilibrium, a single counterfactual experiment is done in which a 50% ad-valorem tax is levied on each unit of capital services employed in the manufacturing sector. Tax revenue is returned as lump-sum income for the two households, 40% to the rich and 60% to the poor.
$title Compare MPSGE and its algebraic representation
set i /M, N/;
set c /RICH, POOR/;
set rich(c) /rich/;
set poor(c) /poor/;
parameters
sigma /m 2, n 0.5/,
phi /m 1.5, n 2/,
delta /m 0.6, n 0.7/,
sig /rich 1.5, poor 0.75/,
capital /rich 25, poor 0/,
labor /rich 0, poor 60/;
table alpha(c,i)
M N
RICH 0.5 0.5
POOR 0.3 0.7 ;
parameter taxrate(i);
$ontext
$model:work
$SECTORS:
Y(i) !Production
$COMMODITIES:
P(i) !Price
W !Price of labor
R !Price of capital
$CONSUMERS:
cons(c) ! Representative agent
$prod:y(i) s:sigma(i)
o:p(i) q:phi(i)
i:w q:1 p:delta(i)
i:r q:1 p:(1-delta(i))
+ a:cons("rich") t:(0.4*taxrate(i))
+ a:cons("poor") t:(0.6*taxrate(i))
$demand:cons(c) s:sig(c)
d:p(i) q:1 p:(alpha(c,i)**(1/sig(c)))
e:r q:capital(c)
e:w q:labor(c)
$offtext
$sysinclude mpsgeset work
taxrate(i) = 0;
work.iterlim = 5000;
$include work.gen
solve work using mcp;
taxrate("M") = 0.5;
$include work.gen
solve work using mcp;
* MCP version
alias(i,j);
taxrate("M") = 0;
parameter pk0(i);
pk0(i) = 1+taxrate(i);
parameter thetak(i), thetal(i);
thetak(i) = 1-delta(i);
thetal(i) = delta(i);
parameter abeta(c,i);
abeta(c,i) = alpha(c,i)**(1/sig(c));
equations
prf_y(i)
mkt_y(i)
mkt_k
mkt_l
i_cons(c);
* Zero profit
prf_y(i)..
(thetak(i)*(r*(1+taxrate(i))/(pk0(i)*(1-delta(i))))**(1-sigma(i))
+ thetal(i)*(w/delta(i))**(1-sigma(i)))**(1/(1-sigma(i)))
=e= p(i) * phi(i);
* Market clearance
mkt_y(i)..
y(i) * phi(i) =e=
sum(c, 1 *
((((sum(j, abeta(c,j) * (p(j)/abeta(c,j))
** (1-sig(c))))**(1/(1-sig(c)))) * abeta(c,i) / p(i)) ** sig(c)) *
cons(c) / ( ((sum(j, abeta(c,j) * (p(j)/abeta(c,j))
** (1-sig(c))))**(1/(1-sig(c))))) );
mkt_k..
sum(c, capital(c)) =e=
sum(i,
((thetal(i)*(w/delta(i))**(1-sigma(i))
+ thetak(i)*(r*(1+taxrate(i))/(pk0(i)*(1-delta(i))))**(1-sigma(i)))
**(1/(1-sigma(i))) / (r*(1+taxrate(i))/(pk0(i)*(1-delta(i)))))
**(sigma(i)) * y(i));
mkt_l..
sum(c, labor(c)) =e=
sum(i,
((thetal(i)*(w/delta(i))**(1-sigma(i))
+ thetak(i)*(r*(1+taxrate(i))/(pk0(i)*(1-delta(i))))**(1-sigma(i)))
**(1/(1-sigma(i))) * delta(i)/w)**(sigma(i)) * y(i));
i_cons(c)..
cons(c) =e= r*capital(c) + w*labor(c)
+ (sum(i, 0.6 * taxrate(i) * y(i) * r *
((thetal(i)*(w/delta(i))**(1-sigma(i))
+ thetak(i)*(r*(1+taxrate(i))/(pk0(i)*(1-delta(i))))**(1-sigma(i)))
**(1/(1-sigma(i)))
/ (r*(1+taxrate(i))/(pk0(i)*(1-delta(i)))))**(sigma(i))))$poor(c)
+ (sum(i, 0.4 * taxrate(i) * y(i) * r *
((thetal(i)*(w/delta(i))**(1-sigma(i))
+ thetak(i)*(r*(1+taxrate(i))/(pk0(i)*(1-delta(i))))**(1-sigma(i)))
**(1/(1-sigma(i)))
/ (r*(1+taxrate(i))/(pk0(i)*(1-delta(i)))))**(sigma(i))))$rich(c);
model obana /prf_y.y, mkt_y.p, mkt_k.r, mkt_l.w, i_cons.cons/;
cons.fx("rich") = 34.3368;
cons.fx("poor") = 60.0000;
obana.iterlim = 5000;
solve obana using mcp;
* Introduce tax
taxrate("M") = 0.5;
cons.fx("rich") = 29.0935;
cons.fx("poor") = 61.3484;
solve obana using mcp;
Appendix 3 contains GAMS programs which can be copied and executed.
The following files are located here.
1 Thanks to Thomas Rutherford and Miles Light for not letting me
sink when I hit the MPSGE iceberg. The author can be reached at:
sergey.paltsev@colorado.edu
2 University of Colorado, Department
of Economics, Boulder, CO 80309-0256, USA.
3 The latest version of this document in PDF
format can be downloaded from
http://debreu.colorado.edu/papers/move.pdf
4 I refer to this
dynamic modelling paper (Lau, Pahlke, Rutherford [1997]) as the
Rutherford's paper. It is available at:
http://nash.colorado.edu/tomruth/primer/paper.htm
5 The discussion in this
paper is limited to a Ramsey model. Another approach to introducing
dynamics is overlapping generations (OLG) models. For OLG formulation
in MPSGE, see Rutherford [1998].
6 To learn more about GAMS, visit
their homepage at
http://www.gams.com
7 As of April 2000, a free
student version of GAMS is available at:
http://www.gams.de/5download/cd.htm
8 Or,
equivalently, a population of identical households.
