Computable General Equilibrium (CGE) Analysis: Lecture Notes by Prof. Dr. Artem Korzhenevych TU Dresden, Summer Term 2016
Introduction General context Three broad types of methods in computational economics: Statistical and econometric analysis Computable general (or partial) equilibrium analysis Agent-based modeling (micro-simulations, based on big data) Major motivation for the use of all these methods: policy evaluation and forecasting Treatment of model parameters or data: stochastic vs. deterministic CGE and econometric approach can be merged: e.g. estimation of elasticity parameters CGE and agent-based modeling can also be merged
Introduction The definitions A CGE model is a system of equations describing the behavior of the economic agents, the structure of the markets and the institutions, and the links between them, one solution to which is believed to be known from the observed data Traditional applications of CGE approach: tax policy, environmental policy, trade policy As one can suspect, a CGE model must have some relation to a general equilibrium model known from a microeconomics textbook ( a theoretical model filled with data ) GE in CGE stands for: covering all agents and markets, with no loose ends left... To better understand this link we will later review the GE theory (microeconomics)
Introduction CGE approach (statics) The micro textbook exercises on GE theory start with specifying the technology and preferences, markets setup, as well as endowments and profit distribution rules In a CGE approach you start from a set of data that records economic transactions, and not from a description of an economy in terms of mathematical functions The basic problem of modeling: how to make use of the data, such that it will be possible to perform policy analysis and forecasting The answer of a CGE approach is: assume that the data describes the equilibrium of some deterministic model, and try to come up with a guess of this model
The algorithm of CGE modeling The algorithm of CGE modeling (1) 1. Formulate a research question and collect raw data 2. Define agents, commodities, and institutions 3. Organize data in a benchmark equilibrium dataset 4. Specify market forms and prices (in case of taxes or markups) 5. Specify technology and preferences, adopt functional forms 6. Specify macroeconomic closure rules (if institutions are present), complete the mathematical formulation
The algorithm of CGE modeling The algorithm of CGE modeling (2) 7. Numerical specification (calibration) 8. Choosing solution (programming) strategy 9. Benchmark replication 10. Run experiments 11. Evaluate the outcomes 12. Sensitivity analysis
The algorithm of CGE modeling New concepts Agents: consumers and firms Commodity: anything, for which there is demand Closure: behavioral rules for institutions Institutions: government, central bank, tax collector Benchmark: initial equilibrium described by data Calibration: solving for free parameters, so that the benchmark equilibrium is reproduced Simple example: partial equilibrium
Basic concepts from microeconomics Consumer behavior A standard microeconomic model describes a toy world inhabited by many rational individuals (consumers) These individuals are capable of making consistent decisions about all kinds of issues, and act independently of all others Preferences of a given individual c are assumed to be described by a utility function u c with certain mathematical properties (e.g. strictly increasing) Consumers are price-takers, no strategic interaction They maximize utility subject to an individual budget constraint The result of their optimization can be expressed as a set of demand functions, which completely characterize the preferences
Basic concepts from microeconomics Representation of preferences A simple consumer s problem: max d c U c (d c ) s.t. p i d ic M c, i where d c is a vector of consumption quantities, p i are given product prices, and M c is income Solution is a set of (Marshalian) demand functions: d ic = d ic (p, M c ) Properties: continuous, homogenous of degree 0 If preferences are strictly monotonic (we will only have such cases) => budget constraint holds with equality
Basic concepts from microeconomics Representation of preferences An alternative (dual) approach: min d c p d c s.t. U c (d c ) U c The result is an expenditure function, which also completely characterizes the preferences E c (p, U c ) Under some additional assumptions (homothetic preferences), it is possible to get the following representation: E c (p, U c ) = e c (p) U c Then, the unit expenditure function e c (p) can be used as a natural price index for the single consumer (we will use it later)
