IS-LM (monetary policy implications) Moon Oulatta, PhD Department of Economics

IS-LM (monetary policy implications) Introduction Hicks (1937) introduced the static IS-LM framework. However, the lack of microeconomics foundations within static IS-LM framework led to the development of new macroeconomic models capable of incorporating forward-looking expectations and the optimizing behavior of ﬁrms and households. Nonetheless, core concepts and models in monetary theory (for example, the liquidity trap, Poole’s 1970 model for optimal monetary policy, or the new Keynesian macroeconomic model) require some basic understanding of the IS-LM framework. Hence the importance of this lecture. Here, we will focus on the static IS-LM model for understanding some key concepts in monetary theory. However, I will provide an introduction to the intertemporal optimization problem.

IS-LM (monetary policy implications) Optimal consumption The dynamic IS curve is an important component of the New Keynesian model and it is derived from a household optimization problem, which we will discuss and derive in this lecture. For the dynamic IS curve, the relationship between aggregate demand and the real interest rate stems from the Euler equation, which is vital for understanding the optimal path of consumption.

IS-LM (monetary policy implications) Optimal consumption The representative household faces an optimization problem that consists of maximizing a lifetime utility function V = Z ∞ 0 e −βt lnc (t ) dt (1) subject to a dynamic budget constraint given by ˙ A(t )= rA(t )+ Y (t ) − C (t ) where C denotes consumption, β denotes the time preference rate for current consumption over future consumption, r denotes the real interest rate, A stands for assets, and Y denotes labor income.

IS-LM (monetary policy implications) Optimal consumption Consumption denotes the control variable and asset represents the state variable in the household optimization problem. Next, we take the following steps to solve the optimization problem Hamiltonian : e −βt lnC (t )+ λ(t )[rA(t )+ Y (t ) − C (t )] (a). ∂ H ∂ C (t ) 99K e −βt C (t ) − λ(t )=0 (b). ∂ H ∂ A(t ) = − ˙ λ(t ) 99K λ(t )r = βe −βt C (t ) + e −βt ˙ C (t ) C (t ) 2 (c ).TVC : lim t →∞ e −r (t ) A(t )=0

IS-LM (monetary policy implications) Optimal consumption Applying steps (a,b,c) allows us to derive the optimal path of consumption as follows (see white board for clinical derivations) ˙ C (t ) C (t ) =(r − β) (2) where equation (2) denotes a dynamic relationship between consumption and the real interest rate: a higher interest rate (r >β) delays current consumption. There is more incentive to save, which leads to more consumption over time. Whereas a higher relative time preference rate (impatience) (β> r ) leads to falling consumption over time. Consumption is constant and smooth when (r =β). In the balanced-growth path, consumption and output grow at the same constant rate, which means that the dynamic IS curve in the new Keynesian model can be derived from the Euler equation.

IS-LM (monetary policy implications) IS-LM (aggregate expenditures) Aggregate demand is a function of aggregate expenditures (E), which depends on private consumption expenditures C (·), private investment expenditures I (·), and ﬁnal government expenditures (G). Here, we are mainly interested in short-run monetary policy implications. Therefore, we make the following assumptions: capital stock (K ) is held constant and we assume that the price level (P) is ﬁxed in the short run.

IS-LM (monetary policy implications) IS-LM (aggregate expenditures) Following Sargent’s 1979 macroeconomic model, we begin by deﬁning the goods market. Total expenditures in the economy can be expressed as follows E = {C · (Y p , r )+ I · (r )+ G } (3) where E denotes total expenditures in the economy, r denotes the real expected interest rate (i − π e ). Y p is disposable income or the diﬀerence between real income (Y ) and net taxes (τ ), which is adjusted for transfer payments.

IS-LM (monetary policy implications) IS-LM (ﬁnancial sector) The static IS-LM model includes a ﬁnancial sector where households can hold their wealth between liquid money and interest-bearing assets. By Walras law, we assume that if the money market clears, then the bonds market must also clear. The overall money supply (M) includes currency and deposits and the central bank conducts open market operations by selling and buying short-term bonds (B ). Nonetheless, an open market purchase requires that the following condition holds dM = −dB 99K dM + dB =0 which implies that when the central bank buys bonds from the public, bond private holdings must fall (dB < 0) and the money supply goes up (dM > 0).