9 It sounds funny
for non-economists that there is a single consumer who owns the firms
and purchases the goods produced by these firms. But for economists
with their ``economic way of thinking'' it seems natural. The
significance of this assumption is that distributional considerations
between different types of consumers are ignored.
10 An expression written as
xT y = 0 means xi yi = 0, for all i = 1,...,n. The variables
xi and yi are called a complementary pair and are
said to be complements to each other
11 For more information on
Shephard's lemma see, for example, Varian [1992].
12 When I saw an
MPSGE code for the first time, I said: ``I need to see an underlying
algebra''. Later on I have realized that MPSGE representation is much
more attractive and intuitive than its algebraic formulation. So now
the knowledgeable person always smiles when I am saying: ``I don't
want those ugly MCP equations, show me an MPSGE code''.
13 MPSGE syntax can be found at
http://debreu.colorado.edu/mainpage/mpsge.htm
14 See
Appendix 1 for the examples of conversion of different functions into
the MPSGE format.
15 In this treatment of simple static
and dynamic models, many MPSGE features are not going to be
discussed. For additional practice, see the Markusen examples
(Markusen and Rutherford [1995]), available at:
http://nash.colorado.edu/tomruth/6433/markusen.htm
.
16 For more information on
converting input-output data into an MPSGE model, see Rutherford and Paltsev
[1999]
17 Make sure
to remember that units here represent the value, i.e. price times
quantity.
18 However, a separate
block for utility is helpful for an introduction of a consumption tax
and for welfare analysis.
19 See
the code of the program move1_2.gms from Appendix 3. You can
run it to compare the results with move1_1.gms.
20 See Markusen examples
for practice (Markusen and Rutherford [1995]).
21 Setting P(t) = 1,
an equation for RK makes perfect economic sense, because if capital
and other loans are perfect substitutes, then households can also
receive interest rate r on lending to other households. But capital
depreciates at the rate d. Then r = RK-d, or RK = d+r , which is the same as equation (33).
22
It should be noted that the transversality condition requires that r
> g, otherwise, there is a possibility of a chain-letter borrowing
(for detailed discussion see, for example, Barro and Sala-i-Martin
(1995)).
23 Because all quantities grow at the same rate, it is possible to use
a growth of total output, output in a certain production sector, or
consumption (or simply 1+g in a steady-state case) as a
right-hand side variable of equation (36).
24 Note that K0*(R+DELTA)=VK, so we could have used
Q:VK as a reference quantity for O:RK(T).
25 An alternative formulation is TK.L = SUM(TLAST,
K.L(TLAST)); and PKT.L = SUM(TLAST, PREF(TLAST));
26 It could be a good practice for a novice in MPSGE to
reproduce the results presented in Table 3
27 Another way of
calibrating to a static data set involves given data on a base year
investment, interest rate, and return to capital. The long-run
steady-state interest rate is determined based on assumed labor
growth rate and calibrated utility discount rate.
28 Equation (37) could be derived from an
equation of tangent:
[(y-yn)/( x-xn)] = f¢(xn). Put y = 0 and yn = f(xn), then
x = xn- [(f(xn))/( f¢(xn)))]. Using
f¢(xn) = [(f(xn)-f(xn-1))/( xn-xn-1)]
we receive equation (37).
29 Note that for f(x) we use an
equation for consumption instead of investment in order to explicitly
separate calculated and given values. Usage of ERROR(ITER) =
I.L("1")*I0 - I0; instead of ERROR(ITER) = C.L("1") - (200-I0);
produces the same result.
Appendix 3. Program listings
move1_1.gms Static model The Wonderland example
move1_2.gms no welfare block
move1_3.gms initial quantities in the listing
move1_4.gms report block
move1_5.gms vector syntax
move2_1.gms Dynamic model on a steady-state in a
baseyear
move2_2.gms alternative formulation of
capital block
move3_1.gms not on a steady-state in a
baseyear
$TITLE: move1_1.gms - A Simple Static Model (The Wonderland example)
$ONTEXT
$MODEL:MOVE1_1
$SECTORS:
X ! Activity level for sector X
Y ! Activity level for sector Y
W ! Activity level for sector W (welfare index)
$COMMODITIES:
PX ! Price index for commodity X
PY ! Price index for commodity Y
PL ! Price index for primary factor L
PK ! Price index for primary factor K
