Steady with patience

In economics, an input-output model uses a matrix representation of a nation's (or a region's) economy to predict the effect of changes in one industry on others and by consumers, government, and foreign suppliers on the economy. Wassily Leontief (1905-1999) is credited with the development of this analysis. Francois Quesnay developed a cruder version of this technique called Tableau économique[经济表]. Leontief won the Nobel Memorial Prize in Economic Sciences for his development of this model. And, in essence, Léon Walras's work Elements of Pure Economics on general equilibrium theory is both a forerunner and generalization of Leontief's seminal concept. Leontief's contribution was that he was able to simplify Walras's piece so that it could be implemented empirically. The International Input-Output Association is dedicated to advance knowledge in the field input-output study, which includes "improvements in basic data, theoretical insights and modelling, and applications, both traditional and novel, of input-output techniques."

Input-output depicts inter-industry relations of an economy. It shows how the output of one industry is an input to each other industry. Leontief put forward the display of this information in the form of a matrix. A given input is typically enumerated in the column of an industry and its outputs are enumerated in its corresponding row. This format, therefore, shows how dependent each industry is on all others in the economy both as customer of their outputs and as supplier of their inputs. Each column of the input-output matrix reports the monetary value of an industry's inputs and each row represents the value of an industry's outputs. Suppose there are three industries. Column 1 reports the value of inputs to Industry 1 from Industries 1, 2, and 3. Columns 2 and 3 do the same for those industries. Row 1 reports the value of outputs from Industry 1 to Industries 1, 2, and 3. Rows 2 and 3 do the same for the other industries.

While most uses of the input-output analysis focuses on the matrix set of interindustry exchanges, the actual focus of the analysis from the perspective of most national statistical agencies, which produce the tables, is the benchmarking of gross domestic product. Input-output tables therefore are an instrumental part of national accounts. As suggested above, the core input-output table reports only intermediate goods and services that are exchanged among industries. But an array of row vectors, typically aligned below this matrix, record non-industrial inputs by industry like payments for labor; indirect business taxes; dividends, interest, and rents; capital consumption allowances (depreciation); other property-type income (like profits); and purchases from foreign suppliers (imports). At a national level, although excluding the imports, when summed this is called "gross product originating" or "gross domestic product by industry." Another array of column vectors is called "final demand" or "gross product product consumed." This displays columns of spending by households, governments, changes in industry stocks, and industries on investment, as well as net exports. In any case, by employing the results of an economic census which asks for the sales, payrolls, and material/equipment/service input of each establishment, statistical agencies back into estimates of industry-level profits and investments using the input-output matrix as a sort of double-accounting framework.

The mathematics of input-output economics is straightforward, but the data requirements are enormous because the expenditures and revenues of each branch of economic activity has to be represented. As a result, not all countries collect the required data and data quality varies, even though a set of standards for the data's collection has been set out by the United Nations through its System of National Accounts (SNA): the replacement for the current 1993 SNA standard is pending. Because the data collection and preparation process for the input-output accounts is necessarily labor and computer intensive, input-output tables are often published long after the year data was collected--typically as much as 5-7 years after. Moreover, the economic "snapshot" the benchmark version of the tables provide of the economy's cross-section are taken only once every few years, at best. Although many developed countries estimate input-output accounts annually and with much greater recency.

In addition to studying the structure of national economies, input-output economics has been used to study regional economies within a nation, and as a tool for national and regional economic planning. Indeed, it may well be that a main use of input-output analysis is that for measuring the economic impacts of events as well as public investments or programs. But it is also used to identify economically related industry clusters and also so-called "key" or "target" industries--industries that are most likely to enhance the internal coherence of a specified economy. By linking industrial output to satellite accounts articulating energy use, effluent production, space needs, and so on, input-output analysts have extended the approaches application to a wide variety of uses.

Despite the clear ability of the input-output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy. In 2003, Mohammad Gani, a pupil of Leontief, introduced Consistency Analysis in his book 'Foundations of Economic Science', which formally looks exactly like the input-output table, but explores the dependency relations in terms of payments and intermediation relations. Consistency analysis explores the consistency of plans of buyers and sellers by decomposing the input-output table into four separate matrices, each for a different kind of means of payment. It integrates micro and macroeconomics in one model and deals with money in a fully ideology-free manner. It deals with the flow of funds vis-a-vis the movement of goods. In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.

 

The Arrow-Debreu model, also referred to as the Arrow-Debreu-McKenzie model suggests that, should the assumptions made about the conditions under which it works hold (i.e. convexity, perfect competition and demand independence), then there will be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.

