Introduction to the Equilibrium Business-Cycle Model

Introduction to the

Equilibrium Business-Cycle Model

Mr. VaughanIncome and Employment Theory (402)

1Updated: 3/2/09

Lecture Outline

• Approaches to Modeling Business Cycle– New Keynesian vs. Equilibrium

• Stylized Business-Cycle Facts

• Building Blocks of an Equilibrium Model– Markets: Goods, Labor, Rental, Bond– Household Budget Constraint (Real vs. Nominal)

Total Slides: 37 2

Business-Cycle ModelsPolar Approaches

New Keynesian– Inspired by work of John Maynard Keynes (The General Theory

of Income, Employment, and Money – 1936)

– Assumes some prices are sticky (i.e., move slowly to equate quantities demanded/supplied after shocks).

– Differs from “Old” Keynesian by emphasizing micro-foundations (i.e., microeconomic explanations for frictions like price stickiness).

Implications: Shocks-cum-frictions can produce deviations from full employment

(potential output) for significant periods of time.

Observed cyclical fluctuations in real GDP are Pareto sub-optimal.

Stabilization policy can (in theory) be Pareto improving.

3Total Slides: 37

Business-Cycle Models Polar Approaches

Equilibrium– Intellectual descendant of pre-Keynesian (macro) economics. – Inspired by microeconomic emphasis of Friedman/Lucas (in monetary-

shock models) and Kydland/Prescott (in “real” business-cycle models).– Emphasizes market-clearing responses to shocks (i.e., prices move

fluidly in all markets to re-establish “general” equilibrium).

Implications: Movements in real output largely reflect changes in potential output. Observed cyclical fluctuation in real GDP are Pareto Optimal. Stabilization policy cannot enhance welfare.

Our Approach: Build an equilibrium model Confront model with data Results will reveal relative importance of New Keynesian, Equilibrium components.

4Total Slides: 37

Stylized Facts Historical Perspective on U.S. Economic Performance

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Long Run (Real Trends)• Mean annual growth,

real GDP (1870-2005) = 3.5% Mean annual growth,

per capita real GDP = 2.0% Mean annual growth,

population = 1.5%

Explanation: Continuous advances in technology

Note: Relative importance of trend vs. fluctuations.

Media emphasize fluctuations: Short-run is “news” (long-run

growth is norm, changes harder to detect).

Easier to blame downturns on current administration.

Graph shows real GDP (GNP before 1929) on logarithmic scale, so each unit on vertical axis corresponds to same percentage change in real GDP.

Sources: Bureau of Economic Analysis (1929-2005); Christina Romer (1869-1928)

Total Slides: 37

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Graph shows annual growth of real GDP (GNP before 1929), calculated from figure 1.1 using:

Growth in real GDP (Yt) = [(Yt – Yt-1)/ (Yt-1)] -1

Aside from years marked by major wars (WWI, WWII, aftermath of WWII, and Korean War), years noted indicate recessions (i.e., negative year-over-year growth rates). Red line denoted zero percent growth. Note: Figure 1.2 labels only 15 of 29 recessions.

Short-Run (Real Cyclical Trends)

• Recessions 1869-2005 (29) Average Length: 17.2 months

• Major recessions (annual growth significantly negative) Pre-WWII (1869-1941): 1893-94, 1907-08,

1914, 1920-21, 1930-33, 1937-38

– Worst: 1930-33 (real GDP declined 8% per year for 4 years)

Post-WWII (1945-2005): 1958, 1974-75, 1982

• Major booms (annual growth significantly positive) Pre-WWII (1869-1941): 1875-80, 1896-1906,

1921-29*, 1933-37

Post-WWII (1945-2005): 1961-73*, 1983-89, 1992-2000

*interrupted by mild recession(s)

Total Slides: 37

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Sources: Bureau of Labor Statistics (1929-2005); Christina Romer (1869-1928). Values for 1933-43 adjusted to classify federal emergency workers as employed. Red line denotes median unemployment rate, 1890-2005=5.5%.

