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 tech-

nology in the short period, such that z and K are given.

1. (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?

2. (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.

3. (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?

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4. (d) Write down the four equations that can be used to solve for these four essential endoge- nous variables, and describe where do they come from. Describe in details the steps of how would you solve for the competitive equilibrium (without actually solving for it ).

5. (e) Setup the Social planner’s problem for this economy. What is the marginal condition that pins down the planner’s choices.

6. (f) Now, describe what would happen to c∗, l∗, and w∗ if the firm had more capital, such that Knew > Kold.

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.