ECON 3020 Intermediate Macroeconomics

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1 ECON 3020 Intermediate Macroeconomics Chapter 5 A Closed-Economy One-Period Macroeconomic Model Instructor: Xiaohui Huang Department of Economics University of Virginia c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 1 / 51

2 Main Topics Government sector Competitive equilibrium How do consumers and firms interact in markets? How is equilibrium obtained in their actions? How can unconstrained markets produce socially efficient outcomes? Analyze macroeconomic issues: Effects of an increase in government spending. Effects of an increase in total factor productivity. An example of inefficient equilibrium: distortionary government tax. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 2 / 51

3 Model Structure Closed economy: markets are restricted to a single country; no trade. The representative consumer Buys goods from the firm. Works for the firm and earns labor income. Trade-off between consuming and working. The representative firm Hires labor to produce consumption goods. Sells consumption goods to the consumer. Trade-off between revenue and cost. Government In equilibrium, both markets (goods and labor) need to be cleared under the given prices. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 3 / 51

4 Government Sector Government spending is financed by taxes. Government budget constraint (in real terms): G }{{} government spending = T }{{} tax revenue Fiscal policy in our one-period model: government choices over spending and taxes. Assume: G is fixed and determined outside the model. G is exogenous. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 4 / 51

Exogenous Variables vs. Endogenous Variables Exogenous variables: determined outside the model; taken as given. Endogenous variables: determined within the model.

5 Exogenous Variables vs. Endogenous Variables Exogenous variables: determined outside the model; taken as given. Endogenous variables: determined within the model. Use the model: how changes in exogenous variables change endogenous variables. In our one-period model: Exogenous: G, z, K. Endogenous: C, N s, N d, T, Y, π, w. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 5 / 51

6 Competitive Equilibrium (CE) Competitive: consumers and firms are price-takers. Equilibrium: interaction among all sectors reaches a stable and consistent state. Consumers and firms have no incentive to change their behavior under given prices. (Optimal decisions) Price does not change. (All markets clear) c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 6 / 51

7 Definition: Competitive Equilibrium A competitive equilibrium is a set of endogenous variables (C, N s, N d, T and Y ) and an endogenous real wage rate (w), such that, given exogenous variables (G, z and K ), the following conditions are satisfied: 1 Given w, T and π, the bundle (C, N s ) maximizes the consumer s utility subject to the budget constraint. 2 Given w, z and K, the labor demand N d maximizes the firm s profits. 3 Government has a balanced budget: G = T. 4 Markets clear: Labor market: N d = N s = N. Goods market: C + G = Y. (I = 0, NX = 0 here) c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 7 / 51

8 Walras Law In our model, if labor market clears, so will the goods market, and vice versa! Show this. Generally, if there are N markets, and N 1 markets clear, then the Nth market clears automatically. This is called the Walras Law. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 8 / 51

9 Study the Competitive Equilibrium Throughout this book, we work with a model in graphical forms (i.e. supply curve and demand curve derived from optimal behavior of consumers and firms). Examine the consumer s decision and the firm s decision in the same diagram. Now, for the one-period model, put consumer s problem and firm s probelm in the same (l, C) plane. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 9 / 51

10 Production Possibilities Frontier (PPF) Figure 5.2(a) Production Function Our production function Y = zf (K, N d ). Maximum level of output is Y = zf(k, h). F(K, 0) = 0. The slope is MP N. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 10 / 51

11 Production Possibilities Frontier (PPF) Figure 5.2(b) Output as a Function of Leisure Change the horizontal axis from N d to l, using l = h N d. Production function: Y = zf (K, h l). When l = 0, N d = h, maximum output Y achieved. When l = h, N d = 0, nothing is produced. The slope is MP N. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 11 / 51

12 Production Possibilities Frontier (PPF) Figure 5.2(c) The Production Possibilities Frontier Change the vertical axis from Y to C, using C = Y G. When l = 0, C = Y G. When l = h, C = G. The shaded area is the production possibilities set. The arc DA is the production possibilities frontier. Points on arc BA are not feasible. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 12 / 51

13 Production Possibilities Frontier (PPF) In equilibrium, C = Y G, we can write PPF as C = zf (K, h l) G The PPF describes what the economy can produce in terms of production of consumption and leisure. Marginal rate of transformation (MRT) MRT l,c is called marginal rate of transformation of leisure into consumption. MRT l,c is the rate at which leisure can be converted into consumption goods. MRT l,c = MP N = slope of the PPF. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 13 / 51

