Lecture 3: A dynamic IS-LM model
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1 Lecture 3: A dynamic IS-LM model José L. Torres Universidad de Málaga Advanced Macroeconomics José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 1 / 26
2 Model economy: m t p t = ψy t i t (1) y d t = β 0 β 1 (i t ṗ e t ) (2) ṗ t = µ(y t y t ) (3) ẏ t = υ(y d t y t ) (4) where m is the logarithm of money, p is the logarithm of prices, y d, the logarithm of aggregate demand, y logarithm of output, y the logarithm of potential outpu, i is the nominal interest rate. All the parameters (greek letters) are defined as positive. A "point" over the variable represents time derivative. José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 2 / 26
3 Steps to follow: 1 Endogenous versus exogenous variables. 2 Selection of the endogenous variables of reference. 3 Setting-up the differential equations. 4 Matrix notation of the system. 5 Steady State. 6 Stability analysis. 7 Graphical representation of the partial equilibrium conditions. 8 Phase diagram. 9 Stable path. 10 Shocks. José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 3 / 26
4 Step 1: Endogenous and exogenous variables. 1 Money 2 Price level 3 Potential output 4 Nominal interest rate 5 Aggregate demand 6 Output José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 4 / 26
5 Step 2: Selection of the endogenous variables of reference. 1 Price level 2 Output José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 5 / 26
6 Step 3: Differential equations. Solving foir the nominal interest rate in equation (1): Substituting (5) in (2): y d t Perfect foresign assumption (ṗ t = ṗ e t ): i t = 1 (m t p t ψy t ) (5) = β 0 β 1 ψ y t + β 1 (m t p t ) + β 1 ṗ e t (6) y d t = β 0 β 1 ψ y t + β 1 (m t p t ) + β 1 ṗ t (7) José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 6 / 26
7 Substituting aggregate demand in the dynamic equation for output: [ ẏ t = υ β 0 ( β 1 ψ + 1)y t + β ] 1 (m t p t ) + β 1 ṗ t [ ẏ t = υ β 0 ( β 1 ψ + 1)y t + β ] 1 (m t p t ) + β 1 µ(y t y t ) [ ẏ t = υ β 0 + (β 1 µ β 1 ψ 1)y t + β ] 1 (m t p t ) β 1 µy t (8) (9) (10) José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 7 / 26
8 Step 4: Model in matrix notation. ṗ t = µ(y t y t ) (11) [ ẏ t = υ β 0 + (β 1 µ β 1 ψ 1)y t + β ] 1 (m t p t ) β 1 µy t (12) ] [ [ṗt = ẏ t 0 µ υβ 1 υ(β 1 µ β 1 ψ 1) ] [pt y t ] [ 0 0 µ + υβ υ 1 υβ 1 µ ] β 0 m t y t (13) José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 8 / 26
9 Step 5: Steady State. Steady state is computed as: [ pt y t Using the above expression we obtain: ] = A 1 Bz t (14) p t = β 0 β 1 + m t (ψ + β 1 )y t (15) y t = y t (16) José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 9 / 26
10 Step 6: Stability analysis. [ ] 0 λ µ Det υβ 1 υ(β 1 µ β 1 ψ = 0 1) λ (17) [ λ 2 λ υ(β 1 µ β 1 ψ ] 1) + υβ 1 µ = 0 (18) [ ] υ(β 1 µ β 1 ψ 1) ± υ(β 1 µ β 1 ψ 2 4υβ 1) 1 µ If β 1 µ β 1 ψ 1 > 0 = λ 1 > 0, λ 2 > 0. If β 1 µ β 1 ψ 1 < 0 = λ 1 < 0, λ 2 < 0. 2 (19) José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 10 / 26
11 Step 7: Graphical analysis. Slope of the differential equation for the first variable (price level), under the assumption that the time derivative of this variable is zero: dp t dy t ṗt =0= µ 0 = (20) Slope of the differential equation for the second variable (output level), under the assumption that the time derivative of this variable is zero: dp t ẏt =0= β 1 µ β1ψ 1 0 (21) dy β t 1 José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 11 / 26
12 Step 7: Graphical analysis. p ṗ t = 0 dp t dy t ṗt =0= µ 0 = ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 12 / 26
13 Step 7: Graphical analysis. p ṗ t = 0 dp t dy t ṗt =0= µ 0 = ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 13 / 26
14 Step 7: Graphical analysis. p dp t dy t ẏt =0= β 1 µ β 1 ψ/ 1 β 1 / < 0 ẏ t = 0 y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 14 / 26
15 Step 7: Graphical analysis. p dp t dy t ẏt =0= β 1 µ β 1 ψ/ 1 β 1 / < 0 ẏ t = 0 y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 15 / 26
16 Step 8: Phase diagram. p p ṗ t = 0 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 16 / 26
17 Step 9: Stable path. Stable path: Equation that represent the "fastest" path to the steady state. Equation only in the case of a saddle path solution. The stable path is associated to the negative eigenvalue. José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 17 / 26
18 Step 10: Shocks analysis. Shock: A rise in the money supply. First thing to do: See in which equation(s) is money supply as an exogenous variable. Money supply only appears in the dynamic equation for output. José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 18 / 26
19 Step 10: Shocks analysis. p p ṗ t = 0 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 19 / 26
20 Step 10: Shocks analysis ( m). p ṗ t = 0 p EE 1 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 20 / 26
21 Step 10: Shocks analysis ( m). p ṗ t = 0 p EE 1 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 21 / 26
22 Step 10: Shocks analysis ( m). p p ṗ t = 0 EE 1 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 22 / 26
23 Step 10: Shocks analysis ( m). p p ṗ t = 0 EE 1 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 23 / 26
24 Step 10: Shocks analysis ( m). p p ṗ t = 0 EE 1 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 24 / 26
25 Step 10: Shocks analysis ( m). p p ṗ t = 0 EE 1 EE 0 ẏ t = 0 ȳ y José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 25 / 26
26 3. A dynamic IS-LM model Step 10: Shocks analysis. Long-run effects: p t = β 0 β 1 + m t (ψ + β 1 )y t (22) y t = y t (23) dp t dm t = 1 dy t dm t = 0 José L. Torres (Universidad de Málaga) Lecture 3: A dynamic IS-LM model Advanced Macroeconomics 26 / 26
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