Question

4. Country A is located on a small island that is isolated from the outside world. The country has one representative consumer, whose preference is represented by:

   U(c,l) = ln(c) + ln(l)

There is one representative firm in the economy 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 owns the capital it uses and it’s production technology is represented by

   F(K,N) = zKαN1−α

where z is the total factor productivity, and α is the capital share of income. The firm pays all of its profit back to consumer as dividend. The government of country A spend a pre-determined G amount of goods to provide public services, and it taxes consumer through a lump-sum tax t to finance the expenditure. the government’s budget is balanced:

G = t

Now, suppose that the firm cannot change its capital stock, or improve its production technology in the short period, such that z and K are given.

(a) When describing this economy as an macroeconomic 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 concept of Competitive Equilibrium?

(b) Define the competitive equilibrium for this economy, be specific about what it is, what conditions have to be satisfied, who solves what problem and etc.

(c) List the set of equations that will be used to determine all of the endogenous variables. Which four of these endogenous variables are essentials, such that once you know these four, all other endogenous variables can be obtained easily?

Answer 1

Here is your answer:-

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).