Macroeconomics: Chapter 5, Closed Economy Model

• The notes cover Chapter 5, focusing on a closed economy, one-period macroeconomic model.

Key Insights

• Government spending increase boosts aggregate output but reduces private consumption due to crowding out.
• Productivity growth enhances aggregate output and elevates living standards.
• Unregulated private markets may not achieve socially efficient outcomes because of income taxation's incentive effects.

Closed vs. Open Economy

• Closed economy involves no external trade.
• Open economy engages in international trade.

Actors in a Closed Economy

• Representative Consumer sells labor and purchases goods.
• Representative Firm buys labor and sells goods.
• Government provides services and collects taxes.

Government's Role

• Government behaviour involves collecting lump-sum taxes (T) to finance government spending (G).
• Government services include roads, national defence, and education.
• Public goods hard for the private sector to provide.
• Government spending uses resources, which is modeled as taking goods from the private sector.
• Output is produced, and the government buys an exogenous amount G of this output.
• The representative consumer consumes the rest.
• Government Budget Constraint states that government purchases (G) must equal taxes (T).
• Fiscal Policy covers decisions on expenditures, taxes, transfers, and borrowing.
• Government Expenditures include purchases of final goods/services.
• Transfers reallocate purchasing power among individuals.
• Government choices constricted by a budget in a one-period model.
• No borrowing for government expenditures occurs, as there is no future to repay.
• No taxing beyond spending is allowed to avoid wastage.
• Budget deficit is always zero in this model.
• Assumptions include the government having a balanced budget with no debt.
• Consumers do not value government spending when utility is defined over leisure (L) and private consumption (C), so spending (G) is wasteful.
• Alternatively, government spending (G) could be included in the utility of consumer; the consumer would think of G as public goods that provide benefits.

Competitive Equilibrium

• The macroeconomic Model determines endogenous variables from exogenous ones.
• Exogenous Variables include government spending (G), total factor productivity (z), capital stock (K), and parameters in utility (U), and production function (F)
• Endogenous Variables include consumption (C), Leisure (L), Labour Supply (Ns), Labour Demand (Nd), Real Dividend income (π), Taxes (T), Aggregate Output (Y), and Real Wage (W)
• Competitive Equilibrium with Lump-sum Taxes is a set of endogenous quantities and an endogenous price (w), given exogenous variables which satisfy the following:
• Consumer Optimization: With wage (w), dividend income (π) and taxes (T), the consumer chooses C, L, and Ns to maximize utility subject to constraints.
• Firm Optimization: With wage (w), and exogenous variables K and z, the firm chooses Nd to maximize profits
• At maximized output Y = zF(K, Nd)
• Maximized profits π = Y – wNd
• Government budget is balanced where G = T
• Markets clear:
• Labour market: Nd = Ns (Excess leads to changes in real wage)
• Goods market: Y = zF(K, Nd) = C + G
• Consumers and firms are price-takers
• π must be equal to profits generated by the firm
• Income-Expenditure Identity follows Y = C + G

Production Possibilities Frontier (PPF)

• PPF gives the pairs (C,L) possible to produce in the economy given production technology, the level of z, h, and G and given the function form for F
• (C,L) pairs that satisfy C ≥ 0 and I ∈ [0, h]
• C + G = zF(K, Nd) → C + G = zKª (Nd)1-a
• PPF Equation: C = zF (K, h-l) – G
• PPF Equation Cobb-Douglas Y = zKª N¹-a : C = zKª (h-I)1-a + G
• PPF can be combined with consumer's indifference curves to determine competitive equilibrium
• Slope of PPF is the derivative of C with respect to I
• PPF = zK° (1-a)¯ª (-1) = -MPN
• If Production function is Y = zN then:
• The PPF is C = z(h-l) – G
• Slope of PPC is derivative of C with respect to I, i.e., -z

Marginal Rate of Transformation (MRT₁,c)

• This is the rate at which one good can be exchanged for another.
• Leisure can be converted into consumption goods through work
• MRTI,C = - (Slope of the PPF) = MPN
• Equilibrium Condition: MRT₁,C = MRSI,C = MPN = w
• Efficiency of Competitive Equilibrium: The rate the consumer trades leisure for consumption matches the rate at which leisure can be converted into goods.