The model (ADM model) is the central model in the General (Economic) Equilibrium Theory and often used as a general reference for other microeconomic models. It is named after Kenneth Arrow, Gerard Debreu and Lionel W. McKenzie.

Compared to earlier models, the Arrow-Debreu model radically generalized the notion of a commodity, differentiating commodities by time and place of delivery. So, for example, 'apples in New York in September' and 'apples in Chicago in June' are regarded as distinct commodities. The Arrow-Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods and in all places.

The ADM model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. Once we can prove the existence of such an equilibrium, it is possible to show that it is unique.

The Arrow-Debreu model specifies the conditions of perfectly competitive markets.

In financial economics the term Arrow-Debreu is most commonly used with reference to an Arrow-Debreu security. A canonical Arrow-Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (a so called "state price"). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow-Debreu securities.

The concept of Arrow-Debreu security is a good pedagogical tool to understand pricing and hedging issues in derivatives analysis. Its practical use in financial engineering, however, has turned out to be very limited, especially in the multi-period or continuous markets.

It is undeniable that options contain invaluable information on investors preferences and beliefs. Since the work of Breeden and Lizenberger (1978), a large number or researchers have used options to extract Arrow-Debreu prices for a variety of applications in financial economics.

The Black Scholes analysis and its extensions, despite their strongly formulated and somewhat questionable assumptions, have proven more successful in practice and have contributed directly to the exponential growth in the size of the global derivatives industry over the past 30 years.

 

Applied General Equilibrium (AGE) models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow-Debreu General equilibrium model with empirical data, to provide "a general method for the explicit numerical solution of the neoclassical model" (Scarf with Hansen 1973: 1)

Scarfs method was an algorithm (based on simplical subdivisions) that would narrow a ‘net’ around the possible solution to the quantified general equilibrium problem. With enough iteration, the net was tightened sufficiently to choose a cut off point, giving a price vector that could clear the market.

Brouwer's Fixed Point theorem states that a continuous mapping of a simplex into itself has at least one fixed point. This paper describes a numerical algorithm for approximating in a sense to be explained below, a fixed point of such a mapping (Scarf 1967a: 1326).

Scarf never built an AGE model, but hinted that “these novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environment” (Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price vector could be solved for any changes in policies (or exogenous shocks), giving the equilibrium ‘adjustments’ needed for the prices. This method was first used by Shoven and Whalley (1972 and 1973), but was used up through the 1970s by Scarf’s students and students’ students.

Scarf’s simplex method was quite appealing to the economic theorists at the time, who wanted constructive proofs for the existence of equilibrium. Lance Taylor noted that as a solution algorithm, it ignored 2nd derivatives and curvature information, thus being much less effective than Newton methods for the highly convex model specifications which AGE modelers at the time were creating.

Furthermore, in a recent article, Velupillai (2006) proves how the AGE models, can not be precisely solved numerically and points out that "From a very elementary (classical) recursion theoretic standpoint it is easy to show the absence of a computable (and constructive) content " (Velupillai 2006: 366).

AGE models developed separately and independently from Computable general equilibrium (CGE) models, as illustrated in Mitra-Kahn (2008). However, today the two terms are used inter-changeably, and the Arrow-Debreu theoretical background of AGE models, is then attributed to CGE models, which are in widespread use across the world. This is a confusion as the AGE literature has no connections with the CGE models until the mid 1980's when AGE models are going out of fashion due to the high cost of implementing them, and the AGE modellers begin to adopt the CGE methods of using Social Accounting Matrices for their data set-ups, and macro balancing equations as opposed to Arrow-Debreu and General Equilibrium Theory.

AGE models, being based on Arrow-Debreu general equilibrium theory works in a different manner than CGE models. The model first establishes the existence of equilibrium through the standard Arrow-Debreu exposition, and then inputs data into all the various sectors, and then apply Scarf’s algorithm (Scarf 1967a, 1967b and Scarf with Hansen 1973) to solve for a price vector that would clear all markets instantly. This algorithm would narrow down the possible relative prices through a simplex method, which kept reducing the size of the 'net' within which possible solutions were found. AGE modelers then consciously choose a cutoff, and set an approximate solution as the net never closed on a unique point through the iteration process.

CGE models, are based on macro balancing equations, and use an equal number of equations (based on the standard macro balancing equations) and unknowns solvable as simultaneous equations, where exogenous variables are changed outside the model, to give the endogenous results. This process is uncontroversial, but also completely separate from AGE modelling and formal general equilibrium theory.

