Chapter 4 AD AS. O. Afonso, P. B. Vasconcelos. Computational Economics: a concise introduction

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1 Chapter 4 AD AS O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 32

2 Overview 1 Introduction 2 Economic model 3 Numerical solution 4 Computational implementation 5 Numerical results and simulation 6 Highlights 7 Main references O. Afonso, P. B. Vasconcelos Computational Economics 2 / 32

3 Introduction The AD AS model, Aggregate Demand Aggregate Supply is an aggregation of the elementary microeconomic demand-and-supply model. The AD curve can be obtained from the IS LM curves, by removing the fixed price level, representing the set of output and price level combinations that guarantee equilibrium of goods and services and monetary markets. The AS curve represents the set of output and price level combinations that maximise profits of firms. The equilibrium levels of the main variables, GDP and price level, P, are determined by the interaction of the AD and AS curves. MATLAB/Octave is used to solve the nonlinear model (Newton method and variants). O. Afonso, P. B. Vasconcelos Computational Economics 3 / 32

4 Economic model The set-up of the typical AD AS model specifies relationships among aggregate variables. This model can be used to study the effect of changes either in policy variables or in the specification of the interaction between endogenous variables: product equals aggregate demand, Y = C + I + G; consumption function, C = C + cy (1 t); investment function, I = I br; public spending function, G = G; income taxes function, T = ty ; money demand, L = L + ky hr; money supply function, M P ; production function, Y = A K α H 1 α. O. Afonso, P. B. Vasconcelos Computational Economics 4 / 32

5 Economic model Endogenous variables are: product, Y ; consumption, C; investment, I; interest rate, R; prices, P. Exogenous variables are: government/public spending, G; independent/autonomous consumption, C; independent/autonomous investment, I; money supply, M; nominal wages, W ; Capital, K ; labour (which is given), H; total productivity of factors, A. O. Afonso, P. B. Vasconcelos Computational Economics 5 / 32

6 Economic model Parameters: 0 < c < 1 is the propensity to consume; b > 0 is the interest sensitivity of investment; k > 0 is the output sensitivity of the demand for money; h > 0 is the interest sensitivity of the demand for money; t 0 is the tax rate; 0 < α < 1 is the share of labour in production; 1 α is the share of capital in production. O. Afonso, P. B. Vasconcelos Computational Economics 6 / 32

7 Economic model AD curve: aggregate demand The AD curve represents the various amounts of real GDP, IS-LM equilibrium output, that buyers will desire to purchase at each possible price level. Y = ( ) ( ) 1 h L M P 1 b C + I + G c(1 t) 1 b k h. (1) Representing (Y, P), respectively, in the x axis and y axis, it can be stated that: the position of the AD curve is affected by any factor that affects the position of IS and LM curves; Y < 0 (negative slope); P points on the left (right) side of the curve imply excess (scarcity) of aggregate demand. O. Afonso, P. B. Vasconcelos Computational Economics 7 / 32

8 Economic model AS curve: aggregate supply The AS curve represents the real domestic output level that is supplied by the economy at different price levels, having three distinct segments Horizontal range: the price level remains constant, P = P, with substantial output variation, and the economy is far from full employment Keynesian (or short-run aggregate supply) curve (implicit in the IS LM model). Intermediate (up sloping) range: the expansion of real output is accompanied by a rising price level, near to a full employment level the hybrid (or intermediate or medium-run aggregate supply) curve (AS mr ), Y = A K α ( W P K α (1 α)a ) α 1 α. (2) Vertical range: absolute full capacity is assumed and full employment occurs at the natural rate of unemployment, Y = Y N Classical AS (or long-run aggregate supply) curve. O. Afonso, P. B. Vasconcelos Computational Economics 8 / 32

9 Economic model AS curve: aggregate supply It can be stated that: AS mr position depends on a change in input prices, change in productivity and change in legal institutional environment; Y P > 0 (positive slope). O. Afonso, P. B. Vasconcelos Computational Economics 9 / 32

10 Economic model Putting the AD and the AS curves together Equilibrium price and quantity are found where the AD and AS curves intersect. If the AD curve shifts right, in the intermediate and vertical AS curve ranges it will cause demand-pull inflation, whereas in the horizontal it will only cause output changes. The multiplier effect is weakened by price level changes in intermediate and vertical AS curve ranges: the more price level increases the smaller the effect on real GDP is. O. Afonso, P. B. Vasconcelos Computational Economics 10 / 32

11 Numerical solution Computing the zeros of a real function f, or equivalently the roots of the equation f (x) = 0, is a frequent problem in economy. Contrary to linear systems, this computation cannot be accomplished in a finite number of operations; thus, one must rely on iterative methods. Iterative methods. Starting from x 0, build a sequence x k, k = 1, 2,..., convergent to a zero of f. The quality of the solution can be measured by the residual, f (x k ), or by the error, x k x, where x is the solution sought. The convergence rate r of the iterative process is linear, super-linear or quadratic, if x k+1 x lim k x k x r = c for a nonzero constant c, respectively, r = 1, r > 1 or r = 2. O. Afonso, P. B. Vasconcelos Computational Economics 11 / 32