Basic concepts from microeconomics Producer behavior Firms are characterized by technology: way of converting inputs into outputs Technology of a firm k is represented by a production function f k that has certain mathematical properties (e.g. non-increasing returns to scale) Firms have to choose both the amount of inputs and the amount of outputs, in order to maximize profits Firms use intermediate goods inputs and factor inputs Factors are owned by the households and are in fixed supply Firms are price-takers, no strategic interaction (else: non-walrasian setup)
Basic concepts from microeconomics Representation of technology Producer s problem: Step 1 - Cost function C k (p, w, g k ) = min a k,b k p a k + w b k s.t. f k (a k, b k ) g k, where a k is the intermediate inputs vector, b k is the factor inputs vector, g k is the activity level (in units of throughput ) Shephard s lemma gives the (conditional) demand functions a ik = a ik (p, w, g k ) = C k(p, w, g k ) p i b jk = b jk (p, w, g k ) = C k(p, w, g k ) w j
Basic concepts from microeconomics Representation of technology Producer s problem: Step 2 - Revenue function R k (p, g k ) = max y k p y k s.t. h k (y k ) s k (g k ), where y k is the output vector, h k (y k ) is a concave input requirement function, s k (g k ) is the scale economies relationship, and p is a given price vector The outcome is a conditional supply function: y ik = y ik (p, g k ) = R k(p, g k ) p i In textbook examples, most often h k ( ) = y k, but in CGE applications, we will often have the multiple output case
Basic concepts from microeconomics Representation of technology Producer s problem: Step 3 - Profit maximization In the end, firms choose the level of activity in order to maximize the profits π k (p, w) = max g k R k (p, g k ) C k (p, w, g k ) The optimality condition is the well known rule MR = MC R k (p, g k ) g k = C k(p, w, g k ) g k This finishes the description of firms s optimization Now, we put two sides of economy together
Basic concepts from microeconomics Consumer income The still missing link between the two parts of economy is the specification of income Individuals get income by providing factor service to the firms and getting a share of the firms profits: M c (p, w) = j w j N cj + k ω ck π k (p, w), where N cj is the endowment of individual c with factor j, ω ck is the share of individual c in the profits of firm k, c ω ck = 1 k Endowments and profit shares are exogenous, fixed parameters
Basic concepts from microeconomics General equilibrium Equilibrium in the market system is achieved when the demands of buyers match the supplies of sellers at prevailing prices in every market simultaneously We define a real-valued aggregate excess demand function for each commodity (product or factor) market: z a i (p, w) = k a ik (p, w, g k ) + c d ic(p, M c(p, w)) k y ik (p, g k ) z b j (p, w) = k b jk (p, w, g k ) c N cj And the aggregate excess demand vector is then: z(p, w) = ( z a 1 ( ),..., za I ( ), zb 1 ( ),..., zb J ( )) Walrasian equilibrium: prices that clear all markets: z(p, w ) = 0
Basic concepts from microeconomics Properties of aggregate excess demand function 1. Continuous at p and w 2. Homogenous of degree 0 It means that the system z(p, w ) = 0 has infinitely many solutions We need one more equation to fix the absolute price level This equation defines the units of account, or the numeraire But then we have more equations then unknowns! Need to drop one equation somehow Solution: Walras law
Basic concepts from microeconomics Properties of aggregate excess demand function 3. Walras law: the value of aggregate excess demand is zero at any set of positive prices: p z a (p, w) + w z b (p, w) = 0 It is a purely mathematical result that can be derived when all budget constraints of the households hold with equality The consequence of this result is that the full system of aggregate excess demands is overidentified Thus, one of the equations can be dropped and we end up with an exactly identified system
Basic concepts from microeconomics Complementarity format The optimization problems faced by firms and households actually have to be solved subject to an additional set of constraints, namely the non-negativity of prices and quantities This was first recognized in the 1930s, and the seminal contributions to the economic theory done by Arrow, Debreu, and McKenzie are all based on a system of weak inequilities, rather than equations. This approach allows for corner solutions, when some prices or quantities may be zero This syntax is not needed if you have a priori knowledge that no price and no quantity will go to zero. However, for the sake of generality, it is better to formulate the model in the complementarity syntax In CGE literature outcomes with zero prices or quantities are modeled quite often