Solving for the Competitive Equilibrium (CE)

• Step 1: Write optimality condition for consumer → Slope of IC = Slope of BC
• Step 2: Write optimality condition for the firm → marginal product of labor (MPN) = w
• Step 3: Combine equations (1) and (2) then use the goods market clearing condition also to solve for L (or N)
• Once N and L are known, use the rest of the equations to back out C, π, Y, T, w

Optimality and Competitive Equilibrium

• Competitive Equilibrium occurs when supply equals demand and agents maximize utility.
• Pareto Optimality is when its impossible to make any individual better off without making another worse off.
• A measure of an efficient allocation is Pareto optimal allocation.
• It is located at the point indifference curve is tangent to the PPF.
• Slope of indifference curve = -MRSI,C
• Slope of PPF = -MRTI,C or -MPN
• Pareto Optimum has the property that MRSI,C = MRT₁,C = MPN
• A condition also met in competitive equilibrium

Social Planner Problem

• Features of the social planner: Decides consumption, work, and firm production

Does not follow prices

Understands opportunity cost

Benevolent: Searches for best possible allocation of resources

• Social planner's objective: Maximize utility or welfare of consumers by choosing how much C and L the consumer gets.
• Bundle (C,L) must be feasible: Must be on the PPF
• Must satisfy the economy’s goods market clearing condition
• AKA economy’s resource constraint or the expenditure identity of GDP

Solution to Social Planner's Problem

• Social planner chooses the Pareto optimum where consumer is as well off as possible given technology for production.
• The problem can be solved by using labour as an input.
• Solution to social planner’s problem satisfies: MRSI,C= MRTI,C (= MPN)
• Where these properties are satisfied in all CE meaning all of them are Pareto optimal

Competitive Equilibrium Pareto Optimality

• competitive equilibrium coincides with the solution for N, L, C, Y from the social planner’s problem.
• the note says if we replaced lump-sum taxation in the one-period model with distortionary taxation (proportional taxes on labor income) the CE is not Pareto optimal.
• competitive equilibrium with distortionary taxes is in the following range (proportional tax t E (0,1)):
• MRSI,C = (1-t)MPN
• For t > 0 we obtain: MRS < MPn = MRT
• Social planner picks (C, L, N, Y) such that: MRS = MRT
• when we have distortionary taxation (t > 0) the CE is not Pareto optimal because the CE satisfies conditions different from those of the social planner
• Fundamental Theorems of Welfare Economics:
• First Fundamental Theorem: Competitive equilibrium is Pareto-optimal under certain conditions Second Fundamental Theorem: Pareto optimum is a competitive equilibrium under certain conditions
• Adam Smith’s “Invisible hand” suggests that self-interested actions in a free market can lead to socially efficient outcomes
• First welfare theorem aligns with this idea, competitive market can achieve a socially optimal distribution of resources

Considerations Beyond the Model

• Pareto-optimality does not consider equity; an economy can be Pareto-optimal but have significant wealth disparities
• Societal values such as equity and compassion may require trade-offs with efficiency
• Sources of Social Inefficiencies:

1. Externality: Effects of actions on others not accounted for by the agent
   • Negative Externalities: Unaccounted costs Positive externality: Uncompensated benefits

• The root cause of externality is market failure - Market Failure: no markets for trading costs or benefits causing inefficiencies
• Distorting Taxes: Creates a difference between effective prices faced by buyers and sellers of some good
• Lump-Sum Tax: Independent of actions, does not distort behavior