PW ! Price index for welfare (expenditure function)
$CONSUMERS:
CONS ! Income level for consumer CONS
$PROD:X s:1
O:PX Q:100
I:PL Q:40
I:PK Q:60
$PROD:Y s:1
O:PY Q:100
I:PL Q:60
I:PK Q:40
$PROD:W s:1
O:PW Q:200
I:PX Q:100
I:PY Q:100
$DEMAND:CONS
D:PW Q:200
E:PL Q:100
E:PK Q:100
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE1_1
MOVE1_1.ITERLIM = 0;
$INCLUDE MOVE1_1.GEN
SOLVE MOVE1_1 USING MCP;
$TITLE: move1_2.gms - A Simple Static Model (no welfare block)
$ONTEXT
$MODEL:MOVE1_2
$SECTORS:
X ! Activity level for sector X
Y ! Activity level for sector Y
$COMMODITIES:
PX ! Price index for commodity X
PY ! Price index for commodity Y
PL ! Price index for primary factor L
PK ! Price index for primary factor K
$CONSUMERS:
CONS ! Income level for consumer CONS
$PROD:X s:1
O:PX Q:100
I:PL Q:40
I:PK Q:60
$PROD:Y s:1
O:PY Q:100
I:PL Q:60
I:PK Q:40
$DEMAND:CONS s:1
D:PX Q:100
D:PY Q:100
E:PL Q:100
E:PK Q:100
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE1_2
MOVE1_2.ITERLIM = 0;
$INCLUDE MOVE1_2.GEN
SOLVE MOVE1_2 USING MCP;
$TITLE: move1_3.gms. Simple Static Model: Formulation for Initial Quantities
$ONTEXT
$MODEL:MOVE1_3
$SECTORS:
X ! Activity level for sector X
Y ! Activity level for sector Y
W ! Activity level for sector W (welfare index)
$COMMODITIES:
PX ! Price index for commodity X
PY ! Price index for commodity Y
PL ! Price index for primary factor L
PK ! Price index for primary factor K
PW ! Price index for welfare (expenditure function)
$CONSUMERS:
CONS ! Income level for consumer CONS
$PROD:X s:1
O:PX Q:1
I:PL Q:0.4
I:PK Q:0.6
$PROD:Y s:1
O:PY Q:1
I:PL Q:0.6
I:PK Q:0.4
$PROD:W s:1
O:PW Q:1
I:PX Q:0.5
I:PY Q:0.5
$DEMAND:CONS
D:PW Q:200
E:PL Q:100
E:PK Q:100
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE1_3
X.L=100;
Y.L=100;
W.L=200;
MOVE1_3.ITERLIM = 0;
$INCLUDE MOVE1_3.GEN
SOLVE MOVE1_3 USING MCP;
$TITLE: move1_4.gms - A Simple Static Model (report block)
$ONTEXT
$MODEL:MOVE1_4
$SECTORS:
X ! Activity level for sector X
Y ! Activity level for sector Y
W ! Activity level for sector W (welfare index)
$COMMODITIES:
PX ! Price index for commodity X
PY ! Price index for commodity Y
PL ! Price index for primary factor L
PK ! Price index for primary factor K
PW ! Price index for welfare (expenditure function)
$CONSUMERS:
CONS ! Income level for consumer CONS
$PROD:X s:1
O:PX Q:100
I:PL Q:40
I:PK Q:60
$PROD:Y s:1
O:PY Q:100
I:PL Q:60
I:PK Q:40
$PROD:W s:1
O:PW Q:200
I:PX Q:100
I:PY Q:100
$DEMAND:CONS
D:PW Q:200
E:PL Q:100
E:PK Q:100
$REPORT:
V:X_OUT O:PX PROD:X
V:K_X I:PK PROD:X
V:L_X I:PL PROD:X
V:Y_OUT O:PY PROD:Y
V:K_Y I:PK PROD:Y
V:L_Y I:PL PROD:Y
V:W_OUT O:PW PROD:W
V:X_W I:PX PROD:W
V:Y_W I:PY PROD:W
V:C D:PW DEMAND:CONS
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE1_4
MOVE1_4.ITERLIM = 0;
$INCLUDE MOVE1_4.GEN
SOLVE MOVE1_4 USING MCP;
parameter report;
report("px","X") = x_out.l;
report("py","y") = y_out.l;
report("pw","w") = w_out.l;
report("pl","X") = -l_x.l;
report("pk","X") = -k_x.l;
report("pl","y") = -l_y.l;
report("pk","y") = -k_y.l;
report("px","w") = -x_w.l;
report("py","w") = -y_w.l;
report("pl","cons") = l_x.l+l_y.l;
report("pk","cons") = k_x.l+k_y.l;
report("pw","cons") = -w_out.l;
display report;
$TITLE: move1_5.gms - A Simple Static Model (vector syntax)
set i Sectors /x,y/,
f Factors /k,l/;
TABLE sam(*,*) Benchmark data
X Y W CONS
X 100 -100
Y 100 -100
W 200 -200
L -40 -60 100
K -60 -40 100 ;
PARAMETER Y0(I) Benchmark sectoral output,
FD0(F,I) Benchmark factor demands,
C0(I) Benchmark consumption demand,
E0(F) Factor endowments,
W0 Benchmark total consumption;
* Extract data from the original format into model-specific arrays:
Y0(I) = SAM(I,I);
FD0(F,I) = -SAM(F,I);
C0(I) = -SAM(I,"W");
W0 = SUM(I, C0(I));
E0(F) = SAM(F,"CONS");
$ONTEXT
$MODEL:MOVE1_5
$SECTORS:
OUT(i) ! Production Activity Level
W ! Welfare Index
$COMMODITIES:
P(i) ! Price index for commodities
PF(f) ! Price index for factors
PW ! Utility Price Index
$CONSUMERS:
CONS ! Income level for consumer CONS
$PROD:OUT(i)
O:P(i) Q:Y0(i)
I:PF(f) Q:FD0(f,i)
$PROD:W s:1
O:PW Q:W0
I:P(i) Q:C0(i)
$DEMAND:CONS
D:PW Q:W0
E:PF(f) Q:E0(f)
$REPORT:
V:OUT_o(i) O:P(i) PROD:OUT(i)
V:INP(f,i) I:PF(f) PROD:OUT(i)
V:OUT_w O:PW PROD:W
V:INP_w(i) I:P(i) PROD:W
V:C D:PW DEMAND:CONS
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE1_5
MOVE1_5.ITERLIM = 0;
$INCLUDE MOVE1_5.GEN
SOLVE MOVE1_5 USING MCP;
parameter report;
report(i,i) = out_o.l(i);
report("w","w") = out_w.l;