Basic concepts from microeconomics Complementarity format So, what you do is you perform the consumer s optimization subject to non-negativity of demand, and the firm s optimization - subject to non-negativity of activity levels Then, you formulate the general equilibrium conditions as: z a (p, w ) 0 z b (p, w ) 0 p 0 w 0 p. z a (p, w ) = 0 w. z b (p, w ) = 0
Basic concepts from microeconomics The complete system The complete system has the following form: M c = j w j N cj + k θ ck π k (p, w) c R k (p,g k ) g k C k (p,w,g k ) g k g k 0 k zi a (p, w) 0 p i 0 i zj b (p, w) 0 w j 0 j If utility functions are continuous, strongly increasing, and strictly quasiconcave, while the production functions are continuous and convex, the solution to this problem exists
Basic concepts from microeconomics Extending the simple setup We will later drop the distinction between goods and factors to save on notation and to allow the consumption of factors (labour-leisure choice) State may collect taxes and make transfers, or provide public goods Mobility of goods and services (trade) as well as of production factors (intenational capital markets, migration) can be introduced More complex extensions: dynamics, uncertainty Equilibrium can also be shown to exist if taxes are introduced, or several regions (time periods), but in general not for the cases with increasing returns to scale or market failures
Some history: GE and CGE Existence The existence of the Walrasian (competitive) general equilibrum was proved by Debreu (1952), Arrow and Debreu (1954), and McKenzie (1954) The first numerical algorithm to compute this equilibrium was proposed by Scarf (1967) Any algorithm to solve for the GE as formulated above would give an approximate solution, because of nonlinearities First computational example based on Scarf algorithm is Shoven and Whalley (1972) => Applied general equlibrium (AGE) In the first AGE papers computational exercises were preceeded by the existence proofs (hard math). The solvable models had to be quite small
Some history: GE and CGE Computability Limitations on the size of the model and long computational times were a drawback of the AGE models In the meanwhile, an alternative approach was gaining popularity, that also allowed to compute the effects of economy-wide policies The major difference of this approach was that the existence of the equilibrium was a-priori assumed, moreover, it was assumed that the economic data collected for a certain year described this equilibrium The sort of equations used was also different: they did not describe individual agents, but aggregates of them: whole income classes and industries The computational procedures used were fast, if not certainly converging (Newton s method)
Some history: GE and CGE Input-output analysis The grounding father of this approach was Leontief (1941, 1951) His goal was to create a national accounting system for the USA that encompassed all branches of industry, agriculture, and services, and also the individual budgets of all private persons This idea was realized in his table of input-output accounts In addition to the table, a model of the underlying economy was needed Leontief reinterpreted Walrasian system as describing the linkages between branches of economy and made some simplifying assumptions: he used fixed coefficients assumption, thus eliminating a lot of price effects
Some history: GE and CGE Input-output analysis The corresponding system of equations was linear, which was very important in the era when the computing capacities were severely limited The assumption of fixed coefficients allowed estimating the unknown technological parameters from a single data point by simple calculations Impact analysis and forecasting essentially involved inverting one single matrix (IO slides next)
Some history: GE and CGE Input-output analysis The corresponding system of equations was linear, which was very important in the era when the computing capacities were severely limited The assumption of fixed coefficients allowed estimating the unknown technological parameters from a single data point by simple calculations Impact analysis and forecasting essentially involved inverting one single matrix (IO slides next) Leontief published the results of several applications of this method, and one of them was particularly fortunate: he was almost the only one who predicted that demand for steel will remain high in the US after the war After that, multisector input-output models were widely adopted for economy-wide analysis
Some history: GE and CGE Johansen Later, Chenery and Clark (1957) extended the original Leontief s framework by adding behavioral equations to explain final demand (consumer demands were derived from an aggregate utility function) The first successful implementation of numerical multisectoral model without the fixed-coefficients assumption of input-output analysis is due to Johansen (1960) He employed linear-logarithmic or Cobb-Douglas production functions in modeling the substitution between production factors, and a nonlinear system of demand functions Beyond this, Johansen basically invented the algorithm of CGE modeling (term CGE coined by Adelman and Robinson (1978))