Distorting Taxations

• Distorting Tax: Depends on actions, distorts behavior
• Proportional Wage Income Tax: Causes a wedge between marginal rate of substitution and marginal product of labour, leading to less work and more leisure
• Competitive equilibrium with distorting tax fails to satisfy MRSI,C= MRTI,C = MPN Instead we have → MRS₁,c< MRT₁,C = MPN
• Firms welfare theorem does not hold
• Real-World Taxes: All cause distortions; lump-sum taxes are not practical Monopoly Power: Large firms may not be price-takers and can restrict output to raise profits, leading to underproduction
• Efficiency in Economic Models: Simplification for Macro Issues: Models without inefficiencies are less complex and behave similarly to those with inefficiencies Competitive Equilibrium = Pareto Optimum: Assumption simplifies analysis to solving the social planner's problem
• Government Policy Considerations: Market Failures: Justify government intervention for economic efficiency Adam Smith’s Invisible Hand: Unrestricted markets tend to produce efficient outcomes, but government policy must balance market failures with the cost of regulation

How to Use the Model

• Equivalence Principle: Competitive Equilibrium = Pareto Optimum: model relies on this equivalence for analysis Social Planner’s Problem: Solving this is akin to finding the competitive equilibrium
• Graphical Analysis: PPF (Production Possibility Frontier): Equilibrium Point: Where indifference curve is tangent to the PPF
• Equilibrium Quantities: • Consumption (C*) • Leisure (1*) • Employment (N*) -- Quantity of employment is N* = h – I* • Output (Y*)
• -• Quantity of output is Y* = C* + G
• Real Wage Determination: • Real wage (w): Determined by negative slope of PPF or indifference curve at equilibrium Firm Optimization: Sets marginal product of labour (MPN) equal to real wage Consumer Optimization: Sets marginal rate of substitution (MRS) equal to real wage Analyzing Changes:
• Exogenous Variables: impact endogenous variables by shifting PPF • Government spending (G) • Total factor productivity (z) • Capital stock (K) Impact on Endogenous Variables: Shifts in PPF affect C,Y,N, and w

Determinants of Economic Activity

• This covers consumer preferences (indifference curves) and firm technology (PPF).
• Aggregate Effects occur when Changes in preferences or technology alter aggregate output, consumption, employment, and real wage
• Here are the Effects of a Change in Government Purchases when Working with the Model
• Government Spending Increase Effects: Increased G: Increases: N, Y, T Decreases: L, C Shift in PPF: Increase in G shifts PPF downward by the change in G • Slope remains unchanged
• Taxes: to finance higher G, gov't increases T by same amount, reducing consumer's disposable income
• How does government spending have a pure income effect when G increases? --Because it is financed by lump-sum taxation
• Negative Income Effect: Higher G leads to reduced disposable income, decreasing consumption (C) and leisure (I), while increasing employment (N) and output (Y)
• Why Consumption and Leisure Decrease is due to the Normal Goods Assumption: a negative income effect from higher G reduces consumption and leisure --Employment and Output Rise: As leisure decreases, employment rises, and with fixed capital, output increases
• Output increases but consumption falls when G goes up? is due to Private consumption spending being “crowded out" by increased government spending, or the Private consumption output ratio C/Y falls
• The Model is missing, where G is bad for consumer since and L decreases, and does not allow gov't debt which allows for smoothing the burden of taxation
• Income-Expenditure Identity: therefore, there is a change for each condition
• Private consumption is reduced (not completely) due to higher G Real Wage Dynamics consist of:
• Real Wage Decrease: As employment rises, the real wage falls because firms only hire more labour at a lower real wage
• Business Cycle Considerations of Employment and Government Spending:
• Procyclical with government spending, aligning with business cycle facts
• Consumption and Real Wage: There is a Counttercyclical in response to G shocks
• -Conclusion to these Cycles: Government spending fluctuations are unlikely the primary cause of business cycles.

