" .. in all cases where the repercussions of proposed policies are widespread, there is no real alternative to CGE."
Nobel Laurette in Economics, Kenneth J. Arrow (2005, p. 13), Personal Reflections on Applied General Equilibrium Models, in Frontiers in Applied General Equilibrium Models, Kehoe, Srinivasan, and Whalley (eds), Cambridge University Press, 2005.

Introduction: The Theory of General Equilibrium Model

Overview

General equilibrium theory is a formalization of the observation that real-world markets are interdependent; changes in supply or demand conditions usually have repercussions on supply and demand conditions, and thus equilibrium prices, on several other markets. Computable general equilibrium (CGE) modeling is an attempt to use general equilibrium theory as an operational tool for empirically oriented analyses of resource allocation issues in market economies.

The objective of this overview is to let readers who are familiar with microeconomics at the (advanced) undergraduate level, know that CGE model is not something new. It is a concept that they are already familiar with. Most who are new, especially students, to CGE modelling or who just heard about it may think that CGE model is something complicated Note_3 . This introduction (and also this book) attempt to give an idea that the principle of CGE modelling is something that is not beyond reach by even an undergradute students in Economics.

Another different and not less important message, however, is that good understanding of microeconomics at rather advance undergraduate levels is a necessary conditions for a good comprehension of CGE modeling. There are many cases, where anyone can carry out analysis using a CGE model especially those with good computing skills. Without good understanding of microeconomics principle, they are like users of a machine without understanding of how the machines work. They punch a button in a PC keyboards, producing some numbers, but can not explain what is going on.

The simplest CGE model is illustrated in figure 1.1, where most (advance) undergraduate students in Economics should be familiar with. This implies that basic understanding of Microeconomics (at advance undergraduate level) is a pre-requisite to understanding the principle of CGE modelling.

Figure

Figure 1. 2-sector, 2-factor closed GE model

Recalling the final year microeconomics, the above figure is a representation of firm's and household's optimization where both firms and households clear their supply and demand for factors (K and L) and commodities at the market place. Recall that the unknown (or endogenous variable) in this model is MATH, while the known (exogenous) variables are $K^{S}$ and $L^{S}$. If we know the production function of $X_{1}$ and $X_{2}$ (or the shape of their isoquant map) and the utility function (or the shape of indifference map), then in principle we can solve this model to find all the unknown (endogenous) variable Note_4 .

We can also say that the job of a CGE model (like the one in figure 1) is to find the resource allocation tha maximize welfare (utility) i.e. how input or resource ($K^{S}$ and $L^{S}$) are allocated among industries or how much are MATH? The shape of isoquant map and indifference map is represented by production function (technology) and utility function (taste). Hence, for example if the technology and taste are represented with MATH, MATH, and MATH, it can be shown that the optimal resource allocation are (see example 1 below for derivation),MATHAs explicitly shown by that (closed-form) solution, resource allocation is determined by economy's endowment of resource ($K^{S}$ and $L^{S}$), parameter of technology ($\alpha $ and $\beta $), and taste parameter ($\gamma $).

The CGE model, however complicated or extended, carry the same principle, of which the biggest magic of all is it can solve for the (unobservable) set of prices.

Using this model, we then can make a comparative static analysis to ask such question as, how much all endogenous variables change if there is an increase in supply of capital or labor, labor or capital-saving technical change, or even increase in indirect taxes with little addition to production functions, or price relations.

Example with Cobb-Douglas Technology and Taste

This example will show that we can even solve a 2 sector and 2 factor, CGE model by hand if the technology and taste are simple enough. Let production function of industy 1 MATH, and industry 2 MATH, and consumer's maximize utility function of MATH.

Industry 1 maximize profit,

MATHwith first order conditionsMATHand these are simplified to,

MATH

Similarly, for industry 2MATH

Since both firm face the same factor prices,MATH

While, household maximize utility subject to budget constraint, Note_5

MATH

The lagrangian function is,

MATHfirst order condition,MATHcombining the first 2 of the first order conditions,MATH

To summarize, now we haveMATH

From 1.3,MATHsubstituting this to 1.1 and 1.2,MATHorMATHorMATHbut we haveMATHso, letMATHby substitutionMATH

MATHOnce we now $K_{1}$ and $L_{1}$, we can findMATHMATHand once, we know MATH then we know output MATH, and MATH.

How do we know prices? First we have to set one price as numeraire, let $P_{1}=1$, so all other prices (including factor prices) is relative to $P_{1} $.

We haveMATHso we know solution for $P_{2}$, we also haveMATHand finallyMATH

Hence we have the solution to all endogenous variables in the GE model. To summarize, the solution for real or quantity variables are MATHwhile for price variables are

MATH

Functional forms, Calibration, and Critiques

To make CGE a tool for empirical analyisis not only model to play with hyphothetically, CGE model needs data.This data is supposed to be a representation of economy's initial (general) equilibrium.

