A Closed-Economy One-Period Macroeconomic Model PDF - University of South Carolina 2025

A Closed-Economy One-Period Macroeconomic Model (Ch. 5) Sang Min Lee University of South Carolina Spring 2025 Sang Min Lee Competitive Equilibrium Spring 2025 1 / 46 Contents 1 Introduction 2 Government...

A Closed-Economy One-Period Macroeconomic Model (Ch. 5) Sang Min Lee University of South Carolina Spring 2025 Sang Min Lee Competitive Equilibrium Spring 2025 1 / 46 Contents 1 Introduction 2 Government 3 Competitive Equilibrium 4 Optimality 5 Comparative Statics – Working with the Model 6 Distorting Taxes on Wages and Laffer Curve 7 Summary Sang Min Lee Competitive Equilibrium Spring 2025 2 / 46  Introduction Section 1: Introduction Sang Min Lee Competitive Equilibrium Spring 2025 3 / 46  Introduction Introduction Having talked about the representative consumer and the representative firm, our goal is to build a working model of macroeconomy. We will start the model by analyzing consumers and firms that interact in markets in a closed economy. A closed economy is a model of a single country that has no interaction with the rest of the world – it does not trade with other countries. In the later half of the course, we will study an open economy where international trade is allowed. Sang Min Lee Competitive Equilibrium Spring 2025 4 / 46  Government Section 2: Government Sang Min Lee Competitive Equilibrium Spring 2025 5 / 46  Government Government Many macroeconomic models have government. The role of government is quite simple in our model. It purchases a given quantity of consumption goods, G and finances the purchase by taxing the representative consumer. The government must abide by the government budget constraint which we write as G = T, or, government purchases equal taxes, in real terms. Sang Min Lee Competitive Equilibrium Spring 2025 6 / 46  Competitive Equilibrium Section 3: Competitive Equilibrium Sang Min Lee Competitive Equilibrium Spring 2025 7 / 46  Competitive Equilibrium Macroeconomic Model We have looked at the behavior of the representative consumer, the representative firm, and the government. Now, it is time to construct the model to show how consistency can be obtained in the actions of all economic agents. Making use of the model is a process of running experiments to determine how changes in the exogenous variables change the endogenous variables. Remember the comparative statics? In doing so, we hope to understand the real-world macroeconomic events and look at macroeconomic policies (positive questions). Sang Min Lee Competitive Equilibrium Spring 2025 8 / 46  Competitive Equilibrium Competitive Equilibrium By consistency, we mean that, given market prices, demand is equal to supply in each market in the economy. (Q: How many markets do we have?) Such state is called a competitive equilibrium. “Competitive” refers to the fact that all consumers and firms are price-takers, and the economy is in “equilibrium” when the actions of all consumers and firms are consistent. When demand equals supply in a markets, we say that the market clears. Let us now look at the full model and then learn how to define a competitive equilibrium. Sang Min Lee Competitive Equilibrium Spring 2025 9 / 46  Competitive Equilibrium Representative Agent Model Exogenous Endogenous Model Variables Variables Figure 1: Macroeconomic Model Exogenous variables are determined outside the model. Given the exogenous variables, the model determines the endogenous variables. The model helps us understand how the endogenous variables change when there are changes in exogenous variables. Sang Min Lee Competitive Equilibrium Spring 2025 10 / 46  Competitive Equilibrium Formal Definition of a Competitive Equilibrium A competitive equilibrium is a set of endogenous variables (prices and allocations) such that given the exogenous variables, the following conditions are satisfied: 1 Given wages w , taxes T , dividend income π, the representative consumer chooses the level of consumption C and labor supply N s to maximize their utility given their budget constraint. 2 Given wages w , TFP z, and capital stock K , the representative firm chooses N d to maximize profits. 3 The profits that the representative firm earns are equal to the dividend income that is received by the consumer. 