Some history: GE and CGE Genesis of CGE The algorithm applied by Johansen was the following: 1. Set up a system of (nonlinear) equations characterizing an imputed macroeconomic equilibrium 2. Calculate the values of exogenous parameters based on the available data 3. Make sure that the system is exactly identified 4. Explicitly linearize the model around the initial equilibrium 5. Perform comparative static analysis by changing the values of exogenous variables Thus, CGE models are not true general equilibrium models if the latter is reserved for models devoted to the interaction of individual optimizing micro units in the economy
Some history: GE and CGE Genesis of CGE Johansen s method was reintroduced by Taylor and Black (1974), who applied it to the model of Chilean economy This method of policy analysis became increasingly popular among international organizations and governments The popularity of CGE modeling further expanded with the development of powerful modeling systems (GAMS, AMPL, GEMPACK, MATLAB) By the end of the 1980s, Walrasian foundations and the classical existence proofs were widely perceived as having no policy significance The new solution methods were further applied in the same procedure as described above, only the linearization step could now be skipped
More CGE basics Aggregation An important distinction between textbook GE and CGE approach is the size of interacting agents The problem of connecting the two (how to derive macroeconomic balancing conditions from individual optimization) is in fact the problem of consistent aggregation A generally used assumption in CGE modeling is that of a representative household and a representative firm The sum of individuals choices is mathematically equivalent to the decision of one individual, only if their demand functions are linear in expenditure and identical up to the addition of a function that is independent of expenditure (Gorman, 1953)
More CGE basics Aggregation For the firms, if the cost function of each individual firm is of generalized linear form, an exact aggregate cost function for the industry exists (Gorman, 1968) Even if these two conditions are fulfilled, the existence proofs are available for a rather short list of models and not available for many cases with market imperfections Having these issues in mind, we thus can assume functional forms for the aggregate of all households and for the aggregate of firms in the industry However, it is important to remember that the claim of micro-foundations of a CGE model is not always valid This however does not make CGE models a less useful policy analysis tool
More CGE basics A CES functional form: intro A function with constant elasticity of substitution is a workhorse of the CGE modeling, and also a victim of constant critisism It implies relatively simple expressions for the demand functions and gives some flexibility in terms of pattern of substitution It plays an important role in the new economic geography (NEG, Krugman) literature, as a basis for Dixit-Stiglitz framework Special cases of CES are the Cobb-Douglas (linear-logarithmic) and Leontief (fixed coefficients) forms
More CGE basics CES: basic forms We will consider the following form of the CES utility function U c = ϕ c ( i γ1 ρc ic and of the production function g k = ψ k ( j β1 ρ k jk d ρc ic ) 1/ρc ) b ρ 1/ρk k jk See the notes in ces.pdf to trace the derivation of other CES forms Elasticity of substitution σ = 1 1 ρ A CET function (e.g. input requirement function h k (y k )): just replace σ by θ, θ > 0
More CGE basics CES: derived forms For brevity we drop the destinction between factors and goods: i counts all commodity markets Unit cost function (Note: small-letter syntax for c, e, r!): ( ) 1 c k (p) = 1 ψ k i β ikp 1 σ 1 σ k k i Input demand function: b ik (p, g k ) = g k β ik ψ σ k 1 k ( ck p i ) σk Unit expenditure function: ( ) 1 e c (p) = 1 ϕ c i γ icp 1 σc 1 σc i Consumer demand function: d ic (p, M c ) = γ ic ϕ σc 1 M c c e c ( ec p i ) σc
More CGE basics CES calibration Calibrated share form syntax We have introduced some shift parameters in the functions that can be removed from the model formulation (thus, we don t have to calibrate them) by applying the following technique: 1. Write down all equations evaluated at the initial equlibrium 2. Express the shift and share parameters as functions of the initial values of variables 3. Substitute the derived expressions for the parameters in the general formulation of the model This is not always equivalent to just dividing both parts of equations by their benchmark values!!!