IS-LM (monetary policy implications) IS-LM (monetary policy framework) For simplicity, we consider two types of monetary policies. The central bank can choose to either ﬁx the money supply or the short-term nominal interest rate. 1 monetary aggregate rule : the central bank ﬁxes the money supply and allows interest rates to adjust freely. Consequently, M is exogenous and (Y ) and (i ) are endogenous. interest rate rule : the central bank ﬁxes the short-term nominal interest rate and allows the money supply to adjust freely. Here, the short-term nominal interest rate (i) is exogenous and (Y ) and (M) are endogenous. 1 We assume that the money multiplier is equal to 1, which implies that the central bank can eﬀectively inﬂuence the composition of the money supply by simply adjusting reserve money.

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) Here, the central bank ﬁxes the money supply and allows the short-term nominal interest rate to adjust freely. This is the case where the demand for real money balances is downward sloping. Consider the following implicit equations Y = E (Y , i − π e ,τ, G ) (4) M P 99K m = m(Y , i ) (5) where equation (4) denotes the goods market equilibrium and equation (5) denotes the monetary sector equilibrium.

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) Total diﬀerentiating equation (4) simpliﬁes the IS relation as follows dY = 1 (1 − E y ) {E r di − E r d π e + E τ d τ + E g dG } (6) 0 < E y < 1; E r =(C r + I r ) < 0; E τ < 0; E g > 0 where the slope of the IS curve (∂ i /∂ y ) can be found by expressing i as a function of Y and diﬀerentiating both sides with respect to Y and i di dY 99K (1 − E y ) E r < 0

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) Figure: Slope of IS curve

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) The LM relation can be totally diﬀerentiated as follows dm = m y dY + m i di (7) m y > 0; m i < 0 where the slope of the LM curve (∂ i /∂ y ) can be found by expressing i as a function of Y and diﬀerentiating both sides with respect to Y and i di dY 99K − m y m i > 0

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) Figure: Slope of LM Curve

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) We are interested in understanding the eﬀects of exogenous monetary policy shocks on the key endogenous variables in the system. This can be achieved by combining equations (4) and (5) into the following system:  1 −ϕ r m y m i  dY di  =  ϕ τ d τ −ϕ r d π e +1 ϕ g dG dm  (8) where ϕ r = {E r /(1 − E y )} < 0; ϕ τ = {E τ /(1 − E y )} < 0; ϕ g = {E g /(1 − E y )} > 0. Stability in the IS-LM system guarantees that an increase i causes aggregate demand to fall. Mathematically, this implies that the determinant of the system has to be negative(△ < 0). This is our stability condition.

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) How does an increase in the money supply aﬀects the equilibrium interest rate and real income in the economy? ∂ i ∂ m 99K 1 m i + ϕ r m y < 0; ∂ y ∂ m 99K ϕ r m i + ϕ r m y > 0 (9) The comparative statics can be interpreted as follows: it follows that an open market purchase of short-term government bonds causes the money supply to increase instantaneously. This causes an excess supply of money and excess demand for bonds at the initial interest rate. Bonds prices rise, interest rates fall, this stimulates aggregate demand and real output increases.

IS-LM (monetary policy implications) IS-LM (ﬁxed money supply rule) Theoretically, monetary policy is relatively more eﬀective when the key components of the goods market (for example, consumption and investment) are relatively more sensitive to changes in interest rates: lim ϕr →∞ ∂ y ∂ m = 1 m y Monetary policy is ineﬀective when the demand for money is highly sensitive to changes in interest rates: lim m i →−∞ ∂ y ∂ m =0

IS-LM (monetary policy implications) IS-LM (application) Empirically, is there a strong empirical relationship between real output, interest rates, and real money balances? There are multiple ways to test the core assumptions of the standard IS-LM model. Here, to test the theoretical results found in Equation (9): we will rely on a simple vector autoregression model with exogenous variables (VARX) to examine the eﬀect of exogenous monetary policy shocks on real output and interest rates.

IS-LM (monetary policy implications) IS-LM (identifying monetary policy shocks) Figure: Money, Interest Rates, and Real Output (Covariances)

IS-LM (monetary policy implications) IS-LM (application) Consider the following VARX (1) model Z t =ΦZ t −1 +ΓX t + u t (10) where Z t denotes a 2 × 1 vector containing the real output gap and the real short-term interest rate (r ) proxy by the federal funds rate, adjusted for inﬂation. X t denotes a 1 × 1 vector containing the exogenous monetary policy shock proxy by the cyclical component of real M2 and u t is a 2 × 1 vector of reduced-form white noise errors. Φ and Γ 1 are matrices of coeﬃcients to be estimated. We estimate the model with a one-lag structure. All variables utilized in the model are I (0). The real interest rate is I (0) in levels, which allows us to estimate a bivariate VAR(X) model without a cointegrating vector between real output and interest rates.