4 Labor market clears: N s = N d. 5 Goods market clears: Y = C + G. 6 Government budget constraint is satisfied: G = T. Q: What are the exogenous variables? In-class exercise: (Walras’s Law) Show that condition #5 automatically holds. (Hint: definition of profits, budget constraints) Sang Min Lee Competitive Equilibrium Spring 2025 11 / 46  Competitive Equilibrium Gross Domestic Product An important property of a competitive equilibrium is Y = C + G. Recall, the income-expenditure identity from the GDP measurement chapter: Y = C + I + G + NX. In this model, NX = 0 as this is a closed economy, and I = 0 as there is no dynamics. (Investment expenditures are for the future.) Sang Min Lee Competitive Equilibrium Spring 2025 12 / 46  Competitive Equilibrium Competitive Equilibrium - Graphical Approach There are many ways to work with a macroeconomic model - algebraically (math + pen/paper), numerically (math + computer), and graphically (graphs + pen/paper). We will use the graphical approach to understanding the competitive equilibrium. In doing so, we will examine the consumer’s and the firm’s decisions in the same diagram, so that we can determine how the aggregate consistency is achieved in competitive equilibrium. Let us start with the production function of the representative firm. Sang Min Lee Competitive Equilibrium Spring 2025 13 / 46  Competitive Equilibrium Production function (a) shows the equilibrium relationship between the quantity of labor employed by the representative firm and aggregate output. (b) is the mirror image of the production function in (a). Sang Min Lee Competitive Equilibrium Spring 2025 14 / 46  Competitive Equilibrium Production Possibilities Frontier (c) shows the production possibilities frontier (PPF). PPF is the technological relationship between C and l, determined by shifting the relationship in (b) down by the amount G. The shaded region represents consumption bundles that are technologically feasible to produce in this economy. Sang Min Lee Competitive Equilibrium Spring 2025 15 / 46  Competitive Equilibrium Marginal Rate of Transformation Notice that the slopes in (b) and (c) are the same and given by −MPN. Another name for the negative of the slope of the PPF is marginal rate of transformation. It tells us the rate at which leisure can be converted in the economy into consumption goods through production. Marginal rate of transformation The marginal rate of transformation is the rate at which one good can be converted technologically into another. MRTl,C = MPN = −(the slope of PPF) Sang Min Lee Competitive Equilibrium Spring 2025 16 / 46  Competitive Equilibrium Adding Consumer Competitive Equilibrium The figure brings together consumer’s preferences and firm’s production technology to determine a competitive equilibrium. J is the equilibrium consumption bundle. ADB is the budget constraint faced by the consumer where the slope AD is equal to the minus real wage. Sang Min Lee Competitive Equilibrium Spring 2025 17 / 46  Competitive Equilibrium Summarizing Equilibrium condition The equilibrium condition for the economy is: MRSl,C = MRTl,C = MPN As the consumer and the firm face the same market real wage in equilibrium, the following two are the same: 1 The rate at which the consumer is just willing to trade leisure for consumption 2 The rate at which leisure can be converted into consumption goods using the production technology In-class exercise: explain why MRSl,C ̸= MRTl,C cannot happen in equilibrium. (Focus on what will happen to market wages if the two differ.) Sang Min Lee Competitive Equilibrium Spring 2025 18 / 46  Optimality Section 4: Optimality Sang Min Lee Competitive Equilibrium Spring 2025 19 / 46  Optimality Pareto-Optimality In economics, socially-optimal ≡ Pareto-optimal. Pareto-optimal An equilibrium is Pareto optimal if there is no way to rearrange production or to reallocate goods so that someone is made better off without making someone else worse off. In a representative consumer setup, there is only one person. So Pareto-optimality implies maximizing the utility of the representative consumer Preview of the result: competitive equilibrium = pareto-optimal equilibrium Why do we care? Philosophical reason: free markets can deliver socially optimal outcomes Technical reason: solving the social optimum is often times easier than solving the competitive equilibrium. Sang Min Lee Competitive Equilibrium Spring 2025 20 / 46  Optimality Social Planner Social planner: hypothetical agent who can simply dictate what agents do: 1 Representative firm: how much labor to hire and how much consumption goods to produce. 