 

Real Business Cycle Theory (or RBC Theory) is a class of macroeconomic models in which business cycle fluctuations to a large extent can be accounted for by real (in contrast to nominal) shocks. (The four primary economic fluctuations are secular (trend), business cycle, seasonal, and random.) Unlike other leading theories of the business cycle, it sees recessions and periods of economic growth as the efficient response to exogenous changes in the real economic environment. That is, the level of national output necessarily maximizes expected utility, and government should therefore concentrate on the long-run structural policy changes and not intervene through discretionary fiscal or monetary policy designed to actively and discretionary smooth economic short-term fluctuations.

It is associated with freshwater economics (the Chicago school of economics).

According to RBC theory, business cycles are therefore "real" in that they do not represent a failure of markets to clear, but rather reflect the most efficient possible operation of the economy, given the structure of the economy. It differs in this way from other theories of the business cycle, like Keynesian economics and Monetarism, which see recessions as the failure of some market to clear.

Economists have come up with many ideas to answer the above question. The one which currently dominates the academic Real Business Cycle Theory literature was introduced by Finn E. Kydland and Edward C. Prescott in their seminal 1982 work “Time to Build And Aggregate Fluctuations.” They envisioned this factor to be technological shocks i.e., random fluctuations in the productivity level that shifted the constant growth trend up or down. Examples of such shocks include innovations, bad weather, imported oil price increase, stricter environmental and safety regulations, etc. The general gist is that something occurs that directly changes the effectiveness of capital and/or labour. This in turn affects the decisions of workers and firms, who in turn change what they buy and produce and thus eventually affect output. RBC models predict time sequences of allocation for consumption, investment, etc. given these shocks.

But exactly how do these productivity shocks cause ups and downs in economic activity? Let’s consider a good but temporary shock to productivity. This momentarily increases the effectiveness of workers and capital. Also consider a world where individuals produce goods they consume. The problem with this reasoning is that on aggregate level, this shock would average out.

Individuals face two types of trade offs. One is the consumption-investment decision. Since productivity is higher, people have more output to consume. An individual might choose to consume all of it today. But if he values future consumption, all that extra output might not be worth consuming in its entirety today. Instead, he may consume some but invest the rest in capital to enhance production in subsequent periods and thus increase future consumption. This explains why investment spending is more volatile than consumption. The life cycle hypothesis argues that households base their consumption decisions on expected lifetime income and so they prefer to “smooth” consumption over time. They will thus save (and invest) in periods of high income and defer consumption of this to periods of low income.

Unlike estimation, which is usually used for the construction of economic models, calibration only returns to the drawing board to change the model in the face of overwhelming evidence against the model being correct; this inverts the burden of proof away from the builder of the model. Since RBC models explain data ex post, it is very difficult to falsify any one model that could be hypothesised to explain the data. RBC models are highly sample specific, leading some to believe that they have little or no predictive power.

Crucial to RBC models, "plausible values" for structural variables such as the discount rate, and the rate of capital depreciation are used in the creation of simulated variable paths. These tend to be estimated from econometric studies, with 95% confidence intervals. If the full range of possible values for these variables is used, correlation coefficients between actual and simulated paths of economic variables can shift wildly, leading some to question how successful a model which achieves a coefficient of 80% really is.

 

Dynamic stochastic general equilibrium modeling (abbreviated DSGE or sometimes SDGE or DGE) is a branch of applied general equilibrium theory that is increasingly influential in contemporary macroeconomics. The DSGE methodology attempts to explain aggregate economic phenomena, such as economic growth, business cycles, and the effects of monetary and fiscal policy, on the basis of macroeconomic models derived from microeconomic principles. One of the main reasons macroeconomists have begun to build DSGE models is that unlike more traditional macroeconometric forecasting models, DSGE macroeconomic models should not, in principle, be vulnerable to the Lucas critique.

Structure of DSGE models
As their name indicates, DSGE models are dynamic, studying how the economy evolves over time. They are also stochastic, taking into account the fact that the economy is affected by random shocks such as technological change, fluctuations in the price of oil, or errors in macroeconomic policy-making. This contrasts with the static models studied in Walrasian general equilibrium theory, applied general equilibrium models and computable general equilibrium models.

Traditional macroeconometric forecasting models used by central banks in the 1970s, and even today, estimated the dynamic correlations between prices and quantities in different sectors of the economy, and often included thousands of variables. Since DSGE models are technically more difficult to solve and analyze, they tend to abstract from so many sectoral details, and include far fewer variables: just a few variables in theoretical DSGE papers, or on the order of a hundred variables in the experimental DSGE forecasting models now being constructed by central banks.