14 Competitive Equilibrium (CE) Firm s PPF: curve FH. If w is an equilibrium wage rate and AD with slope w is tangent to the PPF, then ADB is the budget constraint of consumers. Why? Optimizing conditions: Consumer: MRS l,c = w. Firm: MP N = w. Same tangent point J. Why? c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 14 / 51

15 Competitive Equilibrium (CE) Maximum profit of the firm: π = zf(k, h l ) w (h l ) π : distance DH. Why? Maximum after-tax dividend income paid to the consumer: π T π T : distance DB. Why? c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 15 / 51

16 Competitive Equilibrium (CE) In equilibrium (at point J), we have MRS l,c = MRT l,c = MP N = w Interpretation: The consumer and the firm face the same market real wage in equilibrium, so the rate at which the consumer is just willing to trade leisure for consumption (MRS l,c ) is the same as the rate at which leisure can be converted into consumption goods (MRT l,c ). Graphically, the PPF, the budget line, and the indifference curve are tangent to each other at the same point! c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 16 / 51

17 Competitive Equilibrium and Efficiency Why we study this connection? How free markets can produce socially optimal outcomes. If CE is equivalent to a socially optimal/efficient outcome, then we can study the social optimum instead of CE, which is much easier! Criterion of efficiency Pareto optimality. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 17 / 51

18 Pareto Optimality Pareto Optimal A CE 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. Question: Whether the CE is Pareto optimal? To find efficient allocations, we use a benevolent social planner. Then compare the CE in our model with the social planner outcome. Attention: social planner government. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 18 / 51

19 Pareto Optimum: Social Planner s Choice The social planner chooses a consumption bundle that is On or within the PPF (feasible for the firm to produce) On the highest possible indifference curve (best for the consumer) Pareto optimum: point B, where MRS l,c = MRT l,c = MP N. To the left of B: MRS l,c > MRT l,c. Could make the consumer better off by l. To the right of B: MRS l,c > MRT l,c. Could make the consumer better off by l. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 19 / 51

20 Competitive Equilibrium vs. Pareto Optimum Social planner s choice is the same as the outcome of free markets: under certain conditions, CE PO. Two fundamental theorem of welfare economics. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 20 / 51

21 Fundamental Theorems of Welfare Economics First fundamental theorem of welfare economics: Under certain conditions, a competitive equilibrium is Pareto optimal. (CE PO) Second fundamental theorem of welfare economics: Under certain conditions, a Pareto optimum is a competitive equilibrium. (PO CE) Attention: efficiency equity! c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 21 / 51

22 Fundamental Theorems of Welfare Economics First welfare theorem holds: The CE allocation (C, l ) is Pareto optimal. Why? Second welfare theorem holds: The Pareto efficient allocation (C, l ) can be decentralized as a CE. Why? c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 22 / 51

23 Sources of Social Inefficiencies A CE may fail to be Pareto optimal due to: Externalities The social planner would take externalities into account, but individuals and firms don t. Externalities can be negative or positive. Essentially roots in missing markets. (Market failure) Distortionary taxes For example, a propotional labor income tax t. Consumers and firms do not face the same price (consumers facing w(1 t); firms facing w). The competitive equilibrium is not Pareto optimal, why? In practice, all taxes cause distortions. Market power Firms with market power do not take price as given, as they know they can influence them. Typically leads to under-production. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 23 / 51

24 Comments on Efficiency Why should we analyze an economy that is efficient in the sense that a CE is Pareto optimal? Why don t we model sources of inefficiencies here? With inefficiencies existing in the real world, is it always better to have government regulations? c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 24 / 51

25 Use the Model: Working with Planner s Problem In general, the planner s problem is easier to work with. No worry about prices. Consumption and production choices are determined by preferences and technology. Pareto optimum: tangent point of indifference curve and PPF. We can always find prices to decentralize the allocation as a CE. PO CE. From now on, we work with the social planner s problem instead of CE. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 25 / 51

26 Two Experiments How a change in an exogenous variable (G, z or K ) affects the key endogenous variables (C, Y, N and w). The effects of a change in government purchases, G. The effects of a change in TFP, z. Model mechanism: Changes in exogenous variables Shift or twist PPF in some ways Changes in the PO/CE Changes in endogenous variables c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 26 / 51

