Chapter 5: A Closed-Economy One-Period Macroeconomic Model – Structured Study Notes

Model Overview

Key Features of the One-Period Closed-Economy Model

This model provides a foundational framework for macroeconomic analysis by combining the behaviors of a representative consumer, firm, and government within a single period and a closed economy (no international trade).

• Closed Economy: No imports or exports; all economic activity occurs within one country.

• Key Actors:

  • Representative Consumer: Supplies labor and purchases goods.

  • Representative Firm: Demands labor and produces goods for sale.

  • Government: Purchases goods ($G$) and finances these purchases through taxes ($T$).

• Assumptions:

  • The government budget is always balanced ($G = T$) because there is no future period to repay debt.

  • All consumers and firms are price-takers in competitive markets.

Variables in the Model

Classification of Variables

The model distinguishes between variables determined within the system (endogenous) and those set externally (exogenous).

• Exogenous Variables (Inputs):

  • Government spending ($G$)

  • Total factor productivity ($z$)

  • Capital stock ($K$)

• Endogenous Variables (Outputs):

  • Consumption ($C$)

  • Labor supply ($N^s$)

  • Labor demand ($N^d$)

  • Taxes ($T$)

  • Aggregate output ($Y$)

  • Real wage ($w$)

Competitive Equilibrium

Market Clearing and Optimization

A competitive equilibrium is achieved when all markets clear and economic agents optimize their objectives.

• Market Clearing: The labor market clears when labor supply equals labor demand: $N^s = N^d$.

• Optimizing Behavior:

  • Consumers: Maximize utility subject to their budget constraint: $C = wN^s + \pi - T$

  • Firms: Maximize profit: $\pi = Y - wN^d$

• Income-Expenditure Identity: In equilibrium, total output equals total spending: $Y = C + G$ Note: Investment ($I$) and net exports ($NX$) are zero in this closed, one-period model.

Production Possibilities Frontier (PPF)

Technological Tradeoff Between Consumption and Leisure

The PPF illustrates the maximum feasible combinations of consumption and leisure given the economy's resources and technology.

• Slope: The negative slope of the PPF represents the Marginal Product of Labor ($MP_N$), also called the Marginal Rate of Transformation ($MRT_{l,C}$).

• Feasibility: Only points where output is sufficient to cover government spending ($G$) and allow for positive consumption are feasible.

• Equilibrium Condition: In competitive equilibrium, the rate at which consumers are willing to trade leisure for consumption equals the technological rate of transformation: $MRS_{l,C} = MRT_{l,C} = MP_N = w$

Economic Efficiency & Optimality

Pareto-Optimality and Welfare Theorems

The model assesses whether market outcomes are socially efficient using the concept of Pareto-optimality and the fundamental theorems of welfare economics.

• Social Planner: A hypothetical actor who chooses consumption ($C$) and leisure ($l$) to maximize consumer welfare within the constraints of the PPF.

• Fundamental Theorems of Welfare Economics:

  • First Theorem: Under certain conditions, a competitive equilibrium is Pareto-optimal.

  • Second Theorem: Any Pareto optimum can be achieved as a competitive equilibrium.

• Key Finding: In this model, the competitive equilibrium and the Pareto optimum coincide, occurring where the consumer's indifference curve is tangent to the PPF.

Summary Table: Key Variables and Relationships

Variable	Type	Role in Model
$G$	Exogenous	Government spending; sets fiscal policy
$T$	Endogenous	Taxes; finances government spending ($G = T$)
$z$	Exogenous	Total factor productivity; affects output
$K$	Exogenous	Capital stock; input for production
$C$	Endogenous	Consumption; determined by equilibrium
$N^s$, $N^d$	Endogenous	Labor supply and demand; market clearing ($N^s = N^d$)
$Y$	Endogenous	Aggregate output; $Y = C + G$
$w$	Endogenous	Real wage; determined by equilibrium

Example: Competitive Equilibrium Calculation

• Suppose government spending ($G$) is set at 100, total factor productivity ($z$) is 2, and capital stock ($K$) is 50.

• Consumers and firms optimize their choices, leading to equilibrium values for consumption ($C$), labor ($N$), and real wage ($w$).

• In equilibrium, the following conditions must hold: $Y = C + G$ $N^s = N^d$ $MRS_{l,C} = MP_N = w$

Additional info: The notes expand on the basic points by providing definitions, examples, and a summary table for clarity. The competitive equilibrium and Pareto-optimality concepts are central to understanding efficiency in this model.