What DSGE models give up in sectoral detail, they attempt to make up in logical consistency, because they are founded on microeconomic principles of constrained decision-making. Therefore, DSGE models must spell out the following aspects of the economy.

Preferences: the objectives of the agents in the economy must be specified.

For example, households might be assumed to maximize a utility function over consumption and labor effort. Firms might be assumed to maximize profits.
Technology: the productive capacity of the agents in the economy must be specified.

For example, firms might be assumed to have a production function, specifying the amount of goods produced, depending on the amount of labor and capital they employ. Technological constraints on agents' decisions might also include costs of adjusting the capital stock, the level of employment, or the price level.
Institutional framework: the institutional constraints under which economic agents interact must be specified. In many DSGE models, this might simply mean that agents make their choices within some exogenously imposed budget constraints, and that prices are assumed to adjust until markets clear. It might also mean specifying the rules of monetary and fiscal policy, or even how policy rules and budget constraints change depending on a political process.

 

By specifying preferences (what the agents want), technology (what the agents can produce), and institutions (the way they interact), it is possible (in principle, though challenging in practice) to solve the DSGE model to predict what is actually produced, traded, and consumed. In principle, it is also possible to make valid predictions about the effects of changing the institutional framework.

In contrast, as Robert Lucas pointed out, such a prediction is unlikely to be valid in traditional macroeconometric forecasting models, since those models are based on observed past correlations between macroeconomic variables. These correlations can be expected to change when new policies are introduced, invalidating predictions based on past observations.

Given the difficulty of constructing accurate DSGE models, most central banks still rely on traditional macroeconometric models for short-term forecasting. However, the effects of alternative policies are increasingly studied using DSGE methods. Since DSGE models are constructed on the basis of assumptions about agents' preferences, it is possible to ask whether the policies considered are Pareto optimal, or how well they satisfy some other social welfare criterion derived from preferences.

At present two competing schools of thought form the bulk of DSGE modeling.

Real business cycle (RBC) theory builds on the neoclassical growth model, under the assumption of flexible prices, to study how real shocks to the economy might cause business cycle fluctuations. The paper of Kydland and Prescott (1982) is often considered the starting point of RBC theory and of DSGE modeling in general. The RBC point of view is surveyed in Cooley (1995).
New-Keynesian DSGE models build on a structure similar to RBC models, but instead assume that prices are set by monopolistically competitive firms, and cannot be instantaneously and costlessly adjusted. The paper that first introduced this framework was Rotemberg and Woodford (1997). Introductory and advanced textbook presentations are given by Galí (2008) and Woodford (2003). Monetary policy implications are surveyed by Clarida et al. (1999).

Example
The European Central Bank (ECB) has developed a DSGE model, often called the Smets-Wouters model, which it uses to analyze the economy of the Eurozone as a whole. The model is intended as an alternative to the Area-Wide Model (AWM), a more traditional empirical forecasting model which the ECB has been using for several years. The ECB webpage that describes the Smets-Wouters model also discusses the advantages of building a DSGE model instead of relying on more traditional methods.

The equations in the Smets-Wouters model describe the choices of three types of decision makers:

households, who choose how much to work, to consume, and to invest;

firms, which choose how much labor and capital to employ;

and the central bank, which controls monetary policy.

The parameters in the equations were estimated using Bayesian statistical techniques so that the model approximately describes the dynamics of GDP, consumption, investment, prices, wages, employment, and interest rates in the Eurozone economy. In order to accurately reproduce the sluggish behavior of some of these variables, the model incorporates several types of frictions that slow down adjustment to shocks, including sticky prices and wages, and adjustment costs in investment.

Critics, such as Willem Buiter, have argued that DSGE models can be misleading. In his blog for the Financial Times, Buiter has argued that DSGE models rely excessively on an assumption of complete markets, and are unable to describe the highly nonlinear dynamics of economic fluctuations, making training in 'state of the art' macroeconomic modeling 'a privately and socially costly waste of time and resources'.

N. Gregory Mankiw, regarded as one of the founders of New Keynesian DSGE modeling, has also argued that

'New classical and new Keynesian research has had little impact on practical macroeconomists who are charged with ... policy. ... From the standpoint of macroeconomic engineering, the work of the past several decades looks like an unfortunate wrong turn.'
Replying to Mankiw, Michael Woodford argues that DSGE models are commonly used by central banks today, and have strongly influenced policy makers like Ben Bernanke. However, he argues that what is learned from DSGE models is not so different from traditional Keynesian analysis:

 

The Lucas critique, named for Robert Lucas's work on macroeconomic policymaking, says that it is naïve to try to predict the effects of a change in economic policy entirely on the basis of relationships observed in historical data, especially highly aggregated historical data.