Short-Run (Real Cyclical Trends)

• Mean Unemployment(1890-2005) = 6.3%

• Median Unemployment(1890-2005) = 5.5%

Noteworthy Pre-WWII Average Highs:1931-35 = 18%; 1938-39 = 12%;1894-98 = 11%; 1921-22 = 8%(n.b., lags recessions)

Noteworthy Post-WWII Average Highs:1982-83 = 10%; 1975-76 = 8%;1958,1961,1991-93 = 7%;

Total Slides: 37

Peak = 22%

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Graph shows price deflator for GDP (GNP before 1929). Numbers are on logarithmic scale (2000-100). Red line denotes beginning of WWII.

Long-Run (Nominal Trends)

• Price level fluctuates prior to WWII. Periods of falling prices (deflation):

1869-92; 1920-1933

• Price level rises continually since WWII.

Only rate of increase fluctuations

Total Slides: 37

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Graph shows annual inflation rate based on GDP deflator (GNP before 1929). Inflation rate is annual growth rate of price level shown in figure 1.4 calculated as follows:

Inflation Rate = [(Pt – Pt-1)/ (Pt-1)] -1

Where Pt is price level (GDP deflator) in year “t”. Red line denotes zero percent (i.e., rates below 0% correspond to deflation).

Long-Run (Nominal Trends)

• Inflation rates since WWII positive except 1949.

Inflation rose to mean peak of 8.8% for 1980-81.

Inflation fell to means of 2.5% for 1983-2005 and 2.0% for 1992-2005.

Total Slides: 37

Post-WWI Peak

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Interaction of Nominal/Real Variables(from Friedman and Schwartz’s Monetary History)

Long Run: – Provided monetary growth is moderate and stable, real growth proceeds

independent of price level. – Trends in price level reflect trends in money supply (high-powered money, in

particular) .

Short Run:– Rate of monetary growth slows down before cyclical peak (contraction); speeds

up before cyclical trough (expansion). Absolute declines in money supply precede severe contractions.

– Monetary changes have often had independent origin (i.e., not been simply reflection of changes in economic activity).


Total Slides: 37

Instructive to compare pre-WWI and post-WWII periods (Eliminates “noise” from interwar instability)

Long-term growth rates (per year)– Post-WWII: 1948-2005 (real GDP) = 3.4%– Pre-WWI: 1870-1914 (real GDP) = 3.8% (1890-1914 = 3.4%)

Mean unemployment rates– Post-WWII: 1948-2005 = 5.6%– Pre-WWI: 1890-1914 = 6.4%

Cyclical fluctuations only moderately larger before WWI – (Christina Romer evidence)

Implication: We can test model on post-WWII data (better quality).

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Total Slides: 37

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NOTE

• Real GDP fluctuates around long-run trend, i.e.,

Movement in Real GDP =

Trend Growth in Real GDP + Cyclical Fluctuation in Real GDP

• So,

Cyclical part of real GDP = Real GDP − Trend Real GDP

Post-WWII Stylized Facts Data Patterns to be Explained

Blue line shows U.S. real GDP from 1947:Q1 to 2006:Q1. Data are quarterly, seasonally adjusted, and measured in base year dollars (2000). Proportionate (logarithmic ) scale used, so percentage change along vertical axis represents same percentage change in real GDP. Red line is smooth trend through GDP data (i.e., long-run economic growth).

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NOTE

• Standard deviation of cyclical part of real GDP (1947:Q1-2006:Q1) = 1.7%.

Typical range of fluctuation of real GDP was +/- 1.7% from trend.

• Recession (Barro definition): At least 1.5% below trend.

Mild: 1990-91 (-1.6%) 2001-02 (-1.9%) 1961 (-2.7%)

Severe: 1949 (-6.2%) 1982-83 (-4.7%) 1958 (-4.2%) 1975 (-3.8%)


Blue line plots cyclical portion of real GDP movements (i.e., deviations of real GDP from its trend) from 1947:Q1 to 2006:Q1. Cyclical part is measured in a proportionate sense – e.g., 0.02 means real GDP is 2% above its trend. Recession years noted are those in which cyclical part of real GDP fell short of trend by at least 1.5% (denoted by red line). Note: All data are quarterly, seasonally adjusted, and measured in base year dollars (2000).

Total Slides: 37

14

Business-Cycle Terminology

When variable fluctuates in same direction as real GDP, that variable is pro-cyclical.

– Variable moves tends to be high relative to its trend in boom and low relative to its trend in recession.

When variable fluctuates in opposite direction from real GDP, that variable is counter-cyclical.

Variable with little tendency to move in particular direction during business cycle is a-cyclical.

