EC Exam 2 C9

Chapter 9 Notes

Introduction to the IS-LM / AD-AS Framework

The IS-LM model consolidates three core macroeconomic markets into a unified framework to analyze business cycles, macroeconomic policy effects, and economic forecasting:

• Labor market

• Goods market

• Asset market

Historical Context and Origins

• Sir John Hicks (1937): Developed in his article Mr. Keynes and the Classics: A Suggested Interpretation (published in Econometrica, April 1937, pp. 137–159).

• Originally created as a graphical representation of the economic theory set forth by John Maynard Keynes in his 1936 work The General Theory of Employment, Interest, and Money.

Meaning of the Terminology

• IS: Represents equilibrium in the goods market, where desired investment ($I$) equals desired national saving ($S$).

• LM: Represents equilibrium in the asset market, where real money demand ($L$) equals real money supply ($M / P$).

Theoretical Flexibility

• Hicks originally assumed a fixed price level in the short run, aligning with Keynesian price stickiness.

• The model can be adapted to represent the Classical approach, where wages and prices adjust rapidly to clear all markets.

• The IS-LM model is mathematically and conceptually equivalent to the Aggregate Demand–Aggregate Supply (AD-AS) model.

Methodological Benefits of a Unified Model

• Eliminates the necessity of learning two separate frameworks for Classical and Keynesian economic analyses.

• Highlights fundamental areas of agreement between Classical and Keynesian macroeconomists while precisely isolating their specific points of divergence (such as the speed of price adjustment).

• Serves as a standard framework utilized in policy debates, empirical research, and applied forecasting.

The FE Line: Equilibrium in the Labor Market

Definitions

• Full-Employment Level of Employment ($\bar{N}$): The equilibrium quantity of employment attained after wages and prices have completely adjusted across the economy.

• Full-Employment Output ($\bar{Y}$): The total volume of real output produced when labor input is at its full-employment level ($\bar{N}$), given the existing capital stock ($K$) and total factor productivity ($A$).

Algebraic Representation:

Y_ = A * F(K, N_) = The production function

Graphic Representation of the Full-Employment (FE) Line

• Plotted on a two-dimensional graph with the real interest rate ($r$) on the vertical axis and aggregate real output ($Y$) on the horizontal axis.

• The FE line is vertical at $Y = \bar{Y}$ because labor market equilibrium determines full-employment output independently of the current real interest rate.

• While the real interest rate influences investment and future capital accumulation, it does not alter the current capital stock or current full-employment output.

Factors That Shift the FE Line

• Beneficial Supply Shock (Shifts FE Right):

  1. Increases the output produced for any given combination of capital and labor.

  2. Raises the marginal product of labor ($MPN$), increasing labor demand and raising equilibrium employment ($\bar{N}$).

• Increase in Labor Supply (Shifts FE Right): Raises equilibrium employment ($\bar{N}$), expanding full-employment output ($\bar{Y}$).

• Increase in the Capital Stock ($K$) (Shifts FE Right): Direct capability to produce more output with existing labor; additionally, increased capital may raise $MPN$, raising labor demand and equilibrium employment ($\bar{N}$).

• Adverse Supply Shock, Decline in Labor Supply, or Reduction in Capital Stock: Shifts the FE line to the left by lowering full-employment output ($\bar{Y}$).

The IS Curve: Equilibrium in the Goods Market

Definition

Shows the combinations of output ($Y$) and real interest rate ($r$) for which the goods market is in equilibrium. At every point on the IS curve, desired national saving ($S^d$) equals desired investment ($I^d$), or aggregate output supplied ($Y$) equals aggregate demand for goods ($C^d + I^d + G$).

Graphical Derivation from the Saving-Investment Diagram

• In saving-investment space ($S^d, I^d$ on the horizontal axis, $r$ on the vertical axis), desired saving ($S^d$) slopes upward because a higher real interest rate incentivizes current saving.

• An increase in output ($Y$) raises disposable income and increases desired saving at any real interest rate, shifting the saving curve rightward (e.g., from $S(Y = 4000)$ to $S(Y = 5000)$).

• Desired investment ($I^d$) slopes downward because a higher real interest rate increases the user cost of capital, reducing desired investment. Current output ($Y$) does not directly affect desired investment.

• Market-clearing points:

  • When $Y = 4000$, goods market equilibrium occurs at point $D$, where $r = 7\%$.

  • When $Y = 5000$, goods market equilibrium occurs at point $F$, where $r = 5\%$.

• Plotting $(Y, r)$ pairs $D(4000, 7\%)$ and $F(5000, 5\%)$ yields a downward-sloping IS curve.

Explanation of Downward Slope via Aggregate Supply and Demand

• At any point on the IS curve, $Y = C^d + I^d + G$.

• If real interest rate $r$ rises, desired consumption ($C^d$) and desired investment ($I^d$) fall, creating excess supply of goods at the initial output level.

• To restore equilibrium in the goods market at the higher real interest rate, output supplied ($Y$) must fall.

• A drop in output reduces supply by 1 unit while reducing desired consumption by less than 1 unit because the marginal propensity to consume ($MPC$) is less than 1. Therefore, falling output clears excess aggregate supply.

Factors That Shift the IS Curve

For a constant level of output, any economic change that decreases desired national saving relative to desired investment (or increases aggregate demand for goods) increases the goods-market-clearing real interest rate, shifting the IS curve up and to the right. Conversely, any change that increases desired national saving relative to desired investment shifts the IS curve down and to the left.

• Expected Future Output (Shifts IS Up/Right): Desired saving falls (desired consumption rises), raising the real interest rate that clears the goods market.

• Wealth (Shifts IS Up/Right): Desired saving falls (desired consumption rises), raising the market-clearing real interest rate.

• Government Purchases ($G$) (Shifts IS Up/Right): Desired saving falls ($S^d = Y - C^d - G$) and aggregate demand increases directly, raising the market-clearing real interest rate.