27 Experiment 1: G G PPF parallelly shifts down. At each l, the slope of PPF is the same as before. This is exactly like a negative income change to the consumers, since G T. Pareto optimum: A B. Both C and l are normal goods, we should expect C and l. Now we study the effects of G on C, Y, l/n, and w. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 27 / 51

28 Experiment 1: G Effects of G on C, N and Y. Effects on N and Y : Leisure is a normal good. l N Y Effects on C: C = Y G so C = Y G. Y > 0 C > G C by less amount than G. In graph, AE<AD. Consumption is crowded out by government spending, but not completely. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 28 / 51

29 Experiment 1: G Effects of G on w How can firms hire more labor? It must be w. N MP N w. Individuals still want less leisure even if it is cheaper, why? Exercise: Decompose the total effect into an income effect and a substitution effect for the consumer. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 29 / 51

30 Government Spending and Business Cycles Is fluctuation in G a likely cause of business cycles? Can our model replicate those key business cycle facts in response to a change in G? In our model, if G : Output Employment (pro-cyclical) Consumption (counter-cyclical) Real wage rate (counter-cyclical) Key business cycle co-movements (Ch3): Employment is pro-cyclical Consumption is pro-cyclical Real wage rate is pro-cyclical Conclusion: Business cycles are not likely to be the results of government spending fluctuations. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 30 / 51

31 Experiment 2: z Figure 5.8 Increase in Total Factor Productivity z production function shifts up. z MP N at each N. Maximum amount of output is higher: z zf (K, h). Still starts from the origin point. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 31 / 51

32 Experiment 2: z Figure 5.9 Competitive Equilibrium Effects of an Increase in TFP z the PPF shifts outward from BA to DA. More consumption goods can be produced at any level of leisure. z MP N, PPF is steeper. Equilibrium (efficient) allocation: F H. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 32 / 51

33 Experiment 2: z Effects of z on C, N, Y and w Effects on C: C (normal good). Effects on Y : Y = C + G, G fixed, so Y. Effects on w: w (see later). Effects on N: z MP N w N. w income N. Total effect on N is uncertain. In the figure, N is unchanged. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 33 / 51

34 Experiment 2: z Figure 5.10 IE and SE of an Increase in TFP Draw a PPF 3 : parallel to PPF 2 and tangent to I 1 at D. Substitution effects: A D C, l, N Income effects: D B C, l, N Total effects: C l or N uncertain Welfare c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 34 / 51

Experiment 2: z We want to show: z w regardless the change of l From A to D (SE): MRS D l,c > MRSA l,c w D > w A From D to B (IE): PPF 2 and PPF 3 are parallel same MP N for each l.

35 Experiment 2: z We want to show: z w regardless the change of l From A to D (SE): MRS D l,c > MRSA l,c w D > w A From D to B (IE): PPF 2 and PPF 3 are parallel same MP N for each l. l B > l D (normal good) N B < N D MP B N > MPD N w B > w D From A to B, w no matter SE dominates or IE dominates. Intuition: z MP N demand for labor w. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 35 / 51

36 TFP and Long-Run Trends There have been many important technological innovations since World War II. Should they be responsible for the main economic changes over time? Can our model with an increase in TFP match the key observations in the long-run data? In our model, if z : Since WWII, we have observed: Y C w N ambiguous A rise in output A rise in consumption A rise in the real wage rate Roughly constant hours worked Conclusion: IF IE and SE roughly cancel out with each other, the model is consistent with technological innovations having been key to changes in these economic variables in the long run. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 36 / 51

37 TFP and Business Cycles Could fluctuations in TFP be a cause of business cycles? Can our model replicate those key business cycle facts in response to an increase in z? In our model, if z : Y C (pro-cyclical) w (pro-cyclical) N ambiguous (N if SE dominates) Key business cycle co-movements (Ch3): Consumption is pro-cyclical Real wage rate is pro-cyclical Employment is pro-cyclical Conclusion: Fluctuations in TFP may be the primary cause of business cycles if the substitution effect dominates the income effect in the short run. (Real Business Cycle Theory) c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 37 / 51

38 Example of Inefficiency: Distorting Wage Income Tax Consider a proportional tax on wage income. Now, CE PO. So we can no longer use the social planner s problem to study CE. We study the CE directly: Firm s behavior Consumer s behavior Market clearing Incentive effects of wage income tax on labor supply. Relationship between government s tax revenue and tax rates. ("Laffer curve") c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 38 / 51