Total Slides: 37

15

Real Consumption Expenditure 64% of GDP (1954-2006 mean)

Pro-cyclical Correlation of cyclical part of real

consumption and GDP (1954:Q1-2006:Q1) = 0.88.

Less volatile than real GDP Standard deviation of cyclical part of real

consumption (1954:Q1-2006:Q1) = 1.2%.

Standard deviation of cyclical part of real GDP (1954:Q1-2006:Q1) = 1.6%.

Note:

Real consumption includes consumer durables (automobiles, furniture, appliances), which are household capital. Excluding durables makes consumption even less volatile.

Red line is deviation of real GDP from its trend; blue line is deviation of real consumption expenditure from its trend. Deviations are measured in proportionate sense. Real GDP and consumption are quarterly, seasonally adjusted, and in base year dollars (2000).


Total Slides: 37

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Real Investment Expenditure(gross private domestic investment)

16% of GDP (1954-2006 mean)

Pro-cyclical Correlation of cyclical part of real investment

and GDP (1954:Q1-2006:Q1) = 0.92.

More volatile than real GDP Standard deviation of cyclical part of real

investment (1954:Q1-2006:Q1) = 7.2%.

Standard deviation of cyclical part of real GDP (1954:Q1-2006:Q1) = 1.6%.

Red line is deviation of real GDP from its trend; blue line is deviation of real gross private domestic investment expenditure from its trend. Deviations are measured in proportionate sense. Real GDP and investment are quarterly, seasonally adjusted, and in base year dollars (2000).


Total Slides: 37

17

Real Wage Rate

Pro-cyclical Correlation of cyclical part of real wage

and GDP (1964:Q1-2006:Q1) = 0.58.

Less volatile than real GDP

Red line is deviation of real GDP from its trend; blue line is deviation of real wage from its trend. Deviations are measured in proportionate sense. Real GDP and wage rates are quarterly and seasonally adjusted. Note: Real wage is measured by average hourly nominal earnings of production workers in private economy, divided by GDP deflator. Wage series began in 1964.

Post-WWII Stylized Facts Data Patterns to be Explained by Model

Total Slides: 37

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Real Rental Price of Capital

Pro-cyclical Correlation of cyclical part of real wage

and GDP (1954:Q1-2003:Q4) = 0.52.

Less volatile than real GDP

Red line is deviation of real GDP from its trend; blue line is deviation of real rental price of corporate capital from its trend. Deviations are measured in proportionate sense. Note: Real rental price of capital computed by Casey Mulligan. Rental price series available from 1954:Q1 to 2003:Q4.


Total Slides: 37

In Sum, Model Needs to Explain

• Real: Persistent fluctuations in real GDP– Real Consumption: pro-cyclical, less volatile than GDP– Real Investment: pro-cyclical, more volatile than GDP– Real Wage: pro-cyclical, less volatile than GDP – Real rental price for capital: pro-cyclical, less volatile than GDP

• Nominal: Inflation/deflation before WWI;only inflation since WWII.– Long-run “neutrality” of money– Short-run “non-neutrality of money

• Money has “causal” and endogenous components.

19Total Slides: 37

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Equilibrium Business-Cycle ModelBasic Set-Up

• Note: Model highly stylized (i.e., unrealistic)– Based on M. Friedman’s “Methodology of Positive Economics”

• Assume households perform all functions in macro-

economy. – Each household runs family business and uses labor, L, and capital, K, to

produce goods, Y, through a production function: [Y = A· F( K, L)]

– Household interact with other households in four markets: goods market, labor market, rental market (capital services), and bond market.

– All markets are perfectly competitive (i.e., household is price-taker in

each market).

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• Goods Market– Households sell all goods produced in goods market, then buy back goods they want. – Households buys goods for:

• Consumption. • Investment (i.e., to increase stock of goods in form of capital used for production).

• Labor Market– Households supply labor on labor market. – Assume quantity supplied, Ls, is constant, L.

• Rental Market– Each household rents all capital it owns on rental market.– Capital offered on rental market is supply of capital services, Ks. – Since each household rents all its capital (by assumption), Ks is constant, K.

• Bond Market– Borrowing household receives loan from another household.– Household making loan receives piece of paper called bond.– Market in which households borrow/lend is bond market.

