Question

4. Country A is located on a small island that is isolated from the outside workd. The country has one representative consumer, whose preference is represented by There is one representative firm in the econony which is owned by the consumer, it produces one type of good that can be used for consumption or government expenditure using capital and labour as inputs. The firm ows the capital it-and it's production technology represented by where: is the total factor productivity, and a is the capital shase of income. The firm pays all of its prolfit back to consumer as dividend. The government of country A spend a pre-determined G amount of goods to provide publie services, and it taxes consumer throagh a lump-sum tax t to finance the expenditure the government's badget is balanoed G-t Now, suppone that the finm cannot change its capital stock, or improve its production tech nology in the short period, such that and K are given (a) When escibing this economy as an macroecosomic model, what is the set of exogenous variables? What is the set of endogenous variables that can be determined given the set of exogenous variables using the conoept of Competitive Equilabrinum? conditions have to be satisfied, who solves what problem and etc Which four of these endogenous variables are essentials, such that once you kno these (b) Define the competitive equilibeium for this econony, be specifie about what it is, what (c) List the set of equations that wil be used to determine all of the endogenous variables. four, all other endogenous variables can be obtained easily? (d) Write down the four equations that can be ssed to solve for these Sor essential enxloge nous variables, and describe where do they come from. Describe in details the steps of (e) Setup the Social planner's problem for this ecomoy. What is the marginal condition i the firm had more capital, such how would you solve for the competitive equilibriam (without actually solving for it ). that pins down the planner's choices (f) Now, describe what would happen to e, an that K>Kol

Answer 1

a.An exogenous variable is a variable that is not affected by other variables in the system. For example, take a simple causal system like farming. Variables like weather, farmer skill, pests, and availability of seed are all exogenous to crop production. Exogenous comes from the Greek Exo, meaning “outside” and gignomai, meaning “to produce.” In contrast, an endogenous variable is one that is influenced by other factors in the system. In this example, flower growth is affected by sunlight and is therefore endogenous.

Endogenous variables are used in econometrics and sometimes in linear regression. They are similar to (but not exactly the same as) dependent variables. Endogenous variables have values that are determined by other variables in the system (these “other” variables are called exogenous variables).

Each household and each firm in the economy act independently from each other,seeking their own interest, and taking as given the fact that other agents will also seektheir best. In the previous section we have described the behavior of each agent inthe economy, here we show how all individual actions aggregate into the behavior of the whole economy. For this purpose, we use the notion of equilibrium, meaning that in each market aggregate demand equals aggregate supply, so the correspondingequilibrium price “clears the market”.We make a simplifying assumptions. We assume that all households are the same.Identical consumers behave in identical ways, so it’s enough to analyze the behaviorof one consumer, the representative consumer of the economy. We make the same assumption for firms. We assume all firms have the same CRS technology, and just study the behavior of the representative firm. Representative household (who buysgood and sells labor) and firm (who sells goods and buys labor) play the role of astand-in for all consumers and firms in the economy. The third actor is the governmentthat buys goods and taxes agents to finance such purchases.

Definition of CE

We need to distinguish the exogenous variables determined outside the model, and treated as parameters into the model from the endogenous variables determined inequilibrium.

b.Competitive equilibrium is the traditional concept of economic equilibrium, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated.

Competitive Equilibrium. There are three requirements for a competitive equilibrium, corresponding

to the requirements that producers optimize, consumers optimize, and that ”markets clear” at the equilibrium

prices. An equilibrium will then consist of a production plan yj for each firm, a consumption vector

xi for each consumer, and a price vector p.

ProfitMaximization: For every firm the set of inputs used and outputs produced maximize profit at

those prices given the firms technology.

UtilityMaximization: For each consumer the consumption bundle is maximal for i in the budget

set defined by the initial endowment (valued at the equilibriumprices) and their share of the profits

of the firms in the economy.

Market Clearing: The total consumption of products by consumers is equal to initial endowments

plus the net output of firms.

c.A macroeconometric model like the US model is a set of equations designed to explain the economy or some part of the economy. There are two types of equations: stochastic, or behavioral, and identities. Stochastic equations are estimated from the historical data. Identities are equations that hold by definition; they are always true.

There are two types of variables in macroeconometric models: endogenous and exogenous. Endogenous variables are explained by the equations, either the stochastic equations or the identities. Exogenous variables are not explained within the model. They are taken as given from the point of view of the model. For example, suppose you are trying to explain consumption of individuals in the United States. Consumption would be an endogenous variable-a variable you are trying to explain. One possible exogenous variable is the income tax rate. The income tax rate is set by the government, and if you are not interested in explaining government behavior, you would take the tax rate as exogenous.

Specification

It is easiest to consider what a macroeconometric model is like by considering a simple example. The following is a simple multiplier model. Ct is consumption, It is investment, Yt is total income or GDP, Gt is government spending, and rt is the interest rate. The t subscripts refer to period t.