report(i,"w") = -inp_w.l(i);
report(f,i) = inp.l(f,i);
report(f,"cons") = sum(i, inp.l(f,i));
report("w","cons") = -out_w.l;
display report;
$TITLE: move2_1.gms. A Simple Ramsey Dynamic Model
SET T /1*10/,
TFIRST(T),
TLAST(T);
TFIRST(T) = YES$(ORD(T) EQ 1);
TLAST(T) = YES$(ORD(T) EQ CARD(T));
SCALAR DELTA DISCOUNT FACTOR /0.05/
R INTEREST RATE /0.05/
G GROWTH RATE /0.02/
VK INITIAL CAPITAL EARNINGS /100.0/
K0 INITIAL CAPITAL STOCK
RK0 INITIAL RETURN TO CAPITAL
I0 INITIAL INVESTMENT;
PARAMETER QREF(T) QUANTITIES
PREF(T) PRICES;
RK0 = DELTA+R;
K0 = VK/RK0;
I0 = (DELTA + G) * K0;
QREF(T) = (1 + G) ** (ORD(T) - 1);
PREF(T) = (1/(1 + R)) ** (ORD(T) - 1);
$ONTEXT
$MODEL:MOVE2_1
$SECTORS:
X(T) ! Activity level for sector X
Y(T) ! Activity level for sector Y
W(T) ! Activity level for sector W (welfare index)
I(T) ! Investment Sector
K(T) ! Capital Accumulation
$COMMODITIES:
PX(T) ! Price index for commodity X
PY(T) ! Price index for commodity Y
PL(T) ! Price index for primary factor L
PK(T) ! Price index for primary factor K
PW(T) ! Price index for welfare (expenditure function)
RK(T) ! Rental rate for capital
PKT ! Post-terminal capital constraint
$CONSUMERS:
CONS ! Income level for consumer CONS
$AUXILIARY:
TK ! Terminal Capital Stock
$PROD:X(T) s:1
O:PX(T) Q:100
I:PL(T) Q:40
I:RK(T) Q:60
$PROD:Y(T) s:1
O:PY(T) Q:100
I:PL(T) Q:60
I:RK(T) Q:40
$PROD:K(T)
O:PK(T+1) Q:((1-DELTA)*K0)
O:PKT$TLAST(T) Q:((1-DELTA)*K0)
O:RK(T) Q:(K0*(DELTA+R))
I:PK(T) Q:K0
$PROD:I(T)
O:PK(T+1) Q:I0
O:PKT$TLAST(T) Q:I0
I:PY(T) Q:(0.5*I0)
I:PX(T) Q:(0.5*I0)
$PROD:W(T) s:1
O:PW(T) Q:130
I:PX(T) Q:65
I:PY(T) Q:65
$DEMAND:CONS s:1
D:PW(T) Q:(130*QREF(T)) P:PREF(T)
E:PL(T) Q:(100*QREF(T))
E:PK(TFIRST) Q:K0
E:PKT Q:(-1) R:TK
$CONSTRAINT:TK
SUM(T$TLAST(T), I(T)/I(T-1) - Y(T)/Y(T-1)) =G= 0;
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE2_1
X.L(T) = QREF(T);
Y.L(T) = QREF(T);
W.L(T) = QREF(T);
I.L(T) = QREF(T);
K.L(T) = QREF(T);
TK.L = K0 * (1 + G) ** CARD(T);
PX.L(T) = PREF(T);
PY.L(T) = PREF(T);
PL.L(T) = PREF(T);
PK.L(T) = (1+R)*PREF(T);
PW.L(T) = PREF(T);
RK.L(T) = PREF(T);
PKT.L = SUM(TLAST, PK.L(TLAST)/(1+R));
MOVE2_1.ITERLIM = 0;
$INCLUDE MOVE2_1.GEN
SOLVE MOVE2_1 USING MCP;
$TITLE move2_2.gms Simple Dynamic Model
* No RK0=DELTA+R in O:RK(T), but in initial RK.L and
* in production blocks
SET T /1*10/,
TFIRST(T),
TLAST(T);
TFIRST(T) = YES$(ORD(T) EQ 1);
TLAST(T) = YES$(ORD(T) EQ CARD(T));
SCALAR DELTA DISCOUNT FACTOR /0.05/
R INTEREST RATE /0.05/
G GROWTH RATE /0.02/
VK INITIAL CAPITAL EARNINGS /100.0/
K0 INITIAL CAPITAL STOCK
RK0 INITIAL RETURN TO CAPITAL
I0 INITIAL INVESTMENT;
PARAMETER QREF(T) QUANTITIES
PREF(T) PRICES;
RK0 = DELTA+R;
K0 = VK/RK0;
I0 = (DELTA + G) * K0;
QREF(T) = (1 + G) ** (ORD(T) - 1);
PREF(T) = (1/(1 + R)) ** (ORD(T) - 1);
$ONTEXT
$MODEL:MOVE2_2
$SECTORS:
X(T) ! Activity level for sector X
Y(T) ! Activity level for sector Y
W(T) ! Activity level for sector W (welfare index)
I(T) ! Investment sector
K(T) ! Capital accumulation
$COMMODITIES:
PX(T) ! Price index for commodity X
PY(T) ! Price index for commodity Y
PL(T) ! Price index for primary factor L
PK(T) ! Price index for primary factor K
PW(T) ! Price index for welfare (expenditure function)
RK(T) ! Rental rate for capital
PKT ! Post-terminal capital constraint
$CONSUMERS:
CONS ! Income level for consumer CONS
$AUXILIARY:
TK ! Terminal Capital Stock
$PROD:X(T) s:1
O:PX(T) Q:100
I:PL(T) Q:40
I:RK(T) Q:(60/RK0) P:RK0
$PROD:Y(T) s:1
O:PY(T) Q:100
I:PL(T) Q:60
I:RK(T) Q:(40/RK0) P:RK0
$PROD:K(T)
O:PK(T+1) Q:((1-DELTA)*K0)
O:PKT$TLAST(T) Q:((1-DELTA)*K0)
O:RK(T) Q:K0
I:PK(T) Q:K0
$PROD:I(T)
O:PK(T+1) Q:I0
O:PKT$TLAST(T) Q:I0
I:PY(T) Q:(0.5*I0)
I:PX(T) Q:(0.5*I0)
$PROD:W(T) s:1
O:PW(T) Q:130
I:PX(T) Q:65
I:PY(T) Q:65
$DEMAND:CONS s:1
D:PW(T) Q:(130*QREF(T)) P:PREF(T)
E:PL(T) Q:(100*QREF(T))
E:PK(TFIRST) Q:K0
E:PKT Q:(-1) R:TK
$CONSTRAINT:TK
SUM(T$TLAST(T), I(T)/I(T-1) - Y(T)/Y(T-1)) =G= 0;
$OFFTEXT
$SYSINCLUDE mpsgeset MOVE2_2
X.L(T) = QREF(T);
Y.L(T) = QREF(T);
W.L(T) = QREF(T);
I.L(T) = QREF(T);
K.L(T) = QREF(T);
TK.L = K0 * (1 + G) ** CARD(T);
PX.L(T) = PREF(T);
PY.L(T) = PREF(T);
PL.L(T) = PREF(T);
PK.L(T) = (1+R)*PREF(T);
PW.L(T) = PREF(T);
RK.L(T) = RK0*PREF(T);
PKT.L = SUM(TLAST, PK.L(TLAST)/(1+R));
MOVE2_2.ITERLIM = 0;
$INCLUDE MOVE2_2.GEN
SOLVE MOVE2_2 USING MCP;
$TITLE move3_1.gms. Simple Dynamic Model with off steady-state investment
$ontext
Before dynamics. Static model SAM.