12 Numerical solution Scalar nonlinear equations Bisection method. If f on [a 0, b 0 ] satisfies f (a 0 )f (b 0 ) < 0, then it has at least one zero in the segment ]a 0, b 0 [. Taking m 0 = (a 0 + b 0 )/2, new intervals can be iteratively defined by halving the previous one according to { ]ak, m ]a k+1, b k+1 [ = k [, f (m k )f (b k ) > 0 ]m k, b k [, f (m k )f (b k ) < 0. Advantage: independent of the regularity of f. Disadvantage: low rate of convergence. O. Afonso, P. B. Vasconcelos Computational Economics 12 / 32

13 Numerical solution Scalar nonlinear equations Newton method x k+1 = x k f (x k ) 1 f (x k ), f (x k ) 0, k = 0, 1, 2,... computes the zero by locally replacing f by its tangent line. Advantage: rate of (local) convergence of this method is quadratic. Disadvantage: requires f, expensive to compute and often not explicitly available. O. Afonso, P. B. Vasconcelos Computational Economics 13 / 32

14 Numerical solution Scalar nonlinear equations Secant method x k+1 = x k x k x k 1 f (x k ) f (x k 1 ) f (x k), k = 0, 1, 2,..., where f (x k ) is approximated by finite differences using two successive iterates. (Local) super-linear convergence. Convergence can be improved by considering a higher order degree interpolation polynomial. O. Afonso, P. B. Vasconcelos Computational Economics 14 / 32

15 Numerical solution Nonlinear system of equations Compute the zeros of a system of n nonlinear equations, f (x) = 0 with f : lr n lr n, x k a vector with n components, using Newton method x k+1 = x k f (x k ) 1 f (x k ), k = 0, 1, 2,... f (x k ) = J(x k ) = f 1 f 1 x 1 (x k ) x n (x k )..... f n f x 1 (x k ) n x n (x k ) where the Jacobian matrix at x k (J(x k ) is nonsingular). O. Afonso, P. B. Vasconcelos Computational Economics 15 / 32

16 Numerical solution Nonlinear system of equations Variants of the Newton method: use the same Jacobian several iterations; updating factorisation methods rather than re-factorising the Jacobian matrix. Disadvantage: Newton method (or variants) may not converge for x(0) far from the solution. O. Afonso, P. B. Vasconcelos Computational Economics 16 / 32

17 Numerical solution Nonlinear (system) of equations in practice MATLAB/Octave: zeros of a continuous function of one variable: fzero(f,x0), where f is the function and x 0 an initial approximation; system of nonlinear equations: fsolve(f,x0), where f is the vector function and x 0 an initial approximation. O. Afonso, P. B. Vasconcelos Computational Economics 17 / 32

18 Computational implementation The following baseline values are considered: c = 0.6, b = 1500, k = 0.2, b = 1500, h = 1000, t = 0.2, α = 0.5, A = 1, K = 30000, C = 160, I = 100, G = 200, M = 1000, W = 50 and L = 225. O. Afonso, P. B. Vasconcelos Computational Economics 18 / 32

19 Computational implementation Presentation and parameters %% AD AS model % Medium run e q u i l i b r i u m % Implemented by : P. B. Vasconcelos and O. Afonso function adas global C_bar I_bar G_bar M_bar W_bar L_bar A_bar K_bar... c b t k h alpha aux disp ( ) ; disp ( AD AS model ) ; disp ( ) ; %% parameters c = 0. 6 ; % marginal p r o p e n s i t y to consume b = 1500; % s e n s i b i l i t y of the investment to the i n t e r e s t r a t e k = 0. 2 ; % s e n s i b i l i t y of the money demand to the product h = 1000; % s e n s i b i l i t y of the money demand to the i n t e r e s t r a t e t = 0. 2 ; % tax on consumption alpha = 0. 5 ; % c a p i t a l share i n production O. Afonso, P. B. Vasconcelos Computational Economics 19 / 32

20 Computational implementation Exogenous and endogeneous variables %% exogenous v a r i a b l e s ( autonomous components ) A_bar = 1; % exogenous p r o d u c t i v i t y K_bar = 30000; % stock of c a p i t a l C_bar = 160; % autonomous consumption I_ bar = 100; % autonomous investment G_bar = 200; % government spending M_bar = 1000; % money supply W_bar = 50; % wage L_bar = 225; % autonomous money demand disp ( exogenous v a r i a b l e s ( autonomous components ) : ) f p r i n t f ( G_bar = %d ; M_bar = %d ; W_bar = %d ; L_bar = %d \ n,... G_bar, M_bar, W_bar, L_bar ) ; %% endogenous v a r i a b l e s % Y, product % P, p r i c e O. Afonso, P. B. Vasconcelos Computational Economics 20 / 32