(1) Ct = a1 + a2Yt + et

(2) It = b1 + b2rt + ut

(3) Yt = Ct + It + Gt

Equation (1) is the consumption function, equation (2) is the investment function, and equation (3) is the income identity. Equations (1) and (2) are stochastic equations, and equation (3) is an identity. The endogenous variables are Ct, It, and Yt; they are explained by the model. rt and Gt are exogenous variables; they are not explained.

The specification of stochastic equations is based on theory. Before we write down equations (1) and (2), we need to specify what factors we think affect consumption and investment in the economy. We decide these factors by using theories of consumption and investment. The theory behind equation (1) is simply that households decide how much to consume on the basis of their current income. The theory behind equation (2) is that firms decide how much to invest on the basis of the current interest rate. In equation (1) consumption is a function of income, and in equation (2) investment is a function of the interest rate. The theories behind these equations are obviously much too simple to be of much practical use, but they are useful for illustration. In practice it is important that we specify our equations on the basis of a plausible theory. For example, we could certainly specify that consumption was a function of the number of sunny days in period t, but this would not be sensible. There is no serious theory of household behavior behind this specification.

et and ut are error terms. The error term in an equation encompasses all the other variables that have not been accounted for that help explain the endogenous variable. For example, in equation (1) the only variable that we have explicitly stated affects consumption is income. There are, of course, many other factors that are likely to affect consumption, such as the interest rate and wealth. There are many reasons that not all variables can be included in an equation. In some cases data on a relevant variable may not exist, and in other cases a relevant variable may not be known to the investigator. We summarize the effects of all of the left out variables by adding an error term to the equation. Thus, the error term et in equation (1) captures all the factors that affect consumption other than current income. Likewise, the error term ut in equation (2) captures all the factors that affect investment other than the interest rate.

Now, suppose that we were perfectly correct in specifying that consumption is solely a function of income. That is, contrary to above discussion, suppose there were no other factors that have any influence on consumption except income. Then the error term, et, would equal zero. Although this is unrealistic, it is clear that one hopes that consumption in each period is mostly explained by income. This would mean that the other factors explaining consumption do not have a large effect, and so the error term for each period would be small. This means that the variance of the error term would be small. The smaller the variance, the more has been explained by the explanatory variables in the equation. The variance of an error term is an estimate of how much of the left hand side variable has not been explained. In macroeconomics, the variances are never zero; there are always factors that affect variables that are not captured by the stochastic equations.

Equation (3), the income identity, is true regardless of the theories one has for consumption and investment. Income is always equal to consumption plus investment plus government spending (we are ignoring exports and imports here).

d.that exogenous variables are not explained within the model. Say that we are in period t-1 and we want to use the model to forecast consumption, investment, and income for period t. We must first choose values of government spending and the interest rate for period t. Since t is in the future, we do not know for certain what government spending and the interest rate will be in period t, and so we must make our best guesses based on available information. (For example, we might use projected government budgets.) In other words, our model forecasts what consumption, investment, and income will be in period t if the values we chose for government spending and the interest rate in period t are correct. We must also choose values for the error terms for period t, which in most cases are taken to be zero. Given the exogenous variable values, the values of the error terms, and the coefficient estimates, equations (1), (2), and (3) are three equations in three unknowns, the three unknowns being the three endogenous variables, Ct, It, and Yt. Thus, the model can be solved to find the three unknowns.

A model like equations (1)-(3) is called simultaneous. Income is an explanatory variable in the consumption function, and consumption is a variable in the income identity. One cannot calculate consumption from equation (1) unless income is known, and one cannot calculate income from equation (3) unless consumption is known. We thus say that consumption and income are "simultaneously" determined. (Investment in this model is not simultaneously determined because it can be calculated once the value for the exogenous variable rt has been chosen.)

The above model, even though it is simultaneous, is easy to solve by simply substituting equations (1) and (2) into (3) and solving the resulting equation for Yt. Once Yt is solved, Ct can then be solved. In general, however, models are not this simple, and in practice models are usually solved numerically using the Gauss-Seidel technique. The steps of the Gauss-Seidel technique are as follows:

Guess a set of values for the endogenous variables.

Using this set of values for the right hand side variables, solve all the equations for the left hand side variables.

Step 2 yields a new set of values of the endogenous variables. Replace the initial set with this new set, and solve for the left hand side variables again.

Keep replacing the previous set of values with the new set until the differences between the new set and the previous set are within the required degree of accuracy. When the required accuracy has been reached, "convergence" has been attained, and the model is solved. The right hand side values are consistent with the computed left hand side values.

If a variable computed by an equation is used on the right hand side of an equation that follows, usually the newly computed value is used rather than the value from the previous iteration. It is not necessary to do this, but it usually speeds convergence. The usual procedure for the above model would be to guess a value for Yt, compute Ct from equation (1) and It from equation (2), and use the computed values to solve for Yt in equation (3). This new value of Yt would then be used for the next pass through the model.

e.The social planner maximizes utility of households subject to aggregate resource allocation constraint of the economy..If the solutions coincide, then the outcome is Pareto optimal.

f. if the firm has more capital there would be greater investment and wealth which will improve consumers standard of living as there will be more oppurtunities created so that people can enjoy high economic strata.