Production Sectors Consumers
Markets | X Y W | CONS
------------------------------------------------------
PX | 100 -100 |
PY | 100 -100 |
PW | 200 | -200
PL | -40 -60 | 100
PK | -60 -40 | 100
------------------------------------------------------
After dynamics. SAM is consistent with a steady-state in
a base period.
| X Y W Inv | CONS
------------------------------------------------------
PX | 100 -65 -35 |
PY | 100 -65 -35 |
PW | 200 | -200
PL | -40 -60 | 100
PK | -60 -40 | 100
sav | -70 70 |
After dynamics. SAM is away of a steady-state in base period.
| X Y W Inv | CONS
------------------------------------------------------
PX | 100 -60 -40 |
PY | 100 -60 -40 |
PW | 200 | -200
PL | -40 -60 | 100
PK | -60 -40 | 100
sav | -80 80 |
$OFFTEXT
SET T /1*100/,
TFIRST(T),
TLAST(T);
TFIRST(T) = YES$(ORD(T) EQ 1);
TLAST(T) = YES$(ORD(T) EQ CARD(T));
set trep(t) period labels for investment report /1*5/,
tdecade(t) /1,10,20,30,40,50,60,70,80,90,100/;
SCALAR DELTA DISCOUNT FACTOR /0.05/
R INTEREST RATE /0.05/
G GROWTH RATE /0.02/
KZERO INITIAL CAPITAL EARNINGS /100.0/
K0 INITIAL CAPITAL STOCK
RK0 INITIAL RETURN TO CAPITAL
RKG GUESS ON INITIAL RETURN TO CAPITAL
I0 INITIAL INVESTMENT;
PARAMETER QREF(T) QUANTITIES
PREF(T) PRICES;
* IN THE ON-SS CASE, LABOR AND CAPITAL GROW AT A RATE G.
* INITIAL RETURN TO CAPITAL IS RKO = DELTA+R;
* IN THE OFF-SS CASE LABOR GROWS AT A RATE G, CAPITAL GROWS AT A
* DIFFERENT RATE, DETERMINED BY THE INITIAL RETURN TO CAPITAL
* (DENOTED AS RKG). WE DON'T KNOW RKG, SO WE MAKE A GUESS TRYING
* TO BE CONSISTENT WITH GIVEN INITIAL INVESTMENT.
RK0 = DELTA+R;
K0 = KZERO/RK0;
I0 = (DELTA + G) * K0 +10;
* RK0 is unknown, we use formulas for RK0, K0, I0 for consistency
* with the on SS dynamic model. We could also simply use the
* declarations as K0 = 1000; IO = 80; (we know these values
* from SAM).
RKG = 0.12;
QREF(T) = (1 + G) ** (ORD(T) - 1);
PREF(T) = (1/(1 + R)) ** (ORD(T) - 1);
$ONTEXT
$MODEL:DYNAMIC
$SECTORS:
X(T) ! ACTIVITY LEVEL FOR SECTOR X
Y(T) ! ACTIVITY LEVEL FOR SECTOR Y
W(T) ! ACTIVITY LEVEL FOR SECTOR W
I(T) ! INVESTMENT SECTOR
K(T) ! CAPITAL ACCUMULATION
$COMMODITIES:
PX(T) ! PRICE INDEX FOR COMMODITY X
PY(T) ! PRICE INDEX FOR COMMODITY Y
PL(T) ! PRICE INDEX FOR PRIMARY FACTOR L
PK(T) ! PRICE INDEX FOR PRIMARY FACTOR K
PW(T) ! PRICE INDEX FOR WELFARE
RK(T) ! RENTAL RATE FOR CAPITAL
PKT ! POST-TERMINAL CAPITAL CONSTRAINT
$CONSUMERS:
CONS ! INCOME LEVEL FOR CONSUMER CONS
$AUXILIARY:
TK ! TERMINAL CAPITAL STOCK
$PROD:X(T) S:1
O:PX(T) Q:100
I:PL(T) Q:40
I:RK(T) Q:(60/RKG) P:RKG
$PROD:Y(T) S:1
O:PY(T) Q:100
I:PL(T) Q:60
I:RK(T) Q:(40/RKG) P:RKG
$PROD:K(T)
O:PK(T+1) Q:(1-DELTA)
O:PKT$TLAST(T) Q:(1-DELTA)
O:RK(T) Q:1
I:PK(T) Q:1
$PROD:I(T)
O:PK(T+1) Q:I0
O:PKT$TLAST(T) Q:I0
I:PY(T) Q:(0.5*I0)
I:PX(T) Q:(0.5*I0)
$PROD:W(T) s:1
O:PW(T) Q:130
I:PX(T) Q:65
I:PY(T) Q:65
$DEMAND:CONS s:1
D:PW(T) Q:(130*QREF(T)) P:PREF(T)
E:PL(T) Q:(100*QREF(T))
E:PK(TFIRST) Q:(KZERO/RKG)
E:PKT Q:(-1) R:TK
$REPORT:
V:C(T) D:PW(T) DEMAND:CONS
V:XX(T) O:PX(T) PROD:X(T)
V:YY(T) O:PY(T) PROD:Y(T)
$CONSTRAINT:TK
SUM(T$TLAST(T), I(T)/I(T-1) - Y(T)/Y(T-1)) =G= 0;
$OFFTEXT
$SYSINCLUDE MPSGESET DYNAMIC
X.L(T) = QREF(T);
Y.L(T) = QREF(T);
W.L(T) = QREF(T);
I.L(T) = QREF(T);
K.L(T) = K0*QREF(T);
TK.L = K0 * (1 + G) ** CARD(T);
PX.L(T) = PREF(T);
PY.L(T) = PREF(T);
PL.L(T) = PREF(T);
PK.L(T) = (1+R)*PREF(T);
PW.L(T) = PREF(T);
RK.L(T) = RKG*PREF(T);
PKT.L = SUM(TLAST, PK.L(TLAST)/(1+R));
SET ITER JACOBI-SECANT ITERATIONS / ITER1*ITER6/;
PARAMETER
INVEST INVESTMENT PATH
INTRATE INTEREST RATE
CONSUM CONSUMPTION LEVEL
GROWTH CONSUMPTION GROWTH RATE
ERROR ERROR ESTIMATE BY ITERATION
RK0VAL VALUE OF RK0;
LOOP(ITER,
$INCLUDE DYNAMIC.GEN