The basic idea pre-dates Lucas's contribution, but in a 1976 paper he drove home the point that this simple notion invalidated policy advice based on conclusions drawn from estimated system of equation models. Because the parameters of those models were not structural – that is, not policy-invariant – they would necessarily change whenever policy – the rules of the game – was changed. Policy conclusions based on those models would therefore potentially be misleading. This argument called into question the prevailing large-scale econometric models that lacked foundations in dynamic economic theory.

The Lucas critique suggests that if we want to predict the effect of a policy experiment, we should model the "deep parameters"  (relating to preferences, technology and resource constraints) that govern individual behavior. We can then predict what individuals will do taking into account the change in policy, and then aggregate the individual decisions to calculate the macroeconomic effects of the policy change.

The Lucas critique was influential not only because it cast doubt on many existing models, but also because it encouraged macroeconomists to build microfoundations for their models. Microfoundations had always been thought to be desirable; Lucas convinced many economists they were essential. Later Finn Kydland and Edward Prescott pioneered the use of microfoundations to formulate macroeconomic models. Contemporary macroeconomic models microfounded on the interaction of rational agents are often called dynamic stochastic general equilibrium (DSGE) models.

Examples
One important application of the critique is its implication that the historical negative correlation between inflation and unemployment, known as the Phillips Curve, could break down if the monetary authorities attempted to exploit it. Permanently raising inflation in hopes that this would permanently lower unemployment would eventually cause firms' inflation forecasts to rise, altering their employment decisions.

For an especially simple example, note that Fort Knox has never been robbed. However, this does not mean the guards can safely be eliminated, since the incentive not to rob Fort Knox depends on the presence of the guards.

In other words, with the heavy security that exists at the fort today, criminals are unlikely to attempt a robbery because they know they are unlikely to succeed. But a change in security policy, like eliminating the guards for example, would lead criminals to reappraise the costs and benefits of robbing the fort. So just because there are no robberies under the current policy does not mean this should be expected to continue under all possible policies. Likewise, just because high inflation was associated with low unemployment under early-twentieth-century monetary policy does not mean we should expect high inflation to lead to low unemployment under all alternative monetary policy regimes.

 

Robert Emerson Lucas, Jr. (born September 15, 1937, Yakima, Washington) is an American economist at the University of Chicago. He was named among the 10 best economists, and received the Nobel Memorial Prize in Economic Sciences in 1995. He is married to economist Nancy Stokey. His ex-wife, Rita Lucas, upon their divorce in 1988, had a clause placed in their divorce settlement that she would receive half of any Nobel Prize won by Lucas in the next seven years. When Lucas did win the Nobel Prize in 1995 (falling just within the time limit), she was awarded half of the prize money.

He received his B.A. in History in 1959 and Ph.D. in Economics in 1964, both from the University of Chicago. He taught at the Graduate School of Industrial Administration (now Tepper School of Business) at Carnegie Mellon University until 1975, when he returned to the University of Chicago. Lucas studied Economics for his PhD on "quasi-Marxist" grounds. He believed that economics was the true driver of history, and so he planned to fully immerse himself in economics and then migrate back to the history department.

One of the most influential economists since the 1970s, he challenged the foundations of macroeconomic theory (previously dominated by the Keynesian economics approach), arguing that a macroeconomic model should be built as an aggregated version of microeconomic models (while noting that aggregation in the theoretical sense may not be possible within a given model). He developed the "Lucas critique" of economic policymaking, which holds that relationships that appear to hold in the economy, such as an apparent relationship between inflation and unemployment, could change in response to changes in economic policy. This led to the development of New Keynesian economics and the drive towards microeconomic foundations for macroeconomic theory.

Lucas is also well known for his investigations into the implications of the assumption of rational expectations. He developed the Lucas-Islands model, which suggests that people are tricked by unsystematic parts of monetary policy, the Lucas-Uzawa model (with Hirofumi Uzawa) of human capital accumulation, and stated the "Lucas paradox" why not more capital is flowing from developed countries to developing countries.

大师的结论也是用数学方法得出来，因为不同的参数对测量结果没有影响，然后举例证明，于是就出现了又一个著名悖论。

迷茫中......