Total Slides: 37


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Money• Assume all transactions involve a single medium of exchange

(called money).

• Assume money is just piece of paper, analogous to government-issued fiat currency.– Money is denominated in arbitrary unit (called “dollar”).

– Paper money earns no interest.

– Dollar amounts are nominal terms.

• Sum of household money holdings (money demand) equals aggregate quantity of money in economy (money supply). – Assume aggregate quantity of money is given (constant).

Total Slides: 37


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Closer Look at the Markets (Goods, Labor, Rental, Bond)

Goods Market• Price of goods, denoted P, expresses number of dollars that

exchange for one unit (i.e., P is price level.)• Recall: Y= A·F(K,L)

– Y also represents real quantity of goods sold/bought on goods market, and PY is dollar (nominal) value per year of goods bought/sold on goods market.

– For sellers, price level, P, is number of dollars obtained for each unit of goods sold. For buyers, P is number of dollars paid per unit of goods.

– P dollars buy 1 unit of goods, so $1 buys 1/P units of goods (i.e., expression 1/P is value of $1 in terms of goods).

– M dollars exchange for (M)·(1/P) = M/P units of goods, so M/P is in real terms (real money holdings), whereas quantity like M is in dollar or nominal terms.

Total Slides: 37


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• Labor Market– Households buy/sell labor services (constant, LS) in labor market at dollar or

nominal wage rate, W.– Real wage rate is W/P.

• Rental Market– Households rent capital services (constant KS) in rental market at dollar or nominal

rental price, R.– Real rental price is R/P.

• Bond Market– Each unit of bonds commits borrower to repay $1 to bondholder (i.e., $1 is principal

of each bond.– Each unit of bonds also commits borrower to pay holder flow of yearly interest ($i). – “i” is interest rate (i.e., ratio of interest payment, $i, to principal $1). – Interest rate, i, can vary over time.– All bonds have very short maturity.

Total Slides: 37


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CONSTRUCTING HOUSEHOLD BUDGET CONSTRAINT• Prices/quantities determined in four markets determine household income.

– Flows of income are sources of funds.– Purchases of goods/assets are uses of funds.

• Total sources of funds must equal total uses of funds. This equality is called household budget constraint.

• Sources of Funds: Nominal Household Income = Nominal Profits + Nominal Wage Income

+ Nominal Rental Income + Nominal Interest Income

Profits Households earn profit (excess of revenue over costs) from business activities. Y = A·F( Kd, Ld ) Nominal Profits (π) = PY − (wLd + RKd) Nominal Profits (π) = P·[A·F( Kd, Ld )] − (wLd + RKd) (6.2)

Total Slides: 37

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Sources of Funds (continued): Nominal Household Income = Nominal Profits + Nominal Wage Income

+ Nominal Rental Income + Nominal Interest Income

Wage Income Households supplying quantity of labor Ls to labor market will receive nominal wage income of wLs per year. Quantity of labor supplied is fixed amount, L, so:

nominal wage income = wL.

Rental income Households supplying quantity of capital Ks to rental market receive nominal rental income of RKs per year. Households supply all available capital, K, to rental market, so Ks = K (i.e., nominal rental income is RK).Annual deprecation of capital = δK; dollar value of lost capital is P· δK.

Net nominal rental income (household income from renting capital)

= nominal rental income - value of depreciation

= RK – δ(PK) = (R/P)·PK – δ(PK)

= (R/P - δ)·PK (where “R/P − δ” is rate of return to owning capital)

Total Slides: 37

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Sources of Funds (continued): Nominal Household Income = Nominal Profits + Nominal Wage Income + Nominal Rental Income + Nominal Interest Income

Interest Income If household nominal bond holdings are B, nominal interest income = iB per year. Households hold wealth in financial form (bonds) and physical form (capital). In

equilibrium, rate of return on bonds (i) must equal rate of return on capital (R/P − δ), so:

i = (R/P − δ) (6.6)

***

SUMMING UP:

Nominal Household Income = π + wL + (R/P − δ)·PK + iB (6.5)

which (using 6.6) can be re-written as:

Nominal Household Income = π + wL + i·(PK + B) (6.7)

Total Slides: 37

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Uses of Funds: Nominal Consumption + Nominal Saving

Nominal Consumption Households consume quantity of goods “C” per year at price “P,” so

nominal household consumption = P·C.