• Taxes ($T$): No change if Ricardian equivalence holds. Shifts IS down and to the left if consumers reduce desired consumption in response to higher current taxes.

• Expected Future TFP / MPK (Shifts IS Up/Right): Desired investment and desired consumption increase, raising the market-clearing real interest rate.

• Effective Tax Rate on Capital (Shifts IS Down/Left): Desired investment falls, lowering the market-clearing real interest rate.

The LM Curve: Asset Market Equilibrium

Asset Market Aggregation and Equilibrium Condition

• All assets in the economy are aggregated into two main categories: money (paying nominal interest $i^m$) and nonmonetary assets (paying nominal interest $i$ and expected real interest $r = i - \pi^e$).

• Asset market equilibrium reduces to the condition that real money supply equals real money demand.

Relationship Between Asset Prices and Interest Rates

Example of a 1-year discount bond paying $10,000 at maturity:

• If current purchase price is $9,615, nominal return is $10,000 - \$9,615 = \$385$, yielding a nominal interest rate of ~4% per year.

• If current purchase price falls to $9,524, nominal return is $10,000 - \$9,524 = \$476$, yielding a nominal interest rate of ~5$MS = M / P$) is represented as a vertical line because nominal money supply $M$ is set by the central bank and price level $P$ is taken as given (e.g., $M = 2000, P = 2 \implies M / P = 1000$).</p></li><li><p>Real money demand ($MD = L(Y, r + \pi^e)$) slopes downward relative to real interest rate $r$ because a higher interest rate increases the opportunity cost of holding money.

• An increase in output ($Y$) increases transaction volume, shifting the real money demand curve rightward.

• Equilibrium derivation:

  • At $Y = 4000$, money demand equals real supply at $r = 3\%$ (point $A$).

  • At $Y = 5000$, money demand shifts right; at $r = 3\%$, excess money demand equals $1200 - 1000 = 200$ (point $B$).

  • Wealth holders sell nonmonetary assets to acquire money, driving down asset prices and bidding up the real interest rate to 5% (point $C$), restoring asset market equilibrium.

• Plotting $(Y, r)$ pairs $A(4000, 3\%)$ and $C(5000, 5\%)$ defines an upward-sloping LM curve.

Factors That Shift the LM Curve

• Nominal Money Supply ($M$) (Shifts LM Down/Right): Increases real money supply $M / P$, lowering the market-clearing real interest rate.</p></li><li><p><strong>Price Level ($P$) (Shifts LM Up/Left): Reduces real money supply $M / P$, raising the market-clearing real interest rate.</p></li><li><p><strong>Expected Inflation ($\pi^e$) (Shifts LM Down/Right):</strong> Reduces demand for money by making nonmonetary assets more attractive, lowering the market-clearing real interest rate.</p></li><li><p><strong>Nominal Interest Rate on Money ($i^m$) (Shifts LM Up/Left):</strong> Increases demand for money, raising the market-clearing real interest rate.</p></li><li><p><strong>Other Factors Increasing Real Money Demand (Shift LM Up/Left):</strong> Increase in wealth, increase in risk of alternative nonmonetary assets relative to money, decline in liquidity of alternative assets, or decline in efficiency of payment technologies.</p></li></ul><div data-type="horizontalRule"><hr></div><h2 id="ca5573ff-bc21-4f92-9a4c-5a9a09b050e7" data-toc-id="ca5573ff-bc21-4f92-9a4c-5a9a09b050e7" collapsed="false" seolevelmigrated="true">General Equilibrium in the Complete IS-LM Model</h2><h3 id="2e265aa6-ea11-4ef6-b096-915b5e0bf5fc" data-toc-id="2e265aa6-ea11-4ef6-b096-915b5e0bf5fc" collapsed="false" seolevelmigrated="true">Definition of General Equilibrium</h3><ul><li><p>Occurs when all three economy-wide markets (labor, goods, and assets) are simultaneously in equilibrium.</p></li><li><p>Graphically represented by the unique intersection point ($E$) of the FE line, the IS curve, and the LM curve.

Complete Dynamic Adjustment to General Equilibrium

• The general equilibrium position is pinned down by the intersection of the IS curve and the FE line.

• Price level adjustments cause the LM curve to shift dynamically until it passes directly through the IS-FE intersection point.

Detailed Application: Analysis of a Temporary Adverse Supply Shock

• Initial State: General equilibrium at point $E$, intersecting $FE_1$, $IS$, and $LM_1$.

• Shock: Productivity parameter $A$ drops temporarily due to bad weather or an oil price shock.

• Labor Market Reaction: $MPN$ declines, shifting labor demand downward. Labor supply is unaffected. Equilibrium real wage ($w$) and employment ($N$) fall.</p></li><li><p><strong>FE Line Shift:</strong> Full-employment output ($\bar{Y}$) falls. The FE line shifts leftward from $FE_1$ to $FE_2$.

• IS Curve Reaction: Does not shift because the shock changes current output without altering functional saving or investment curves. The economy moves along the existing IS curve.

• LM Curve & Price Level Adjustment: At point $F$ ($FE_2$ and $IS$ intersection), aggregate demand exceeds the lower full-employment output level ($\bar{Y}_2$). Excess demand causes the price level ($P$) to rise. Rising $P$ contracts real money supply ($M / P$), shifting the LM curve upward and leftward from $LM_1$ to $LM_2$ until it intersects $FE_2$ and $IS$ at point $F$.</p></li></ul><h4 id="09c3993c-d002-4bfe-9129-f735f182f865" data-toc-id="09c3993c-d002-4bfe-9129-f735f182f865" collapsed="false" seolevelmigrated="true">Summary of Macroeconomic Outcomes:</h4><ul><li><p>Real output ($Y$) falls from $Y_1$ to $Y_2$.</p></li><li><p>Real interest rate ($r$) rises.</p></li><li><p>Price level ($P$) increases, creating a temporary surge in inflation.</p></li><li><p>Employment ($N$) and real wage ($w$) fall.</p></li><li><p>Consumption ($C$) and investment ($I$) decline due to higher real interest rates and lower income.