IS-LM (monetary policy implications) IS-LM (identifying monetary policy shocks) (IS) (LM) Yt rt Yt−1 0.773 ∗∗∗ 0.151 ∗ (0.0381) (0.0793) rt−1 0.0111 0.727 (0.0200) (0.0444) mt 0.0681 ∗∗∗ -0.134 ∗∗∗ (0.0208) (0.0432) observations 1959q1-2024q4 1959q1-2024q4 R 2 .61 0.57 Standard errors in parentheses ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p <.01 Table: Equation (10)

IS-LM (monetary policy implications) IS-LM (identifying monetary policy shocks) The VARX model allows us to eﬃciently test the predictions of the standard the IS-LM model: the real output gap and the real interest rate are both endogenous, but the real money supply shock is treated as exogenous. For the United States, we ﬁnd that an exogenous shock in real money balances puts downward pressure on the real interest rate, which provides an explanation to why the real output gap rises as the real money supply increases. These empirical ﬁndings are consistent with the theoretical ﬁndings provided in equation (9). More importantly, we ﬁnd that the slope of the LM curve is relatively ﬂat. This explains why the eﬀect of monetary policy on real output is extremely small. A ﬂat LM curve implies that the real interest rate’s response to an increase in real money balance is extremely small: this is shown empirically here.

IS-LM (monetary policy implications) What is a liquidity trap? An economy approaches the liquidity trap when the short-term nominal interest rate approaches zero or the zero-lower bound. In the liquidity trap, households are indiﬀerent between holding money or bonds: the demand for real money balances becomes perfectly elastic; which means that any further increases in the quantity of money has no eﬀect on the short-term nominal interest rate. People are willing to hold any quantity of real money balances at any given level of real income. Monetary policy is categorically ineﬀective in stimulating the economy.

IS-LM (monetary policy implications) Liquidity trap (practical evidence) Figure: Zero-Lower Bound Evidence (US Economy)

IS-LM (monetary policy implications) What is the liquidity trap? What happens to aggregate demand when the economy is in the liquidity trap? Derive the AD relation by using equations (4) and (5) (see white board for clinical derivations). It follows that the aggregate demand curve is independent of the price level in the liquidity trap. Conventional monetary policy (↑ M) cannot push the nominal interest rate below the zero-lower bound. In this environment, ﬁscal policy is relatively more useful in stimulating aggregate demand (this was the premise behind Keynes argument in the General Theory of Employment, Interest, and Money).

IS-LM (monetary policy implications) What is the liquidity trap? First, total diﬀerentiate equation (5) and solve for di as follows di =  1 Pm i  dM −  m y m i  dY −  M P 2 m i  dP Plug di into equation (4), then solve for dP as a function of dY in order to derive the aggregate demand curve as follows dP =Φ a − Φ b dY (11) where Φ a = n P 2 M  m i Eτ Er d τ − m i d π e + m i Eg Er dG  + P M dM o ; Φ b = P 2 M n m i (1−Ey )+Er my Er o dY .

IS-LM (monetary policy implications) What is the liquidity trap? Figure: Liquidity Trap (Aggregate Demand)

IS-LM (monetary policy implications) IS-LM (ﬁxed interest rate rule) Instead of ﬁxing the money supply, the central bank can ﬁx the short-term nominal interest rate (i = i ∗ ) and allow the money supply to adjust freely. This type of monetary policy is very common in emerging and advanced economies. A ﬁxed interest rate rule provides the perfect cushion against money demand shocks. More importantly, under this regime, the eﬀect of ﬁscal policy is relatively larger, because there is no crowding-out eﬀect.

IS-LM (monetary policy implications) IS-LM (ﬁxed interest rate rule) Figure: IS-LM (Fixed Interest Rate Rule)

IS-LM (monetary policy implications) IS-LM (conclusion) Despite the lack of microeconomics foundations, the standard IS-LM model remains an eﬀective framework for understanding the short-run eﬀects of monetary policy. Empirically, as shown in this lecture, we ﬁnd a strong empirical relationship between the money supply, interest rates, and real output. The static IS-LM model provides an adequate framework for understanding the monetary transmission mechanism in a closed economy. Clearly, the model provides a starting point for our next theoretical discussion on optimal monetary policy instruments.