Nominal Saving Nominal household saving is change over time in nominal value of household assets (i.e.,

nominal saving= ∆nominal assets).– Household assets include money (M), bonds (B), and physical capital (P K), so nominal asset holdings

= M + B + P·K and ∆nominal household assets = ∆M + ∆B + P· ∆K.– Recall, household nominal demand for money is fixed (by assumption), so ∆M = 0.

Nominal household saving = ∆B + P·∆K

***SUMMING UP:

Uses of Funds = P·C + ∆B + P·∆K (6.5)

Total Slides: 37


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Household Budget Constraint (Nominal Terms) Uses of Funds = Sources of Funds

P·C + ∆B + P·∆K = π + wL + i·(PK + B) (6.11)

Household Budget Constraint (Real Terms) Divide (6.11) through by price of goods (P)

C + ∆B/P + ∆K = π/P + (w/P)·L + i·(K + B/P)(6.12)

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Clearing of Markets for Labor and Capital Services Demand for Labor/Capital Services Derived from Profit Maximization

(i.e., firms take prices of labor/capital as given, set output at profit-maximizing level, and hire quantity of labor/capital services necessary to produce that output.)

Profit Maximizing Labor Demand:

Real Profit = π/P = A·F(Kd,Ld) − (W/P)·Ld − (R/P)·Kd

– Hiring more labor raises revenues by A·F(Kd,ΔLd) – marginal revenue (MR)

– Hiring more labor raises costs by W/P·(ΔLd) – marginal cost (MC)

– Note: A·F(Kd,ΔLd) = Marginal Product of Labor (MPL)

– Real profit maximized when last unit of labor (ΔLd = 1) results in MR = MC, so profit-maximizing level of labor input is:

MPL = W/P


Total Slides: 37

33Total Slides: 37


Demand for Labor (MPL)

At L1, MPL > W/P (i.e., last unit of labor added more to revenues than to costs, thereby boosting profits.) So, firm hires more labor (and MPL falls) until MPL = W/P.

At L2, MPL < W/P (i.e., last unit of labor added more to costs than to revenues, thereby reducing profits.) So, firm lays off labor (and MPL rises) until MPL = W/P.

Implications: Profit-maximizing labor input (Ld)

wherever W/P cuts MPL curve. MPL curve is demand for labor

curve.

34Total Slides: 37


Labor MarketPutting Labor Demand/Labor Supply Together

Supply of Labor: – Recall: Each household supplies fixed

quantity of labor market (i.e., market supply of labor, Ls, is perfectly inelastic curve, L).

Market Clearing:– W/P adjusts to equate aggregate quantity

of labor demanded, Ld, to aggregate quantity supplied, L.

– Equilibrium real wage = W/P* (= MPL)

35Total Slides: 37


Rental Market

Demand for Capital Services: – As with labor, households maximize real

profits by renting capital services up to point where marginal product of capital (MPK) equals real rental price of capital (R/P).

– Demand for capital services given by MPK curve.

Supply of Capital Services: – Recall: Each household supplies all services

from existing capital stock to rental market (i.e., market supply of capital services, Ks, is perfectly inelastic curve, K.)

Market Clearing:– R/P adjusts to equate aggregate quantity of

capital services demanded, Kd, to aggregate quantity supplied, K.

– Equilibrium real price of capital = R/P* (= MPK)

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Closing Observations: Recall: In asset-market equilibrium, rate of return on bonds = rate of

return on ownership of capital (i.e., i = R/P − δ or R/P = i - δ). Also, rental market equilibrium implies R/P = MPK. So,

MPK = i + δ (6.16)

In words, interest rate cannot change unless MPK changes.– For given technology (A) MPK depends on inputs of capital (K) and labor services

(L). In current set-up, “K” and “L” are fixed, so “i” is as well. Put another way, “i” cannot change – clearly at odds with real world (where rates change constantly).

– Later model will be extended to allow fluctuations in MPK (and hence “i”)

When labor and rental markets clear, real economic profits = 0. – Economic profit is return to running business. All households (by assumption) are

similar, and running business is riskless (no uncertainty in model).

Total Slides: 37

Introduction to the

Equilibrium Business-Cycle Model?

Mr. VaughanIncome and Employment Theory (402)

37Total Slides: 37

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Introduction to the Equilibrium Business-Cycle Model