Case Study: The Oil Price Shock of 2008

• Oil prices surged from $51/barrel (Jan 2007) / $96 (Jan 2008) to a record peak of $145 in July 2008.

• Combined with the collapse of the housing market and financial sector crisis, this adverse supply shock pushed the economy into a deep recession.

• Subsequent collapse in global oil demand dropped prices to $30 by December 2008, converting the supply shock from adverse to beneficial, though financial crisis drag dominated overall real activity.

Applied Econometric Forecasting Models (FRB/US Model)

• Federal Reserve Board utilizes the FRB/US econometric model (introduced in 1996 to replace the MPS model) for policy analysis and macroeconomic projection.

• Integrates microfoundations across three sectors (households, firms, financial markets) and incorporates explicit forward-looking expectations and general equilibrium market clearing.

Price Adjustment and the Attainment of General Equilibrium

Speed of Market Adjustments

• Asset Market (LM): Adjusts almost instantaneously (minutes).

• Goods Market (IS): Adjusts at moderate speed.

• Labor Market (FE): Adjusts slowest due to search friction and periodic wage re-contracting.

Detailed Analysis of a Nominal Monetary Expansion (10% Increase in $M$)

• Initial State: General equilibrium at point $E$ ($Y = 1000, r = 5\%, P = 100$).

• Short-Run Equilibrium:

  • Central bank increases nominal money supply $M$ by 10%.

  • Price level $P$ is fixed at 100 in the short run; real money supply ($M / P$) increases by 10%.

  • LM curve shifts down and right from $LM_1$ to $LM_2$.

  • Short-run equilibrium occurs at point $F$ (IS-$LM_2$ intersection), where $r$ falls to 3% and aggregate output demanded rises to 1200.

  • Asset and goods markets clear at $F$, but labor market is in disequilibrium ($1200 > 1000$).

• Long-Run Adjustment:

  • Producing 1200 requires firms to operate above full-employment capacity.

  • Aggregate demand exceeds desired supply, prompting firms to raise prices.

  • Price level $P$ rises by exactly 10% (from 100 to 110).

  • Restores real money supply ($M / P$) to its original level, shifting the LM curve back up and left to $LM_1$.

  • General equilibrium is fully restored at point $E$ ($Y = 1000, r = 5\%, P = 110$).

Classical vs. Keynesian Perspectives on Price Adjustment

• Self-Correcting Mechanism: Both schools agree price adjustments eventually return the economy to general equilibrium.

• Classical View: Wages and prices are fully flexible and adjust rapidly. The economy self-corrects almost immediately, staying continuously on or near the FE line.

• Keynesian View: Wages and prices are sticky and adjust slowly. The economy can remain out of general equilibrium at short-run point $F$ for extended periods (months or years), during which output is aggregate demand-driven.

Monetary Neutrality

• Definition: Money is neutral if a change in nominal money supply $M$ causes a proportional change in price level $P$ without altering real economic variables ($Y, N, r, C, I, w/P$).

• In the long run, monetary neutrality holds in both Classical and Keynesian models.

• In the short run, money is non-neutral under Keynesian price stickiness.

Trend Money Growth and Inflation

If nominal money supply $M$ and price level $P$ grow at a steady trend rate (e.g., 7% per year), real money supply ($M / P$) remains constant, and the LM curve does not shift.</p><div data-type="horizontalRule"><hr></div><h2 id="52be536a-cac1-4445-bfa2-fb504477463f" data-toc-id="52be536a-cac1-4445-bfa2-fb504477463f" collapsed="false" seolevelmigrated="true">Aggregate Demand and Aggregate Supply (AD-AS) Model</h2><h3 id="397f145d-eca7-43a6-8b31-311e595699ff" data-toc-id="397f145d-eca7-43a6-8b31-311e595699ff" collapsed="false" seolevelmigrated="true">Relationship to IS-LM Model</h3><ul><li><p>Equivalent representation of the same underlying macroeconomic structure.</p></li><li><p>IS-LM maps output ($Y$) against real interest rate ($r$); AD-AS maps output ($Y$) against price level ($P$).</p></li></ul><h3 id="de9f4eee-866e-42c8-bf7c-86199be997c2" data-toc-id="de9f4eee-866e-42c8-bf7c-86199be997c2" collapsed="false" seolevelmigrated="true">Aggregate Demand (AD) Curve</h3><ul><li><p>Shows aggregate quantity of output demanded ($C^d + I^d + G$) at any given price level $P$.

Aggregate Supply Curves

• Short-Run Aggregate Supply (SRAS): Horizontal line at the prevailing fixed price level $P_{sr}$. Firms supply whatever output is demanded at this price level in the short run.

• Long-Run Aggregate Supply (LRAS): Vertical line at full-employment output $Y = \bar{Y}$. In the long run, wages and prices fully adjust, clearing labor markets.

Monetary Neutrality in AD-AS Model

1. A 10% monetary expansion shifts the AD curve up and right from $AD_1$ to $AD_2$.

2. Short-run equilibrium moves from $E$ to $F$ along $SRAS_1$ ($Y$ increases to $Y_2$, $P$ remains fixed at $P_1$).</p></li><li><p>Higher output ($Y_2 > \bar{Y}$) pushes firms to raise prices, shifting SRAS upward from $SRAS_1$ to $SRAS_2$.