SOLVE DYNAMIC USING MCP;
INVEST(TREP,ITER) = I.L(TREP) * I0;
INTRATE(TFIRST,ITER) = RKG-DELTA;
INTRATE(T+1,ITER)$PX.L(T+1) = PX.L(T)/PX.L(T+1) - 1;
CONSUM(T,ITER) = C.L(T);
* ERROR(ITER) = C.L("1") - (200-I0);
* ERROR(ITER) = I.L("1")*I0 - I0;
ERROR(ITER) = C.L("1") - (XX.L("1")+YY.L("1")-I0);
RK0VAL(ITER) = RKG;
IF (ORD(ITER)-1,
RKG = RKG + ERROR(ITER)/(ERROR(ITER-1)-ERROR(ITER))
* (RK0VAL(ITER)-RK0VAL(ITER-1));
ELSE
RKG = RKG + 0.01;
);
);
INVEST(TREP,"I0") = I0;
DISPLAY INVEST, INTRATE, CONSUM;
$SETGLOBAL labels TREP
$LIBINCLUDE PLOT INVEST
CONSUM(T,ITER)$(ORD(T) GT CARD(T)-5) = NA;
GROWTH(T,ITER)$(CONSUM(T+1,ITER) NE NA)=
CONSUM(T+1,ITER)/CONSUM(T,ITER)-1;
INTRATE(T,ITER)$(ORD(T) GT CARD(T)-5) = NA;
$SETGLOBAL labels TDECADE
$LIBINCLUDE PLOT CONSUM
$LIBINCLUDE PLOT GROWTH
$LIBINCLUDE PLOT INTRATE
Appendix 4. Dynamic GTAP
Appendix 4 contains Rutherford's version of dynamic GTAP.
$title A Forward-Looking GTAP Model
scalar debug Switch for debugging with steady-state calibration /0/;
set t Time periods (annual steps) /1995*2080/
t0(t) First period of the model /1995/,
tl(t) Final period of the model /2080/,
decade(t) Decade identifiers /2000, 2010, 2020, 2030, 2040, 2050,
2060, 2070, 2080 /;
parameter year(t); year(t) = 1995 + (ord(t)-1);
scalar
esubt Intertemporal elasticity of substitution /0.5/
depr Annual depreciation rate /0.07/
gterm Baseline (long-run) growth rate /0.02/
r0 Benchmark interest rate /0.05/,
pk0 Benchmark capital price
srvshr Single period survival share
termcap Terminal period capital;
* ======================================================================
* Calibration of savings and investment to a uniform steady-state
* equilibrium across all regions:
pk0 = 1 + r0;
* ======================================================================
* Read the dataset and define regional affiliations and baseline
* growth rates:
$libinclude mrtdata shock
SCALAR
esubdm Elasticity of substitution - domestic vs. imports / 4 /,
esubmm Elasticity of substitution - imports / 8 /;
parameter gr0(r) Base year growth rate /
DNK 0.02, USA 0.02, OOE 0.01, ASI 0.08, JPN 0.04, CHN 0.05, FSU 0.05, ROW 0.04/,
gr(t,r) Growth rates through the model horizon;
* First pass: check steady-state consistency:
if (debug, gr0(r) = gterm;);
parameter
qref(t,r) Reference quantity path
pref(t) Reference path of present value prices
a0(i,r) Armington supply
m0(i,r) Aggregate import supply
d0(i,r) Aggregate domestic supply
c0(i,r) Final demand by commodity
tc(i,r) Tax rate on final demand
ct0(r) Total final demand
pc0(i,r) Reference price for final demand
ld0(i,r) Base year labor demand by sector
kd0(i,r) Base year capital earnings by sector
gdp0(r) Base year GDP
k0(r) Base year capital stock;
* ======================================================================
* Adopt some notation which is different from the core GTAPinGAMS model:
a0(i,r) = vipm(i,r) + vdpm(i,r) + vigm(i,r) + vdgm(i,r) + vifm(i,r) + vdfm(i,r);
d0(i,r) = vdpm(i,r) + vdgm(i,r) + vdfm(i,r);
m0(i,r) = vipm(i,r) + vigm(i,r) + vifm(i,r);
c0(i,r) = vpm(i,r) + vgm(i,r);
tc(i,r)$c0(i,r) = (tp(i,r)*vpm(i,r) + tg(i,r)*vgm(i,r)) / c0(i,r);
ct0(r) = sum(i, c0(i,r)*(1+tc(i,r)));
pc0(i,r) = 1 + tc(i,r);
ld0(i,r) = vfm("LAB",i,r);
kd0(i,r) = vfm("CAP",i,r);
gdp0(r) = sum(i, ld0(i,r) + kd0(i,r));
parameter vk(r) Value of capital earnings
vi(r) Value of investment
viratio(r) Investment ratio
rk0(r) Benchmark rental price of capital
taxrev(r) Present value of tax revenue;
vk(r) = sum(i, kd0(i,r));
* Initially assume that Tobin's q=1 in all regions:
rk0(r) = r0 + depr;
* Then we infer the capital stock from capital earnings:
k0(r) = vk(r) / rk0(r);
* and we fix investment from the capital stock and growth rate:
vi(r) = (gr0(r)+depr) * k0(r);
parameter profit;
profit(r,i) = vom(i,r) * (1-ty(i,r)) - sum(j, vafm(j,i,r)*(1+ti(j,i,r)))
- ld0(i,r) - kd0(i,r);
display profit;
parameter c0adj Increment in consumption for investment,
i0adj Increment in investment to match benchmark
lamda Adjustment scaling factor;
c0adj(r) = max(0, vom("cgd",r)-vi(r));
i0adj(r) = max(0, vi(r)-vom("cgd",r));
lamda(r) = (1-ty("cgd",r))*i0adj(r)/sum(i, c0(i,r)*pai0(i,"cgd",r));
vafm(i,"cgd",r) = vafm(i,"cgd",r) + lamda(r) * c0(i,r);
c0(i,r) = c0(i,r) * (1 - lamda(r));
* Investment exceeds the target -- move some of the investment
* demand into final demand:
lamda(r) = (1-ty("cgd",r)) * c0adj(r) / sum(i, vafm(i,"cgd",r)*pai0(i,"cgd",r));
c0(i,r) = c0(i,r) + lamda(r) * vafm(i,"cgd",r);
vafm(i,"cgd",r) = vafm(i,"cgd",r) * (1 - lamda(r));
ct0(r) = sum(i, c0(i,r) * pc0(i,r));
vom(cgd,r) = vi(r);
profit(r,i) = vom(i,r) * (1-ty(i,r)) - sum(j, vafm(j,i,r)*(1+ti(j,i,r)))
- ld0(i,r) - kd0(i,r);
display c0adj, i0adj;
display profit;
parameter gdp(t,r) Relative GDP index
eff(t,i,r) Efficiency index (adjustment factor)
ieff(t,r) Investment efficiency index (shock parameter)
keff(t,r) Capital efficiency index (shock parameter);
ieff(t,r) = 1;
keff(t,r) = 1;
$ontext
$model:gtap
$sectors:
c(t,r) ! Private consumption
y(i,t,r)$vom(i,r) ! Output
a(i,t,r)$a0(i,r) ! Armington aggregation of domestic and imports
k(t,r) ! Capital stock
inv(t,r) ! Investment
yt(t) ! Transport
$commodities:
pc(t,r) ! Final demand
py(i,t,r) ! Output price price
pa(i,t,r)$a0(i,r) ! Armington composite price
pl(t,r) ! Wage rate
pk(t,r) ! Cost of capital
rk(t,r) ! Return to captital
ptk(r) ! Price of a unit of capital in terminal period
pt(t) ! Transport services
ptax(r) ! Present value of tax revenue
$consumers:
ra(r) ! Representative agent
govt(t,r) ! Government agent
$auxiliary:
kt(r) ! Terminal capital
theta(r) ! Steady-state share of capital stock
ta(r) ! Terminal asset
* Final demand
$prod:c(t,r) s:esubt c:1
o:pc(t,r) q:ct0(r)
i:pa(i,t,r) q:c0(i,r) p:pc0(i,r) a:govt(t,r) t:tc(i,r)
* Non-fossil fuel production (includes electricity and refining):
$prod:y(i,t,r) s:0 va:1
o:py(i,t,r) q:vom(i,r) a:govt(t,r) t:ty(i,r)
i:pa(j,t,r) q:vafm(j,i,r) p:pai0(j,i,r) a:govt(t,r) t:ti(j,i,r)
i:pl(t,r) q:ld0(i,r) va:
i:rk(t,r) q:kd0(i,r) va:
$prod:inv(t,r)
o:ptk(r)$tl(t) q:(ieff(t,r)*vi(r))
o:pk(t+1,r) q:(ieff(t,r)*vi(r))
i:py(cgd,t,r) q:vi(r)
$prod:k(t,r)
o:ptk(r)$tl(t) q:((1-depr)*k0(r))
o:pk(t+1,r) q:((1-depr)*k0(r))
o:rk(t,r) q:(keff(t,r)*rk0(r)*k0(r))
i:pk(t,r) q:k0(r)
* Armington aggregation over domestic versus imports:
$prod:a(i,t,r)$a0(i,r) s:esubdm m:esubmm s.tl(m):0
o:pa(i,t,r) q:(eff(t,i,r)*a0(i,r))
i:py(i,t,r) q:(gdp(t,r)*d0(i,r))
i:py(i,t,s) q:(gdp(t,s)*vxmd(i,s,r)) p:pmx0(i,s,r) s.tl:
+ a:govt(t,s) t:tx(i,s,r) a:govt(t,r) t:(tm(i,s,r)*(1+tx(i,s,r)))
i:pt(t)#(s) q:(gdp(t,s)*vtwr(i,s,r)) p:pmt0(i,s,r) s.tl:
+ a:govt(t,r) t:tm(i,s,r)
* International transport services (Cobb-Douglas):
$prod:yt(t) s:1
o:pt(t) q:(sum((i,r), vst(i,r)))
i:py(i,t,r) q:vst(i,r)
* Final demand:
$demand:ra(r) s:esubt
d:pc(t,r) q:(ct0(r)*qref(t,r)) p:pref(t)
e:ptk(r) q:(-k0(r)) r:kt(r)
* value of base year trade balance:
e:pc(t,"usa") q:(vb(r)*qref(t,r))
e:pl(t,r) q:(evoa("lab",r)*qref(t,r))
e:pl(t0,"usa") q:(-1) r:ta(r)
e:pk(t0,r) q:k0(r)
e:ptax(r) q:taxrev(r)
$demand:govt(t,r)
d:ptax(r)
$constraint:kt(r)
sum(t$tl(t+1), c(t,r)*inv(t+1,r) - c(t+1,r)*inv(t,r)) =e= 0;
* Region r share of aggregate assets:
$constraint:theta(r)
theta(r) * sum((tl,rr), c(tl,rr)*pc(tl,rr)*ct0(rr)
- evoa("lab",rr)*qref(tl,rr)*pl(tl,rr)
- vb(rr)*qref(tl,rr)*pc(tl,"usa"))
=e= sum(tl, c(tl,r)*pc(tl,r)*ct0(r)
- evoa("lab",r)*qref(tl,r)*pl(tl,r)