Working with the Model

• Changes in Total Factor Productivity is summarized as
• Increase in TFP (z): Better tech or conditions like weather, or deregulation.
• This Increases Y, C, and w Income and Substitution Effects from change in z determines what happens to N and L
• Affect of Long Run: if Income and Substitution Effects cancel out over the long run with: Increases: Y, C, and w, no change in N
• Short run effect of increased z": Increases: Y, C, w Ambigous N
• There are key business cycles co-movements, consumption is procyclical, the Real wage rate is procyclical and Employment is procyclical as well.
• Real Business Cycle Theory states that : -- Fluctuations in z may be a significant source of business cycle fluctuations if the substitution effect dominates the income effect so that changes in employment are procyclical
• Possible that short run consumers react to changes in productivity while long run TFP growh leaves hours worked unchanged
• If wages are high today, consumers decide to work more today and take more time off in the future (intertemporal substitution)
• Production Function Shift with Higher TFP, meaning more output from the same labor and increased labor productivity
• How does an increased z effect PPF (C = zF(K, h – L) – G?
• With shifts like: Shifts out PPF: Can produce more with same amount of labor
• • Increasing C unambiguously
• Increase in L and N will depend on IE & SE
• How does an increased z effect real wage (w)?:
• -Increase in z leads to an increase in w
• Substitution effect, leisure is relatively more expensive, increasing Consumption C, decrease L -There is a effect called the income effect: When disposable income of consumer goes up,
• C increases, while I increases as well (normal goods)
• there no effect on work time (N) and leisure time (L) if the income and subsitituion effects on L cancel out: -Since L doesn't change with z, N= h- L wont either Equilibrium Changes consists when higher TFP, output and consumption rise, real wages increase, but employment's direction is uncertain due to mixed effects on labor supply Income vs. Substitution Effects occurs when: --More work due to higher value per hour (consumption up, leisure down) with Substitution Effect
• -- Less work needed for same standard (both consumption and leisure up) with Income Effect • The note also states that:
• Real Wage Increase: Must increase due to steeper PPF and higher labor productivity
• Consumer Welfare: Consumers will Always improves; consumers reach a higher indifference curve with higher TFP

Distorting on Wage Income

• This section covers Tax Rate Changes, and the Laffer Curve, which covers consumer behaviour and tax revenue
• with a tax rate (t) as the leisure function L(t): Quantity of leisure the consumer chooses if the after-tax real wage is z(1-t), with z constant
• Tax revenue (REV): REV = tax rate z[h-L(t)]
• Tax base: Quantity of labour being taxed, z[h-L(t)] Changes of Tax Rate (t) on Tax Revenue consists when tax base constant, increase in t increases REV
• Increasing the tax rate (t) actually decrease total tax revenue (REV) when: Tax base constant, increase in t increases REV If increase in I(t) significantly reduces tax base If increase in I(t) significantly reduces tax base, it will occur:
• A large substitution effect (reduced labour due to lower after-tax wage) --- is needed for REV to fall with higher t Increase in t must cause a substantial decrease in labour supply for REV to decrease -A larrfer curve relationship: Shape depends on labour supply response to after-tax wages
• -REV = 0 at t = 0 and t = 1 (no work at 100% tax)
• -t = 1 implies L(1) = h and REV = 0 Max REV (REV*) at tax rate (t*)
• government chooses tax rateto finance spending G, where Government Budget Constraint the Laffer Curve is: tax revenue * z[h-L(t)]
• G < REV* , lower (tax rate) is on good curve/lower tax side lower (high) on lower side with to equilibrium tax
• To collect enough REV to finance spending when not G < REV* isn't possible
• Competitive Equilibrium and Tax Rates states for lower tax rate (t1), we obtain: High Consumption (C) High labour supply (h-l) Lower leisure (I) Higher aggregate output (Y = C + G) "GPT SAYS: Higher tax rate (t2): Opposite effects” is this true?
• Consumer Welfare and Tax Rates: Better off at lower tax rate due to higher C and Y, and lower I
• Government Choice of tax rates consist when: sensible governments would choose lower tax rate to get same REV with better consumer welfare