3. Long-run equilibrium reaches point $H$, where output returns to $\bar{Y}$ and price level rises by 10% to $P_2$.

Worked-Out Numerical Exercise for Solving the IS-LM/AD-AS Model

Structural Model Equations:

• Desired consumption: $C^d = 300 + 0.75(Y - T) - 300r$

• Tax function: $T = 100 + 0.2Y$

• Desired investment: $I^d = 200 - 200r$

• Real money demand: $L = 0.5Y - 500i$

• Exogenous Parameters: $\bar{Y} = 2500$, $G = 600$, $M = 133,200$, $\pi^e = 0.05$, $P_{sr} = 120$

Step 1: Derive Equation for the IS Curve

Goods market equilibrium: $Y = C^d + I^d + G$

Substitute $T$ into $C^d$:

C^d = 300 + 0.75(Y - 100 - 0.2Y) - 300r = 225 + 0.6Y - 300r

Substitute $C^d, I^d, G$ into equilibrium condition:

Y = (225 + 0.6Y - 300r) + (200 - 200r) + 600

Y = 1025 + 0.6Y - 500r

0.4Y = 1025 - 500r

r = 2.05 - 0.0008Y

Step 2: Derive Equation for the LM Curve

Nominal interest rate equation: $i = r + \pi^e = r + 0.05$

Money demand equation:

L = 0.5Y - 500(r + 0.05) = 0.5Y - 500r - 25

Asset market equilibrium:

133,200 / P = 0.5Y - 500r - 25

Solve for $r$ in terms of $P$ and $Y$:

500r = 0.5Y - 25 - 133,200 / P

r = 0.001Y - 0.05 - 266.4 / P

Short-run LM curve with fixed price $P = P_{sr} = 120$:</p><p>r = 0.001Y - 0.05 - 266.4 / 120 = 0.001Y - 2.27</p><h3 id="46c18828-e27c-4744-ae89-77ceb58ea847" data-toc-id="46c18828-e27c-4744-ae89-77ceb58ea847" collapsed="false" seolevelmigrated="true">Step 3: Solve for Short-Run Equilibrium ($P = 120$)

Equate IS and short-run LM curves:

2.05 - 0.0008Y = 0.001Y - 2.27

4.32 = 0.0018Y ==> Y = 2400

• Real interest rate: $r = 2.05 - 0.0008(2400) = 0.13$ (13%)

• Taxes: $T = 100 + 0.2(2400) = 580$

• Consumption: $C = 300 + 0.75(2400 - 580) - 300(0.13) = 1626$

• Investment: $I = 200 - 200(0.13) = 174$

• Aggregate expenditure check: $C + I + G = 1626 + 174 + 600 = 2400 = Y$

• Nominal interest rate: $i = r + \pi^e = 0.13 + 0.05 = 0.18$ (18%)

Step 4: Solve for Long-Run General Equilibrium ($Y = \bar{Y} = 2500$)

• Real interest rate from IS curve: $r = 2.05 - 0.0008(2500) = 0.05$ (5%)

• Taxes: $T = 100 + 0.2(2500) = 600$

• Consumption: $C = 300 + 0.75(2500 - 600) - 300(0.05) = 1710$

• Investment: $I = 200 - 200(0.05) = 190$

• Nominal interest rate: $i = 0.05 + 0.05 = 0.10$ (10%)

• Real money demand: $L = 0.5(2500) - 500(0.10) = 1200$

• Long-run price level $P$ from money market clearing:

  133,200 / P = 1200 ==> P = 133,200 / 1200 = 111

Step 5 & 6: Derive AD Curve and Verify Equilibria

Equate IS and general LM equations:

2.05 - 0.0008Y = 0.001Y - 0.05 - 266.4 / P

0.0018Y = 2.10 + 266.4 / P

Y = 1166.67 + 148,000 / P

• Short-run output check ($P = 120$): $Y = 1166.67 + 148,000 / 120 = 2400$</p></li><li><p><strong>Long-run price check ($Y = 2500$): $2500 = 1166.67 + 148,000 / P \implies 1333.33 = 148,000 / P \implies P = 111$

Algebraic Version of the IS-LM/AD-AS Model

Labor Market Equations

• Production Function: $Y = A(f_1 N - 0.5 f_2 N^2)$</p></li><li><p><strong>Marginal Product of Labor ($MPN$): $MPN = A(f_1 - f_2 N)$

• Labor Demand Equation: $w = A(f_1 - f_2 N^D)$

• Labor Supply Equation: $N^S = n_0 + n_w (1 - t) w$</p></li></ul><p><strong>Equilibrium Real Wage ($w$):</strong></p><p>w = (A(f_1 - f_2 <em> n_0)) / (1 + (1 - t) </em> A <em> f_2 </em> n_w)</p><p><strong>Full-Employment Employment ($\bar{N}$):</strong></p><p>N_bar = (n_0 + (1 - t) <em> A </em> f_1 <em> n_w) / (1 + (1 - t) </em> A <em> f_2 </em> n_w)</p><p><strong>Full-Employment Output ($\bar{Y}$):

  Y_bar = A(f_1 N_bar - 0.5 f_2 * N_bar^2)

  Goods Market Equations

  • Desired Consumption: $C^d = c_0 + c_Y (Y - T) - c_r r$

  • Income Tax Function: $T = t_0 + t Y$

  • Desired Investment: $I^d = i_0 - i_r r$

  • Equilibrium Condition: $Y = C^d + I^d + G$</p></li></ul><p><strong>Algebraic IS Curve Equation:</strong></p><p>r = alpha_IS - beta_IS * Y</p><p>Where:</p><ul><li><p>$\alpha_{IS} = (c_0 + i_0 + G - c_Y * t_0) / (c_r + i_r)$</p></li><li><p>$\beta_{IS} = (1 - (1 - t)c_Y) / (c_r + i_r)$

  Asset Market Equations

  • Real Money Demand: $M^d / P = \eta_0 + \eta_Y Y - \eta_r (r + \pi^e)$</p></li></ul><p><strong>Algebraic LM Curve Equation:</strong></p><p>r = alpha_LM - (1 / eta_r) <em> (M / P) + beta_LM </em> Y</p><p>Where:</p><ul><li><p>$\alpha_{LM} = (\eta_0 / \eta_r) - \pi^e$</p></li><li><p>$\beta_{LM} = \eta_Y / \eta_r$

  General Equilibrium Solutions

  • Real Interest Rate: $r_bar = \alpha_{IS} - \beta_{IS} * Y_bar$</p></li><li><p><strong>Price Level ($P$):

    P = M / (eta_0 + eta_Y Y_bar - eta_r (alpha_IS - beta_IS * Y_bar + pi^e))

  Aggregate Demand (AD) and Supply Solutions

  Aggregate Demand Curve Equation:

  Y = (alpha_IS - alpha_LM + (1 / eta_r) * (M / P)) / (beta_IS + beta_LM)

  • Short-Run Aggregate Supply (SRAS): $P = P_{sr}$

  • Long-Run Aggregate Supply (LRAS): $Y = \bar{Y}$

  Chapter 9 Lecture Notes

  The Expectations Hypothesis and the Term Structure of Interest Rates

  Investment Horizon and Asset Substitution

  When choosing a two-year investment horizon, an investor can either:

  • Purchase a single two-year bond.