2 Representative consumer: how much labor to supply and how much consumption goods to consume. However, the social planner has resource constraints. They cannot create consumption goods or human hours out of thin air. Social planner is benevolent, and they chooses quantities to make the representative consumer as well off as possible. Sang Min Lee Competitive Equilibrium Spring 2025 21 / 46  Optimality Social Planner’s Constrained Optimization Problem The social planner wants to maximize the representative consumer’s utility subject to the feasibility constraints (= resource constraints). max U(C , l) C ,l subject to C + G = Y Y = zF (K , h − l) C ≥0 0≤l ≤h Sang Min Lee Competitive Equilibrium Spring 2025 22 / 46  Optimality Solution (for Students Comfortable with Calculus) Let λ1 and λ2 denote Lagrange multipliers for the first two constraints, respectively. Then, the solutions to the problem is as follows:   ∂U MUc ≡ − λ1 = 0, ∂C     ∂U ∂ (zF ) MUl ≡ − λ2 MPN ≡ = 0, and ∂l ∂(h − l) λ1 − λ2 = 0 The three equations yield MUl MRSl,C = = MPN. MUC The above solution is equivalent to the solution of the competitive equilibrium. Sang Min Lee Competitive Equilibrium Spring 2025 23 / 46  Optimality Graphical Solution The Pareto optimum is a point that a social planner would choose where the representative consumer is as well off as possible given the technology. The Pareto optimum is point B where an indifference curve is tangent to the PPF. Sang Min Lee Competitive Equilibrium Spring 2025 24 / 46  Optimality Welfare Theorems First Welfare Theorem The first fundamental welfare theorem of welfare economics states that, under certain conditions, a competitive equilibrium is Pareto optimal. Second Welfare Theorem The second fundamental welfare theorem of welfare economics states that, under certain conditions, a Pareto optimum is a competitive equilibrium. The two theorems are referring to the set of competitive equilibria and the set of Pareto optima. The equivalence of the competitive equilibrium and the Pareto optimum in the classroom model enables us to solve the model easily. We will be solving for the Pareto optimum, instead of the competitive equilibrium. Sang Min Lee Competitive Equilibrium Spring 2025 25 / 46  Optimality Inefficiencies Our model so far was efficient and socially optimal. In real world, several factors result in inefficiency of a market economy. 1 Externalities 1 Can be either positive or negative 2 Essentially a problem of market incompleteness (goods or services that are not being priced.) 3 Positive externality in colleges? 2 Distorting taxes: pro rata income tax vs lump sum taxes 3 Monopolistic behavior 4 Problems with price mechanism – price rigidities Remember, in our model, MRSl,C = MRTl,C. Social benefit = social costs. Inefficiencies tend to break down these equations. Sang Min Lee Competitive Equilibrium Spring 2025 26 / 46  Comparative Statics – Working with the Model Section 5: Comparative Statics – Working with the Model Sang Min Lee Competitive Equilibrium Spring 2025 27 / 46  Comparative Statics – Working with the Model Effect of G ↑ Suppose the government in this economy decides to increase government spending. (Change in an exogenous variable) We want to find the effect of such an increase on the variables of the economy. (Change in endogenous variables) Recall, that in equilibrium G = T. That is, an increase in G implies an increase in T. Thus, by income effect – the consumption and leisure for the consumer will decrease. Because, leisure goes down =⇒ employment will increase. (Recall, N s = h − l) Because, employment increases =⇒ the quantity of output increases. (Recall, MPN > 0) Sang Min Lee Competitive Equilibrium Spring 2025 28 / 46  Comparative Statics – Working with the Model Effect of G ↑ (Con’t) The income-expenditure identity is Y = C + G. Therefore, C = Y − G. Therefore, we have: ∆C = ∆Y − ∆G where, ∆ denotes the “changes in”. ∆G > 0 crowds out consumption: ∆C < 0 However, from the previous slide, we have that, ∆Y > 0. Therefore, private consumption is not completely crowded out. Sang Min Lee Competitive Equilibrium Spring 2025 29 / 46  Comparative Statics – Working with the Model What happens to real wage when G ↑? An ↑ in G leads to an ↑ in employment. The equilibrium employment rises, therefore, The representative firm would hire more labor only in response to a reduction in the market real wage. Thus, the real wage w ↓. Sang Min Lee Competitive Equilibrium Spring 2025 30 / 46  Comparative Statics – Working with the Model Effect of G ↑ - Graphical Approach Since competitive equilibrium is Pareto-optimal, we will use Social Planner’s problem. Initial equilibrium - Point A where indifference curve I1 is tangent to the PPF PPF1. If G goes up, the PPF shifts to PPF2 and the new equilibrium point is at B. Consumption and leisure are both lower. (Assumption: they are both normal goods.) ∆C is minus the distance AE. The PPF becomes steeper as l increases. Therefore, the real wage falls as a result of the increase in government spending. Sang Min Lee Competitive Equilibrium Spring 2025 31 / 46  Comparative Statics – Working with the Model Effect of TFP ↑ Suppose the total factor productivity (z) increases in this economy. What happens to endogenous variables? An increase in z shifts the production function up: more output can be produced given the same quantity of labor input. It also increase the marginal product of labor for each quantity of labor input. (Real wages go up.) Thus, more consumption is attainable given the better technology, for any quantity of leisure consumed. Workers therefore have now more income, given the number of hours worked =⇒ C ↑ and l ↑ (Recall, income effect). But, the increase in wage =⇒ the consumers substitute away from leisure to consumption. Therefore, C ↑ and l ↓ (Recall, substitution effect). The effect on leisure is ambiguous as we do not know which effect dominates (income or substitution effect). Sang Min Lee Competitive Equilibrium Spring 2025 32 / 46  Comparative Statics – Working with the Model Effect of TFP ↑ - Graphical Approach Initial equilibrium - Point A where indifference curve I1 is tangent to the PPF PPF1. If z goes up, the PPF shifts to PPF2 and the new equilibrium is point B. Consumption goes up. The change in leisure is ambiguous. Can decompose the change in two effects – income and substitution effect. The shift from point A to point D is the substitution effect. Notice that the new equilibrium is on a higher indifference curve I2. Sang Min Lee Competitive Equilibrium Spring 2025 33 / 46  Comparative Statics – Working with the Model Interpretations The interpretation of an increase in z and the resulting effect depends on what we take one period in the model to represent relative to time in the real world. One period could be interpreted as: 1 “years” – in which case we are capturing the long-run effects 2 “month”, “quarter”, or “year” – in which case we are studying the short-run effects Once we look at the effects, we provide interpretation in terms of the short-run and long-run economic implications. Sang Min Lee Competitive Equilibrium Spring 2025 34 / 46  Distorting Taxes on Wages and Laffer Curve Section 6: Distorting Taxes on Wages and Laffer Curve Sang Min Lee Competitive Equilibrium Spring 2025 35 / 46  Distorting Taxes on Wages and Laffer Curve Distorting Taxes on Wages So far, comparative statics in the competitive equilibria (=Pareto-optimal equilibria) Now, a version of the model with distorting tax. Distorting taxes are taxes that distort the allocation of resources by changing the relative prices, thereby affecting supply and demand. With distorting taxes, a competitive equilibrium generally is not Pareto-optimal. Therefore, we cannot use the social planner’s problem as before. The distorting tax we consider: a proportional tax on wages. We are interested in discussing fiscal policy issues, including the effects of income taxation. Sang Min Lee Competitive Equilibrium Spring 2025 36 / 46  Distorting Taxes on Wages and Laffer Curve Simplified Model Proportional