- vb(r)*qref(tl,r)*pc(tl,"usa"));
* Terminal asset adjustment:
$constraint:ta(r)
sum(t0, pl(t0,"usa")) * ta(r) =e=
theta(r) * sum(rr,ptk(rr)*kt(rr)*k0(rr)) - ptk(r)*kt(r)*k0(r);
$offtext
$sysinclude mpsgeset gtap
* Growth rates through the horizon:
scalar ssyear /2060/;
set tss(t); tss(t) = yes$(year(t) ge ssyear);
gr(t,r)$(not tss(t)) = gterm * (year(t)-1995)/(ssyear-1995) +
gr0(r) * (ssyear-year(t))/(ssyear-1995);
gr(tss,r) = gterm;
*$setglobal gp_xl decade
*$libinclude gnuplot gr
qref(t,r) = 1;
loop(t, qref(t+1,r) = qref(t,r) * (1 + gr(t,r)););
* Generate an index to control Armington demand structure:
gdp(t,r) = ( qref(t,r) * gdp0(r) / sum(rr, qref(t,rr)*gdp0(rr)) ) /
( gdp0(r) / sum(rr, gdp0(rr)) );
eff(t,i,r)$a0(i,r) = ( gdp(t,r)*d0(i,r) +
sum(s, gdp(t,s) *
(vxmd(i,s,r)*pmx0(i,s,r)+ vtwr(i,s,r)*pmt0(i,s,r))) ) / a0(i,r);
display gdp, eff;
pref(t) = (1/(1+r0))**(ord(t)-1);
pref(t) = pref(t) / sum(t0, pref(t0));
pc.l(t,r) = pref(t);
py.l(i,t,r) = pref(t);
pa.l(i,t,r) = pref(t);
pl.l(t,r) = pref(t);
pk.l(t,r) = pk0 * pref(t);
rk.l(t,r) = pref(t);
ptk.l(r) = sum(t$tl(t),pk.l(t,r))/(1+r0);
pt.l(t) = pref(t);
c.l(t,r) = qref(t,r);
y.l(i,t,r) = qref(t,r);
a.l(i,t,r) = qref(t,r);
k.l(t,r) = qref(t,r);
inv.l(t,r) = qref(t,r);
yt.l(t) = qref(t,"usa");
kt.l(r) = sum(tl, qref(tl,r))*(1 + gterm);
* Replicate the 1995 calibrated benchmark -- the deviation should be
* on the order of 1e-5:
if (debug,
gtap.iterlim = 0;
else
vb(r) = 0;
);
ta.lo(r) = -inf;
theta.lo(r) = -inf;
theta.l(r) = sum(tl, c.l(tl,r)*pc.l(tl,r)*ct0(r)
- evoa("lab",r)*qref(tl,r)*pl.l(tl,r)
- vb(r)*qref(tl,r)*pc.l(tl,"usa")) /
sum((tl,rr), c.l(tl,rr)*pc.l(tl,rr)*ct0(rr)
- evoa("lab",rr)*qref(tl,rr)*pl.l(tl,rr)
- vb(rr)*qref(tl,rr)*pc.l(tl,"usa"));
taxrev(r) = sum(t, qref(t,r) * pref(t) *
(sum((i,s), gdp(t,s) * tm(i,s,r)*( vtwr(i,s,r) +
vxmd(i,s,r)*(1+tx(i,s,r)) )
+ gdp(t,r) * tx(i,r,s) * vxmd(i,r,s) ) +
sum((i,j), ti(j,i,r)*vafm(j,i,r)) +
sum(i, ty(i,r) * vom(i,r)) +
sum(i, tc(i,r) * c0(i,r)) ));
$include gtap.gen
solve gtap using mcp;
abort$debug "Benchmark precision:", gtap.objval;
parameter capital(t,r) Capital growth rate
consum(t,r) Consumption level
interest(t,r) Interate rate (%);
capital(t,r) = 100 * (k.l(t+1,r)/k.l(t,r) - 1);
consum(t,r) = 100 * (c.l(t+1,r)/c.l(t,r) - 1);
interest(t,r)$(not tl(t)) = 100 * (pc.l(t,r)/pc.l(t+1,r)-1);
capital(tl,r) = na;
consum(tl,r) = na;
interest(tl,r) = na;
parameter tasum Check sum for terminal assets;
tasum = sum(r,ta.l(r)); display tasum;
display capital, consum, interest;
*.$setglobal gp_xl decade
*.$libinclude gnuplot capital
*.$libinclude gnuplot consum
*.$libinclude gnuplot interest
parameter
assets(t,*) Beginning of period assets
borrow(*,*) Net borrowing by period
pvdebt(*,*) Present value of debt
fvdebt(*,*) Future value of debt;
* Net borrowing from abroad equals
pvdebt(t,r) = 0;
assets(t0,r) = pk.l(t0,r)*k.l(t0,r)*k0(r);
borrow(t,r) = pc.l(t,r)*c.l(t,r)*ct0(r) + py.l("cgd",t,r)*inv.l(t,r)*vi(r)
- pl.l(t,r)*evoa("lab",r)*qref(t,r)
- rk.l(t,r)*rk0(r)*k0(r)*k.l(t,r)
- govt.l(t,r);
loop(t,
pvdebt(t+1,r) = pvdebt(t,r) + borrow(t,r);
assets(t+1,r) = pk.l(t+1,r)*k.l(t+1,r)*k0(r) - pvdebt(t,r);
);
fvdebt(t,r) = pvdebt(t,r)/pc.l(t,"usa");
loop(t0,
assets(t,r) = assets(t,r)/pc.l(t0,"usa");
borrow(t,r) = borrow(t,r)/pc.l(t0,"usa");
pvdebt(t,r) = pvdebt(t,r)/pc.l(t0,"usa");
);
*.$libinclude gnuplot assets
*.$libinclude gnuplot borrow
*.$libinclude gnuplot pvdebt
*.$libinclude gnuplot fvdebt
borrow(t,"chk") = sum(r, borrow(t,r));
pvdebt(t,"chk") = sum(r, pvdebt(t,r));
display assets, borrow, pvdebt, fvdebt;
borrow(t,"chk") = 0;
pvdebt(t,"chk") = 0;
Footnotes:
File translated from TEX by TTH, version 2.20.
On 6 Aug 2000, 13:59.