  • Purchase a one-year bond today and roll over the principal and interest into a new one-year bond next year.

  Under the pure expectations theory and assuming risk neutrality, investors are indifferent between these two strategies.

  Derivation of the Two-Period Bond Yield

  • i2: Nominal yield on a two-year bond.

  • i1: Current nominal yield on a one-year bond.

  • i1_e: Expected nominal yield on a one-year bond one year in the future.

  • Total returns compared:

    • Two-year bond: (1 + i2)^2 = 1 + 2(i2) + (i2)^2

    • Consecutive one-year bonds: (1 + i1) (1 + i1_e) = 1 + i1 + i1_e + (i1 i1_e)

  • Approximation for low rates: When nominal interest rates are low (e.g., 1% or 4%), squared and cross-product terms become negligible (e.g., 0.04^2 = 0.0016). Canceling the 1s and ignoring the negligible products simplifies the equation to:

    • 2(i2) = i1 + i1_e

    • i2 = (i1 + i1_e) / 2 (The two-year yield is the arithmetic average of the current and expected future one-year yields).

  Multi-Period Generalization and Term Premium

  For an n-period asset, the yield is the average of the current and expected future one-year yields over n periods, plus a Term Premium (TP).

  • in = [ (Sum of 1-year yields from period 1 to n) / n ] + TPn

  • Term Premium (TP): Extra compensation demanded by investors for locking up capital in longer maturities, which carry higher default and inflation risks. TP generally increases with maturity.

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  Calculating Yields and the Yield Curve

  Numerical Scenario Parameters

  • Current 1-year yield (i1): 6%

  • Expected 1-year yield in year 2 (i1_e2): 4%

  • Expected 1-year yield in year 3 (i1_e3): 3%

  • 2-year Term Premium (TP2): 0.5%

  • 3-year Term Premium (TP3): 1.0%

  Step-by-Step Yield Computations

  • 1-Year Yield: 6.0%

  • 2-Year Yield: Average (6% + 4%) / 2 + 0.5% TP = 5.5%

  • 3-Year Yield: Average (6% + 4% + 3%) / 3 + 1.0% TP = 4.33% + 1.0% = 5.33%

  Yield Curve Implications

  • Inverted Curve (from example): 1yr = 6.0%, 2yr = 5.5%, 3yr = 5.3%. Yields drop as maturity increases. Historically signals an economic slowdown or recession due to expectations of sharply dropping future interest rates.

  • Normal Curve: Slopes upward. Short-term yields are lower than long-term yields. Arises from expectations of higher future inflation and naturally increasing term premiums.

  • Flattening Curve: Transition phase, often seen ahead of a recession.

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  Money Demand, Velocity, and Elasticity

  Circulation and Velocity

  • Velocity is the annual frequency a single unit of currency circulates to finance transactions.

  • Velocity = Nominal Transaction Value / Money Supply

  • Example: $1,000,000 nominal GDP / $100,000 money supply = Velocity of 10.

  Functional Form and Elasticity

  • Money Demand (Md) = P * L(Y, i) (Depends on Price level, Real GDP, and nominal interest rate).

  • Growth Rate Formula: %Change M = %Change P + (nY %Change Y) + (ni %Change i)

    • %Change M: Money supply growth

    • %Change P: Inflation rate

    • nY: Income elasticity of money demand (positive value)

    • %Change Y: Real GDP growth

    • ni: Interest rate elasticity (between -1 and 0, approx -0.1)

    • %Change i: Relative percentage change in nominal interest rate

  Absolute vs. Relative Interest Rate Changes

  Elasticity equations require relative percentage growth, not absolute changes.

  • Absolute Change: 3% - 2% = 1 percentage point difference.

  • Relative Change: (3% - 2%) / 2% = 50% growth rate.

  • Application: With an interest elasticity of -0.1, a relative interest rate increase of 50% causes a 5% drop in money demand (-0.1 * 50%).

  Long-Run Money Neutrality

  • In the long run, money is neutral: money supply changes do not alter real variables (real output, real returns).

  • %Change P = %Change M (e.g., a 10% increase in money supply yields exactly 10% inflation).

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  Comparative Statics and Monetary Policy

  Output Growth and Deflation (Fixed Money Supply)

  • If real GDP grows (e.g., 6%), and both money supply and interest rates are constant (0% change), price levels must drop to maintain equilibrium.

  • Example (Income elasticity = 0.5): 0 = Inflation + (0.5 * 6%) + 0 -> Inflation = -3% (Deflation).

  • Policy Implication: Central banks must expand the nominal money supply proportionally with real output growth to prevent deflation and maintain price stability.

  Monetary Policy Transmission

  The Federal Open Market Committee (FOMC) targets the Federal funds rate.

  • Expansionary Policy: Lowers targets, increases money supply.

  • Contractionary Policy: Raises targets, decreases money supply.

  Short-Run Dynamics (Sticky Prices)

  • Prices are fixed. Expanding money supply shifts the real money supply curve right.

  • Real interest rates drop, reducing the user cost of capital.

  • Lower costs induce firms to increase investment. Lower rates reduce saving incentives, increasing consumption.

  • Aggregate demand rises, driving short-run output above potential.

  Long-Run Dynamics (Flexible Prices)

  • Excess demand in the goods market eventually causes firms to raise prices.