21 Computational implementation Compute, show and plot the solution %% model s o l u t i o n : compute the endogenous v a r i a b l e s x0 = [500 5 ] ; % i n i t i a l approximation f o r Y and P, resp. aux = ( c (1 t ) 1) / b k / h ; % a u x i l i a r y v a r i a b l e f o r AD curve x = f s o l v e (@ADAS_system, x0 ) ; %% show the s o l u t i o n disp ( computed endogenous v a r i a b l e s : ) f p r i n t f ( product, Y : %6.2 f \ n, x ( 1 ) ) ; f p r i n t f ( price, P : %6.2 f \ n, x ( 2 ) ) ; % show v a r i a b l e of i n t e r e s t R = 1/ h ( L_bar M_bar / x ( 2 ) +k x ( 1 ) ) ; f p r i n t f ( i n t e r e s t r a t e (%%), R: %6.2 f \ n, R 100) ; %% p l o t the s o l u t i o n P = 0 : 0. 1 : 1. 5 x ( 2 ) ; AS = A_bar K_bar^ alpha ( W_bar. / P... K_bar.^( alpha ) /((1 alpha ) A_bar ) ). ^ ( ( alpha 1) / alpha ) ; AD = 1/ h ( L_bar M_bar. / P) / aux 1/b ( C_bar+ I_bar+G_bar ) / aux ; plot (AS, P, b,ad, P, r ) ; xlim ([ ]) ; xlabel ( product ) ; ylabel ( p r i c e ) ; legend ( AS, AD ) ; O. Afonso, P. B. Vasconcelos Computational Economics 21 / 32

22 Computational implementation System of nonlinear equations %% AD AS system function f = ADAS_system ( x ) global C_bar I_ bar G_bar M_bar W_bar L_bar A_bar K_bar b h alpha aux ; f = [ x ( 1 ) 1/h ( L_bar M_bar / x ( 2 ) ) / aux +1/b ( C_bar+ I_bar+G_bar ) / aux ; x ( 1 ) A_bar K_bar^ alpha (W_bar / x ( 2 )... K_bar^( alpha ) /((1 alpha ) A_bar ) ) ^ ( ( alpha 1) / alpha ) ; ] ; O. Afonso, P. B. Vasconcelos Computational Economics 22 / 32

23 Numerical results and simulation AD-AS model exogenous variables (autonomous components): G_bar = 200; M_bar = 1000; W_bar = 50; L_bar = 225 computed endogenous variables: product, Y: price, P: 2.73 interest rate (%), R: 2.27 O. Afonso, P. B. Vasconcelos Computational Economics 23 / 32

24 Numerical results and simulation AS AD price product AD AS diagram O. Afonso, P. B. Vasconcelos Computational Economics 24 / 32

25 Numerical results and simulation Expansion of governmental spending (G increases 20%) G_bar from to initial eq. eq. after shock product, Y: price, P: interest rate, R (%): O. Afonso, P. B. Vasconcelos Computational Economics 25 / 32

26 Numerical results and simulation interest rate equilibrium after expansion of governamental spending LM IS LM new IS new product price AS AD AD new product Increase in G O. Afonso, P. B. Vasconcelos Computational Economics 26 / 32

27 Numerical results and simulation Expansion of money supply (M increases 20%) M_bar from 1000 to 1200 initial eq. eq. after shock product, Y: price, P: interest rate, R (%): O. Afonso, P. B. Vasconcelos Computational Economics 27 / 32

28 Numerical results and simulation interest rate equilibrium after expansion of governamental spending LM IS LM new product price AS AD AD new product Increase in M O. Afonso, P. B. Vasconcelos Computational Economics 28 / 32

29 Numerical results and simulation A positive supply shock (A increases 5%) A_bar from 1 to 1.05 initial eq. eq. after shock product, Y: price, P: interest rate, R (%): O. Afonso, P. B. Vasconcelos Computational Economics 29 / 32

30 Numerical results and simulation interest rate equilibrium after expansion of governamental spending LM IS LM new IS new product price AS AD AS new AD new product Increase in A O. Afonso, P. B. Vasconcelos Computational Economics 30 / 32

31 Highlights The AD AS model explains price level and output considering the relationship between aggregate demand and aggregate supply. The AD curve is defined by the IS LM equilibrium and the AS curve reflects the labour market. Iterative numerical methods for nonlinear problems are introduced. Newton and quasi-newton methods for the approximate solution of nonlinear (system of) equation(s) are briefly explained. O. Afonso, P. B. Vasconcelos Computational Economics 31 / 32

32 Main references M. Burda and C. Wyplosz Macroeconomics: a European text Oxford University Press (2009) G. Dahlquist and Å Björck Numerical methods in scientific computing Society for Industrial Mathematics (2008) R. J. Gordon Macroeconomics Pearson Education (2011) C. T. Kelley Iterative Methods for Linear and Nonlinear Equations Society for Industrial Mathematics (1995) N. G. Mankiw Macroeconomics Worth Publishers (2009) O. Afonso, P. B. Vasconcelos Computational Economics 32 / 32