39 A Simplified Model with Proportional Income Tax Assume: labor is the only input for production. Production function for the representative firm: Y = zn d No lump-sum tax (T = 0). Income tax rate t. Budget constraint for the representative consumer: C = w(1 t)(h l) + π w(1 t): the effective wage rate facing the consumer. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 39 / 51

40 Firm Optimization Firm s profit maximization problem: max π = Y wn d = (z w)n d N d Firm s behavior: If w < z, firm wants to hire infinite labor. If w > z, firm would hire no labor. As long as w = z, any N d is optimal. Requirement for existence of equilibrium: Equilibrium exists if and only if w = z. If w z, no equilibrium. In equilibrium, w = z, π = 0, N d is set at the value that clears the labor market. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 40 / 51

41 Labor Demand Curve Firm s demand for labor is infinitely elastic at wage w = z. Figure 5.16 The Labor Demand Curve c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 41 / 51

42 Consumer Optimization Consumer s budget constraint in equilibrium: C = z(1 t)(h l) Note that w = z in equilibrium. π = 0 and T = 0. Budget line: DF. Optimal bundle: H. (MRS l,c = z(1 t)) c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 42 / 51

44 Competitive Equilibrium Figure 5.17 Competitive Equilibrium with a Proportional Income Tax Competitive equilibrium: H. Check conditions for CE: Firm optimizes. Why? Consumer optimizes. Why? Government s balanced budget: G = zt(h l), which implies that AB intersects DF at H. Labor market clears. Why? Goods market clears. Why? c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 44 / 51

45 Inefficiency of Competitive Equilibrium Pareto optimum: E. Competitive equilibrium: H. E is better than H. Reason for inefficiency: Income tax distorts private decisions. Disincentive to work: labor supply is lower in CE than in PO. Both C and Y are lower in CE. Welfare loss: utility difference between I 1 and I 2. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 45 / 51

46 Income Tax Revenue Tax revenue collected by the government in equilibrium: REV (t) = tw[h l(t)] = tz[h l(t)] In equilibrium, w = z. l(t): consumer s choice of leisure depends on t. z[h l(t)] is called the tax base. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 46 / 51

47 Tax Rate and Tax Revenue REV (t) When t, there are two opposing effects on l(t): Substitution effect: t after-tax real wage z(1 t) leisure is relatively cheaper l(t) (disincentive to work) Income effect: t after-tax income l(t) (normal good) Total effect on l(t) is ambiguous. Total effect on z[h l(t)] is also ambiguous. If SE > IE, higher tax rate might lead to lower tax revenue! c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 47 / 51

48 Laffer Curve Laffer curve: graphically shows the relationship between t and REV (t). (curve AB) Shape of Laffer curve depends on labor supply behavior. End points A and B will always be the intersection points. Point A: t = 0, so REV = 0. Point B: t = 1, no one would work, so REV = 0. Maximum tax revenue REV is reached at t. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 48 / 51

49 Government s Choice of Tax Rate Government s budget: G = REV (t). If G > REV, impossible for government to collect enough tax revenue. If G < REV, there are at least two feasible tax rates. Suppose there are two possible equilibrium tax rates to finance G < REV, t 1 and t 2. Which tax rate would the government choose? c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 49 / 51

50 Government s Choice of Tax Rate Suppose t 2 > t 1, and REV (t 1 ) = REV (t 2 ) = G. With low tax rate t 1, CE is F. With high tax rate t 2, CE is H. F must be on a higher indifference curve than H. Why? The consumer is better off in the equilibrium with lower tax rate t 1. Government should choose tax rate t 1 rather than t 2. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 50 / 51

51 Laffer Curve and Supply-Side Economists Supply-side economists view about income tax: Large incentive effects of income taxes on labor supply. Tax rate reductions will largely increase labor supply and thus the tax base. Example: Ronald Reagon s campaign in 1980 election. Question against this argument: Whether the U.S. economy in 1980 was on the upward-sloping side (the "good" side) or the downward-sloping side (the "bad" side) of the Laffer curve? In general, economists believe that U.S. economy is on the good side of the Laffer curve. c Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 51 / 51

A Closed Economy One-Period Macroeconomic Model

A Closed Economy One-Period Macroeconomic Model Chapter 5 Topics in Macroeconomics 2 Economics Division University of Southampton February 21, 2008 Chapter 5 1/40 Topics in Macroeconomics Closing the Model