  • Higher prices shift the real money supply curve back to the left.

  • Real interest rates and real GDP return to baseline equilibrium. The net result is a purely nominal price level increase.

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  General Equilibrium Framework: The FE Line

  General equilibrium unifies the three macroeconomic markets via the real interest rate, real output, and price level.

  The Three Markets

  1. Labor Market: Real wage vs. Employment. Intersection dictates market-clearing employment (N*).

  2. Goods Market: Real interest rate vs. Saving/Investment.

  3. Asset Market: Real money supply vs. Real money demand.

  The Full-Employment (FE) Line

  • Traces combinations of real interest rate and real output that maintain labor market equilibrium.

  • Vertical shape: The FE line is perfectly vertical because full-employment output does not depend on current real interest rates.

  • Current capital and technology are based on historical accumulation, not current rates. While current rates alter future capital through investment, they do not impact current labor supply, labor demand, or current full-employment output.

  Foundations of the IS-LM General Equilibrium Model

  The macroeconomic IS-LM framework integrates equilibrium conditions across three interrelated markets (goods, money, and labor) to analyze how changes in real output (Y) and real interest rates (r) respond to economic shocks. It accounts for 22 distinct economic variables:

  • 6 labor market shocks that shift the Full-Employment (FE) line.

  • 6 structural shocks that shift the goods market (IS) curve.

  • 7 asset/money market shocks that shift the LM curve.

  • Additional real and nominal variables governing equilibrium transitions.

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  The IS Curve (Goods & Loanable Funds Market)

  The IS curve represents all combinations of real interest rates (r) and real output (Y) where the goods market clears, meaning National Saving equals Planned Investment.

  • Deriving the Downward Slope:
    In the loanable funds market, saving acts as supply and investment as demand. When real GDP (Y) increases, national saving rises, shifting the supply of loanable funds to the right. This puts downward pressure on the equilibrium real interest rate (r). This inverse relationship between Y and r generates the downward slope of the IS curve.

  • Movements vs. Shifts:
    Changes in real output (Y) represent a movement along the IS curve. Shifts of the entire curve occur when exogenous variables alter saving or investment while Y is held fixed.

  Shift Factors of the IS Curve

  Six structural variables shift the IS curve directly:

  1. Future expected output

  2. Government purchases (G)

  3. Lump-sum taxes (T)

  4. Effective tax rate on capital

  5. Future marginal productivity of capital

  6. Household wealth / autonomous consumption

  Transmission Mechanisms (Shifting IS Up and to the Right):

  • Higher Government Purchases (G): Increases in G directly reduce national saving. The saving curve shifts left in the loanable funds market, driving r upward at any given Y.

  • Tax Cuts (Decreased T): Households save a fraction of a tax cut, but private saving doesn't fully offset the resulting public deficit. Total national saving falls, driving r up.

  • Investment Incentives: A lower capital tax rate or higher expected future productivity raises the expected return on capital. Firms increase investment demand, raising r at every output level.

  • The "Good News" Heuristic: Any event perceived as positive economic news or an autonomous expansion in aggregate demand (e.g., fiscal stimulus, equity market boom) shifts the IS curve up and to the right.

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  The LM Curve (Asset Market Equilibrium)

  The LM curve represents all combinations of Y and r that equate the supply of real money balances to real money demand.

  • Deriving the Upward Slope:
    Real money supply is fixed (nominal money stock [M] divided by the price level [P]). Real money demand is positively related to output (Y) for transaction purposes and negatively related to the nominal interest rate (the opportunity cost of holding non-interest-bearing cash). If Y increases, transaction demand for money rises. With a fixed real money supply, r must rise to clear the market, creating an upward slope.

  Shift Factors of the LM Curve

  Seven structural shocks shift the LM curve by altering asset market dynamics:

  1. Nominal money supply (M)

  2. Price level (P)

  3. Expected inflation

  4. Nominal interest rate paid on money balances

  5. Riskiness of alternative assets

  6. Liquidity of alternative assets

  7. Financial technology / payment efficiency

  Transmission Mechanisms:

  • Expected Inflation: Higher expected inflation acts as a depreciation risk tax on holding cash. Money demand drops, reducing the interest rate needed to clear the market, shifting the LM curve down and to the right.

  • Asset Riskiness: When non-monetary assets (like stocks) become risky, agents shift to safe, liquid cash. This spikes money demand, shifting the LM curve up and to the left. If alternatives become secure, money demand drops, shifting LM down and to the right.

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  Market Clearing Dynamics & Equilibrium

  The macroeconomic system features distinct adjustment velocities across its three markets:

  1. Money/Asset Market (Fastest): Prices and yields of financial assets adjust continuously and near-instantaneously.

  2. Goods/Loanable Funds Market (Moderate): Adjusts via changes in orders, inventories, and real yields.

  3. Labor Market (Slowest): Wage setting, institutional contracts, and nominal rigidities cause protracted adjustments.

  Short-Run vs. Long-Run Equilibrium:

  • Short-Run: Defined by the intersection of the IS and LM curves. Because the goods and asset markets clear rapidly while the labor market is slow to adjust, the labor market remains in disequilibrium. Output can temporarily deviate from full employment.

  • Long-Run: Defined by the simultaneous intersection of the IS curve, LM curve, and vertical Full-Employment (FE) line. Prices and wages fully adjust, the labor market clears, and output is fixed at potential.

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  Monetary Policy & Price Stickiness (Contraction Example)

  Short-Run Dynamics of a Money Supply Contraction:

  1. The central bank reduces the nominal money supply (M).

  2. Because the price level (P) is sticky in the short run, the real money supply (M/P) plummets, causing a liquidity shortage.

  3. Agents liquidate bonds to get cash, depressing bond prices and driving interest rates up (LM curve shifts up/left).

  4. Simultaneously, as the economy moves along the IS curve to a lower Y, national saving decreases, reinforcing the upward pressure on interest rates.