Calibration is finding or calculating parameter of the CGE model, for example technological and taste parameters (MATH), such that they reflect the economy's initial equilibrium as represented by data (such as Input-Output Table, or Social Accounting Matrix).

Let's rewrite the first order condition of industry 1MATHSince in constant returns to scale technology, or by zero profit competitive condition, MATH, $\alpha $ is in fact the share of capital payment for industry 1, and accordingly $1-\alpha $ is the share of labor payment for industry 1MATHSimilary for industry 2, its cost share areMATHTherefore to find technological parameter in the economy i.e. $\alpha $ and $\beta $, we only need to know every industry labor and capital cost share Note_6 , and this is for example, relatively easy to obtain from Input-Output Table that record labor and capital payment of every industry. This way of calculating $\alpha $ and $\beta $ is called calibration in the CGE modelling, and it guarantee that the parameters calculated represent theoretically-consistent and reflect the economy's data, or in CGE term reflect the economy's initial equilibrium.

Calibration for taste parameter is similar. Let's rewrite equation 1.3MATHsubstituting this into budget constraint,MATHsimilary for $X_{2}$MATHHence, $\gamma $ is expenditure share of $X_{1}$ and $1-\gamma $ is expenditure share of $X_{2}$ and this as well can easily obtained (calibrated) using, for example, Input-Output Table.

Reading. Good article in the Economist, "Economic models: Big questions and big numbers", Jul 13th 2006 (Attached).

Example of calibration and numerical solution to GE model

Suppose from Input-Output Table, we have the following data $WL_{1}=68.05$, $WL_{2}=340.13$, $RK_{1}=278.55$, and $RK_{2}=613.17$. From these, we can calculate MATH, and MATH. Now, we can do the calibration to calculate $\alpha ,\beta ,$ and $\gamma $, i.e.MATH

Now, how do we know $K^{S}$ and $L^{S}$? First we know that MATH, and MATH, then we may assume that initially $W=R=1$ (initialization like this does matter to the final solution, but it will not matter for comparative static analysis, i.e. percentage change in all variables in the comparative static is totally independent to this initialization). Then now we know that $K^{S}=891.72$, and $L^{S}=408.18$. Using the solution we derived earlier, we can calculate all the endogenous variables

Real Variables

$K_{1}$

$278.55$

$K_{2}$

$613.17$

$L_{1}$

$68.05$

$L_{2}$

$340.13$

$X_{1}$

$211.22$

$X_{2}$

$496.9$

Price Variables

$P_{1}$

$1.00$

$P_{2}$

$1.17$

$W$

$0.61$

$R$

$0.61$

As mentioned earlier calibration ensure that the solution represent the initial equilbirum and it has to be able to reproduce the data we have i.e. MATH. A quick look of the solution suggest that it won't. However, we have to remember that all price variables are relative to $P_{1}$ that is why we set $P_{1}=1$. We called $P_{1}$ numeraire, such that if we multiply this by any amount all price variables will change by the same proportion and all real variables are intact. So, let's multiply $P_{1}$ by $1.64$. Our solution will become,

Real Variables

$K_{1}$

$278.55$

$K_{2}$

$613.17$

$L_{1}$

$68.05$

$L_{2}$

$340.13$

$X_{1}$

$211.22$

$X_{2}$

$496.9$

Price Variables

$P_{1}$

$1.64$

$P_{2}$

$1.92$

$W$

$1$

$R$

$1$

Then, now we can confirm that our solution re-produce the data (initial equilibrium), orMATH

More General Representation

Firm's Optimization

More specifically

Firm $i$'s problem, where $i\in 1,2$ isMATHand this will produce demand for factorsMATH

Household's Optimization

Household's problem isMATHwhereMATHand this will produce demand for commoditMATH

Market Clearing for Commodities and Factors

These demand have to be equal to their supply at the market for goodsMATH

In factor market, demands have to be equal to their supplyMATH

Definition

Walras' Law. Walras' law states that if markets for all but one good are in equilibrium, then all markets must be in equilibrium and the economy is in general equilibrium. If there is $n$ market in the economy, and we know that there is equilibrium in $n-1$ market, that we this law guarantee that all the $n$ market is in equilibrium.

Definition

Numeraire. Numeraire is a price of a commodity such that all prices (including factor prices) are relative to this price. When a numeraire is multiplied by a factor, it will produce a proportional increase in all nominal variabes in the GE model, while leaving all real (quantity) variables unchanged. This result is called 'price' or 'nominal' homogeneity of GE model.

Definition

The First Fundamental Theorem of Welfare Economics. In welfare economics, the first welfare theorem is that a competitive market economy will simultaneously lead to a Pareto efficient equilibrium and general competitive equilibrium. This was first demonstrated mathematically by economists Kenneth Arrow and Gerard Debreu