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Moving from Static to Dynamic CGE MODEL
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Moving from Static to Dynamic General Equilibrium Economic Models
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MIT Joint Program on the
Science and Policy of Global Change
Moving from Static to Dynamic
General Equilibrium Economic Models
(Notes for a beginner in MPSGE)
Sergey Paltsev
Technical Note No. 4
June 2004
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Printed on recycled paper
Moving from Static to Dynamic General Equilibrium
Economic Models (Notes for a beginner in MPSGE)
Sergey Paltsev
Joint Program on the Science and Policy of Global Change
Massachusetts Institute of Technology
June 1999 (Revision April 2004)
Abstract
The document is intended to serve as a guide for beginners in MPSGE. It starts with a short introduction to
the class of economic problems which can be solved with MPSGE, followed by a detailed description of step-by-step
transformation of a simple static general equilibrium model into a dynamic Ramsey model. The model is based on
a simplified dataset. Two cases are considered: the first dataset represents an economy on the steady-state growth
path and the second dataset is off the steady-state growth path in a base year. The paper includes GAMS-MPSGE
codes which can be copied and used as a starting point for further exploration of dynamic economic modelling.
Address for correspondece:
Sergey Paltsev
77, Massachusetts Ave., E40-429
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
Phone: (617) 253-0514
Fax: (617) 253-9845
e-mail: paltsev@mit.edu
http://web.mit.edu/paltsev/www
From Static to Dynamic Mo dels 2
1 Introduction
Originally, this document has been written as notes for myself during my struggle with understand-
ing how to build a dynamic general equilibrium model based on Lau, Pahlke, and Rutherford (1997)
paper
1
. I subsequently discovered that the paper could be quite helpful to other novice modellers.
The paper is organized around a simple two-sector, two-factor model. 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 model
2
. 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 based on the example of
a simple Shoven-Whalley (1984) model, 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. The author is greatly thankful to
Thomas Rutherford for correcting endless errors.
2 MPSGE and General Economic Equilibrium Models
MPSGE (Mathematical Programming System for General Equilibrium Analysis) is a programming
language designed by Thomas Rutherford (1999) in the early 80s for solving Arrow-Debreu eco-
nomic equilibrium models. MPSGE uses the programming language GAMS (General Algebraic
Modeling System) as an interface
3
. As a result, if one wants to use MPSGE, he needs to learn
basic GAMS syntax (Brook et al, 2003) as well. As it follows from the name, MPSGE solves com-
putable general equilibrium (CGE) economic models. These models consist of economic agents
who interact among each other through prices which emerge from markets for goods and factors
of production. 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 mo del, 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 consumer
4
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 con-
sumer’s utility
5
. The solution to this problem (which is a simple Arrow-Debreu (1954) problem)
1
The previous version of the document and its Spanish translation kindly made by Juan Carlos Segura can be
found (as of April 2004) at: http://web.mit.edu/paltsev/www/current.html
2
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) and Rasmussen
and Rutherford (2001).
3
To learn more about GAMS, visit their homepage at http://www.gams.com
4
Or, equivalently, a population of identical households.
5
It may sound strange 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.
From Static to Dynamic Mo dels 3
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.
max W (X, Y ) s.t p
x
X + p
y
Y = wL + rK
X = F (K
x
, L
x
)
Y = G(K
y
, L
y
)
L = L
x
+ L
y
K = K
x
+ K
y
where W is a utility function; p
x
and p
y
are the prices of goods X and Y ; r is a price of capital
K; w is a price of lab or L; K
x
and L
x
are capital and labor used in the production sector X; K
y
and L
y
are capital and labor used in the sector Y . This is a standard microeconomic textbook
optimization problem, and a usual technique for finding the solution is the method of Lagrange
multipliers. This problem can be solved in GAMS as a non-linear programming (NLP).
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.
For the purposes of the models presented in this paper, complimentarity really plays no role.
Therefore, I just briefly describe my understanding of MCP. An interested reader is referred to
Rutherford (1995) paper. I consider a nonlinear complimentarity problem (NCP), which is also a
special case of MCP. It can be written as:
Given: f : R
n
→ R
n
Find: z ∈ R
n
s.t. f(z) ≥ 0, z ≥ 0, z
T
f(z) = 0.
I have tried to convert the mathematical symbols into simple English, but it has not helped me
very much. 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 condition
6
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, I found out that these words is just a usual scientific way of hiding something rather
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 = (x
1
, x
2
, ..., x
n
), then there is a
system of n equations like ¯xf(¯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 formu-
lated 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 solv-
ing 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
6
An expression written as x
T
y = 0 (when x ≥ 0 and y ≥ 0) means x
i
y
i
= 0, for all i = 1, ..., n. The variables x
i
and y
i
are called a complementary pair and are said to be complements to each other.
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