  5. The economy enters a temporary recession (intersecting below full employment).
     Note: Because rate hikes induce these short-run recessions when prices are sticky, they frequently draw political criticism.

  Transition to the Long Run:

  1. With output depressed below potential, downward pressure builds on nominal wages and prices.

  2. Eventually, the price level (P) adjusts downward.

  3. This increases the real money supply (M/P) back to normal, shifting the LM curve back outward until full general equilibrium is restored at the FE intersection.

  Core Foundations of the IS-LM Business Cycle Framework

  The IS-LM framework serves as a unifying macroeconomic model to analyze aggregate fluctuations, business cycle dynamics, and general equilibrium across three primary sectors of the economy:

  • The Labor Market: Represented by the Full-Employment Line (FE).

  • The Goods Market: Represented by the Investment-Saving Curve (IS).

  • The Asset/Money Market: Represented by the Liquidity Preference-Money Supply Curve (LM).

  Broader Analytical Scope of Business Cycle Theory:

  • Foundational mechanics of business cycles.

  • The IS-LM model as an integrated framework for macroeconomic analysis.

  • The Aggregate Demand-Aggregate Supply (AD-AS) framework.

  • Classical analysis: Emphasizes market clearing and rapid price adjustments.

  • Keynesian analysis: Emphasizes nominal rigidities, slow wage and price adjustment, and short-run output deviations.

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  1. Labor Market Equilibrium and the Full-Employment (FE) Line

  Labor Market Equilibrium:

  • Full employment is the level of employment attained after all wages and prices have completely adjusted to clear markets.

  • The market clears where aggregate labor supply (NS) equals aggregate labor demand (ND).

  • The downward-sloping ND curve (reflecting the marginal product of labor, MPN) intersects the upward-sloping NS curve at the equilibrium point. This determines the market-clearing current real wage (w_bar) and the equilibrium employment level (N*).

  Aggregate Production and Full-Employment Output:

  • Full-employment output (Y*) is determined directly by the aggregate production function evaluated at full-employment labor (N*) and the prevailing capital stock (K).

    • Formula: Y* = A F(K, N), where A represents total factor productivity.

  • In per-worker terms: output per worker (y_t) is a function of the capital-labor ratio (k_t). At an initial capital-labor ratio of k_1, output per worker is y_1.

  Independence of Full-Employment Output from the Real Interest Rate:

  • The existing capital stock (K) is predetermined by past investment decisions and past real interest rates.

  • The current real interest rate (r) influences current investment and future capital accumulation, but does not alter the current K.

  • Neither N* nor Y* depends on current r. Output remains fixed at Y* regardless of r.

  Graphical Construction & Shifts of the FE Line:

  • Plotted in a graph with Output (Y) on the horizontal axis and Real Interest Rate (r) on the vertical axis, the FE line is perfectly vertical at Y = Y*.

  • Rightward Shifts of the FE Line (Increases in Output):

    • Beneficial Supply Shock: (e.g., permanent/temporary reduction in energy prices). More output is produced with the same quantities of K and N. If MPN increases, labor demand shifts rightward, increasing both N* and Y*.

    • Increase in Labor Supply (NS): An outward shift lowers the equilibrium real wage, expanding total employment (N*) and raising Y*.

    • Increase in Capital Stock (K): Directly expands production capacity. It typically raises MPN, stimulating labor demand and elevating N*.

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  2. Goods Market Equilibrium and the IS Curve

  Goods Market Equilibrium Condition:

  • The goods market clears when desired national saving (Sd) equals desired investment (Id).

  • National saving is defined as national income minus desired consumption (Cd) and government purchases (G). Assuming net factor payments are zero: Sd = Y - Cd - G.

  • The real interest rate (r) is the market price that adjusts to clear the goods market.

  Derivation and Slope of the IS Curve:

  • The IS curve illustrates all combinations of Y and r that clear the goods market.

  • Example: If Y increases from 4000 to 5000, national income rises, causing households to increase desired saving. Desired investment (Id) depends on future returns and capital costs, not current Y, so its schedule remains unchanged. At the original 7% interest rate, higher income creates excess saving. To restore equilibrium, r must decline to 5% (where Sd = Id).

  • Tracing these points yields a downward-sloping IS curve.

  Comprehensive Determinants of IS Curve Shifts:

  • Increase in Government Purchases (G): Directly reduces Sd. At an unchanged output level (e.g., 4500), the real interest rate must rise (e.g., 6% to 7%) to clear the market. Shifts IS upward/rightward.

  • Expected Future Output/Income Increases: Households feel wealthier, raising Cd and decreasing Sd, which elevates r for any given output. Shifts IS upward/rightward.

  • Wealth Increases: Higher financial or real wealth increases Cd, depressing saving and driving up r. Shifts IS upward/rightward.

  • Expected Future Marginal Product of Capital Increases: Firms demand more capital, shifting Id to the right and driving up r. Shifts IS upward/rightward.

  • Taxes (T) Increase:

    • Under Ricardian Equivalence: No shift. Forward-looking consumers realize current tax changes will be offset in the future; Cd and Sd remain unchanged.

    • Without Ricardian Equivalence: Higher current taxes reduce disposable income and Cd. This increases Sd and depresses r. Shifts IS downward/leftward.

  • Effective Tax Rate on Capital Increases: Post-tax profitability falls, dampening Id and lowering r. Shifts IS downward/leftward.

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  3. Asset Market Equilibrium and the LM Curve

  Asset Market Aggregation and Equilibrium:

  • Equilibrium occurs when aggregate asset demand equals aggregate asset supply.

  • Walras' Law: By aggregating assets into monetary and nonmonetary categories, if the money market clears, the nonmonetary asset market must simultaneously clear.

  • The overall price level (P) adjusts to ensure long-run money market clearance.

  Inverse Relationship Between Bond Prices and Interest Rates:

  • The market price of a nonmonetary asset is inversely related to its yield.