Income Taxation 3 agents - representative consumer, representative firm, and government. The representative firm produces according to the production function: Y = zN d Exercise Show that this production function exhibits CRS. Representative consumer’s budget constraint: C = w (1 − t)(h − l) + π The government uses tax revenues for government spending, G = wt(h − l). Sang Min Lee Competitive Equilibrium Spring 2025 37 / 46  Distorting Taxes on Wages and Laffer Curve Labor Demand Curve The representative firm’s profits are given by π = zN d − wN d = (z − w )N d The firm chooses N d to make π as large as possible. (z − w ) is the profit that the firm makes for each unit of labor input. If z = w , then the profits are zero for the firm no matter what it does, so the firm is indifferent concerning how much labor to hire. Therefore, the firm’s demand curve for labor is infinitely elastic at the wage w = z. Sang Min Lee Competitive Equilibrium Spring 2025 38 / 46  Distorting Taxes on Wages and Laffer Curve Production Possibility Frontier Recall, the PPF is a graph in consumption-leisure space. Moreover, in equilibrium we have: Nd = Ns = h − l C +G =Y Therefore, we can write the PPF as: C = z(h − l) − G The PPF is, therefore, a straight line denoted by AB. Sang Min Lee Competitive Equilibrium Spring 2025 39 / 46  Distorting Taxes on Wages and Laffer Curve Competitive Equilibrium The competitive equilibrium is point H. Point E is the Pareto-optimum. Thus, a proportional tax is distorting in the sense that the income tax distorts the private decisions. And, therefore, the competitive equilibrium is not socially efficient. Q: Why is H the point where the PPF and B.C. intersect? (Hint: definition of the two curves & G = wt(h − l)) Sang Min Lee Competitive Equilibrium Spring 2025 40 / 46  Distorting Taxes on Wages and Laffer Curve Government Revenue Suppose that the goal of the government is to maximize G. What t should it set? The tax revenue that the government can collect at income tax rate t is REV = tz [h − l(t)] where, REV is total revenue from the income tax, t is the tax rate, and z[h − l(t)] is the tax base. The total tax collected depends not only on the tax rate t but also the size of the tax base. It is possible for tax revenue to go down when t increases (if the size of the tax base goes down). Sang Min Lee Competitive Equilibrium Spring 2025 41 / 46  Distorting Taxes on Wages and Laffer Curve Laffer Curve Laffer curve shows the quantity of tax revenue generated by the government as a function of tax rate. The curve AB is called the Laffer curve. If G is the amount of government spending that needs to be financed, then in equilibrium: G = tz[h − l(t)] When t = 0 and t = 1, the government cannot raise any money. The government can maximize the revenue by setting t = t ∗. If the government wishes to finance government spending, it can set the tax rate to be either t1 (the good side) or t2 (the bad side). Sang Min Lee Competitive Equilibrium Spring 2025 42 / 46  Distorting Taxes on Wages and Laffer Curve Multiple Equilibria Given that there are two possible tax rates for the government, t1 and t2 for any given G , let us look at the CE again. The CE with low tax rate t1 is given by point F. The CE with high tax rate t2 is given by point H. In the low tax-rate equilibrium consumption and output are higher and leisure is lower, than in the high tax-rate equilibrium. Since, F is on a higher indifference curve, the consumer is better off in the equilibrium with low tax rate than in the one with high tax rate. Sang Min Lee Competitive Equilibrium Spring 2025 43 / 46  Summary Section 7: Summary Sang Min Lee Competitive Equilibrium Spring 2025 44 / 46  Summary Summary We covered the idea of a competitive equilibrium with three agents (consumer, firm, and government) and two markets (goods and labor). We learned the equilibrium conditions and their implication on Pareto optimality. You should be able to conduct comparative statics with the model. (What are the effects of changes in G or z?) You should be able to understand the effects of distorting taxes and the Laffer Curve. Sang Min Lee Competitive Equilibrium Spring 2025 45 / 46  Summary Questions? Sang Min Lee Competitive Equilibrium Spring 2025 46 / 46