  • Example: For a bond that pays 10,000 dollars in one year:

    • If price is 9,615 dollars, yield is: (10,000 - 9,615) / 9,615 = 4%

    • If price declines to 9,524 dollars, yield rises to: (10,000 - 9,524) / 9,524 = 5%

  • Falling asset prices = rising interest rates.

  Derivation and Slope of the LM Curve:

  • The LM curve traces all combinations of Y and r that clear the money market.

  • Real money demand (Md / P) = L(Y, r + expected inflation).

  • Real money supply (M / P) is fixed by the central bank and money multiplier (plotted as a vertical schedule, e.g., at 1000).

  • Example: When Y rises from 4000 to 5000, transaction volume expands, shifting Md outward. At the initial 3% interest rate, there is an excess demand for money (e.g., 1200 demand vs. 1000 supply). People sell nonmonetary assets to get liquid money. This liquidating pressure drives down bond prices, bidding up r to 5%.

  • Tracing these points yields an upward-sloping LM curve.

  Comprehensive Determinants of LM Curve Shifts:

  • Nominal Money Supply (M) Increases: Increases real money supply. Example: M/P rises from 1000 to 1200, creating excess cash at the prevailing 3% interest rate. Individuals buy nonmonetary assets, bidding asset prices up and reducing r to 2%. Shifts LM downward/rightward.

  • Expected Inflation Increases: Raises the opportunity cost of holding non-interest-bearing currency, dampening real money demand and lowering r. Shifts LM downward/rightward.

  • Price Level (P) Increases: Real money supply contracts, creating excess demand for money and raising r. Shifts LM upward/leftward.

  • Nominal Interest Rate on Money Increases: Higher yields on liquid deposits make holding money more attractive, increasing Md and bidding up r. Shifts LM upward/leftward.

  • Other Factors Increasing Real Money Demand (e.g., higher wealth, greater nonmonetary asset risk): Heightened demand for money raises r. Shifts LM upward/leftward.

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  4. General Equilibrium in the Complete IS-LM Model

  Definition of General Equilibrium:

  • Occurs when the labor market, goods market, and asset/money market clear simultaneously.

  • Graphically, it is the unique intersection point where the vertical FE line, downward-sloping IS curve, and upward-sloping LM curve intersect at a single (Y, r) combination, producing Y = Y*.

  Mathematical System (Three Equations, Three Unknowns: Y, r, P):

  1. Labor Market: Y = A F(K, N). Determines full-employment output (Y) given existing K.

  2. Goods Market: Y - Cd - G = Id. Determines the market-clearing real interest rate (r) given output Y.

  3. Asset/Money Market: M / P = L(Y, r + expected inflation). Determines the equilibrium price level (P) given Y, r, and expected inflation.

  Price Adjustment and the Path to General Equilibrium:

  • Speeds of Adjustment: Asset Markets adjust most rapidly (instantly). Goods Market adjusts at an intermediate speed. Labor Market adjusts most slowly (due to frictions, contracts).

  • Short-Run vs. Long-Run: The economy initially moves to the short-run intersection of IS and LM. If this occurs at a point where the labor market is in disequilibrium, the general price level (P) serves as the primary adjusting variable. Firms adjust prices, which shifts the LM curve until it intersects both IS and FE.

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  5. Dynamic Macroeconomic Shocks and Applications

  Monetary Expansion and the Principle of Monetary Neutrality:

  • Initial Position: Equilibrium at r = 5%, P = 100, Y = 1000.

  • Short-Run Dynamics: Central bank increases nominal money supply (M) by 10%. If prices are sticky, real money supply expands to 1200. LM shifts downward/rightward. The new temporary equilibrium sees r drop to 3% and Y expand to 1200.

  • Transition to Long-Run: At Y = 1200, demand exceeds full-employment capacity (1000). Excess liquidity lowers r, temporarily boosting consumption and investment. Firms face depleted inventories and bid up output prices. P increases by exactly 10% (to 110). This contracts the real money supply back to its initial level, shifting the LM curve back to its original position. The economy returns to r = 5% and Y = 1000.

  • Core Takeaways:

    • Monetary Neutrality: Changes in nominal money supply have no long-run impact on real economic variables (Y, N, r, w, C, I, G). Nominal variables (M, P, nominal wage) adjust in exact proportion to the expansion.

    • Continuous money growth causes continuous proportional inflation, keeping real balances constant.

    • Classical view: Fast adjustment, neutral even in short-run. Keynesian view: Sticky prices create an extended short-run period with powerful real effects.

  Temporary Adverse Supply Shock (e.g., Severe Drought):

  • Impact on Labor & FE: Temporarily reduces total factor productivity (A), depressing MPN. Labor demand shifts leftward. Labor supply is unaffected. Equilibrium real wage and total employment decline. Potential output falls, shifting the vertical FE line leftward.

  • Initial IS/LM Status: Non-income determinants (wealth, future expectations) are unchanged, so IS stays static. Money demand determinants are initially unchanged, so LM stays fixed.

  • Transition Dynamics: At the initial IS-LM intersection, desired spending exceeds the new reduced productive capacity. Firms raise prices to ration scarce supply. Rising P contracts the real money supply. Simultaneously, lower output reduces transaction money demand, but rising interest rates further depress it. P rises until the real money supply clears the market at the new intersection of IS and the shifted FE line. LM shifts up/left.

  • Final Long-Run Consequences:

    • Lower full-employment output.

    • Lower employment and equilibrium real wage.

    • Higher equilibrium real interest rate.

    • Lower consumption and investment due to reduced income and higher borrowing costs.

    • Permanently higher price level (generating a temporary spike in inflation).

  Historical Application: The 1973–1974 Oil Price Shock:

  • Model Predictions: Predicted lower employment, lower real wages, lower aggregate output, lower consumption/investment, higher price level (inflation), and higher real interest rate.

  • Empirical Alignment: Historical data strongly validated nearly all model predictions (employment, wages, output plummeted; inflation soared).

  • Anomaly: Real interest rates actually declined during 1973-1974, deviating from the basic supply-shock model prediction of higher real interest rates.