%************************************************************************************************** % EXCHANGE RATE DYNAMICS IN NEW-KEYNESIAN DSGE MODELS * % Yamin Ahmad, UW-Whitewater * % Ming Chien Lo, St. Cloud State University * % Olena Mykhaylova, University of Richmond * % June 2010 * %************************************************************************************************** % Most variables are expressed in logs, except for % optimal price and wage setting variables and dispersion terms; % interest rates int and ints; % governments transfers (tr and trs); % Lagrange multipliers ksi and ksis; % home bias parameters mu(i). %periods 5500; % For simulated variables var % Foreign variables end in "s" y, ys, % GDPs c, cs, % Consumption invt, invts, % Capital investment g, gs, % Government spending ch, cf, % Home consumption of home and foreign goods, respectively cfs, chs, % Foreign consumption of foreign and home goods, respectively invh, invf, % Home investment of home and foreign goods, respectively invfs, invhs, % Foreign investment of foreign and home goods, respectively gh, gf, % Home gov't spending on home and foreign goods, respectively ghs, gfs, % Foreign gov't spending on foreign and home goods, respectively infl, infls, % CPI inflations ph, pfs, % Real price of H goods, delated by H CPI, and F goods, deflated by F CPI infh, inffs, % Growth rates of ph and pfs, respectively (PPI inflations) phn, pbh, pah, % Home optimal price for sale in H (phn) and its components pfsn, pafs, pbfs, % Foreign optimal price for sale in F (pfsn) and its components DPh, DPfs, % Price dispersions rer, delS, % RER (from perspective of H) and NER (growth rate) tot, % Terms of trade (from perspective of H) w, ws, % Wages wn, wb, wa, % Home optimal wage and its components wsn, wbs, was, % Foreign optimal wage and its components l, ls, % Labor efforts winfl, winfls, % Wage inflations r, rs, % Returns on capital k, ks, % Capital stocks z, zs, % Solow residuals mc, mcs, % Marginal costs yh, yhs, % Output of H good for sale in H and F, respectively yfs, yf, % Output of F good for sale in F and H, respectively int, ints, % Interest rates (central bank instruments) tr, trs, % Government transfers (lump-sum tax) lam, lams, % Marginal utilities of comsumption ksi, ksis, % Lagrange multipliers in capital accumulation decision mu1, mu2, % Time-varying weights of H households on H and F goods, respetively mus1, mus2, % Same for foreign households prefshock, % Shock to home biases (mu1, mu2) - demand shocks pmark, wmark, % Price and wage markups ikr, ikrs, % Auxiliary variables, capital adjustment costs klr, klrs, % Capital-labor ratios (for steady state computations) ilr, ilrs, % Investment-labor ratios (for steady state computations) ylr, ylrs, % Output-labor ratios (for steady state computations) clr, clrs, % Consumption-labor ratios (for steady state computations) wrate, crate, % Ratios of H-to-F wages and consumption, respectively relc, rely, % Ratios of H-to-F consumption and output, respectively cr, crs, % Consumption-to-GDP ratios ir, irs, % Investment-to-GDP ratios kr, krs, % Capital-to-GDP ratios gr, grs, % Gov't spending-to-GDP ratios trr, trrs, % Transfers-to-GDP ratios ishock, ishocks, infdif; varexo hepsz, fepsz, % Shocks to home and foreign technologies epsp, % Preference shock (to home biases) hepsg, fepsg, % Shock to government purchases (demand) hepsi, fepsi; % Shocks to monetary policy parameters ssmu, ssmus, eta, theta, thetas, delta, psi, psis, beta, nu, A11, A12, A21, A22, sig, alpha, alphas, omega, omegas, chi, phi, rhoint, gshare, gshares, ssy, ssys, M, Ms, rhopi, rhogap, rhoi, trials, per; % load Param.mat; ssmu = 0.50; % Consumption home bias in the Home country ssmus = 0.50; % Consumption home bias in the Foreign country eta = 1.5; % Elasticity in the final goods aggregators theta = 5; % Risk aversion in the home consumer utility function thetas = 5; % Risk aversion in the foreign consumer utility function delta = 0.021; % Capital depreciation rate psi = 95; % Coefficient in adjustment cost for home investment psis = 95; % Coefficient in adjustment cost for foreign investment beta = 0.99; % Discount factor nu = 0.33; % Share of capital in production A11 = 0.95; % Tech.: home AR(1) coefficient A12 = 0.00; % Tech.: link from foreign to home technology level A21 = 0.00; % Tech.: link from home to foreign technology level A22 = 0.95; % Tech.: foreign AR(1) coefficient sig = 10; % Elasticity in the goods varieties aggregators alpha = 0.75; % Home price stickiness parameter omega = 0.75; % Home wage stickiness parameter alphas = 0.75; % Foreign price stickiness parameter omegas = 0.75; % Foreign wage stickiness parameter chi = 3; % Inverse of Frisch labor supply elasticity phi = 7.7; % Elasticity in labor aggregators rhoint = 0.79; % AR(1) coefficient in the Taylor rule rhopi = 2.15; % Taylor coefficient in the Taylor Rule rhogap = 0.23; % Coefficient on output gap in the Taylor Rule gshare = 0.20; % Share of home government purchases in home output gshares = 0.20; % Share of foreign government purchases in foreign output ssy = 0.794913; % Steady state level of home output ssys = 0.794913; % Steady state level of foreign output M = 1; % Size of the home country Ms = 1; % Size of the foreign country rhoi = 0.9; % Persistence in the monetary policy shock trials = 10; % Number of data-generating trials per = 5500; % Number of periods (same as in declaration above) model; %************************************ PREFERENCE SHOCKS ********************************* %prefshock = 0.9 * prefshock(-1) + epsp; % Zero in the steady state prefshock = 0; % For version with no demand shocks mu1 = ssmu * exp(prefshock); mu2 = 1 - mu1; mus1 = 1 - mus2; mus2 = (1 - ssmus) * exp(prefshock); %***************************** AGGREGATION AND MARKET CLEARING ************************** exp(yh) = M * (exp(ch) + exp(invh) + exp(gh) ); exp(yhs) = Ms * (exp(chs) + exp(invhs) + exp(ghs) ); y = log(exp(yh) + exp(yhs)); exp(yfs) = Ms * (exp(cfs) + exp(invfs) + exp(gfs) ); exp(yf) = M * (exp(cf) + exp(invf) + exp(gf) ); ys = log(exp(yfs) + exp(yf)); ch = log(mu1) - eta*ph + c; cf = log(mu2) - eta*(rer + pfs) + c; invh = log(mu1) - eta*ph + invt; invf = log(mu2) - eta*(rer + pfs) + invt; gh = log(mu1) - eta*ph + g; gf = log(mu2) - eta*(rer + pfs) + g; cfs = log(mus1) - eta*pfs + cs; chs = log(mus2) - eta*(ph - rer) + cs; invfs = log(mus1) - eta*pfs + invts; invhs = log(mus2) - eta*(ph - rer) + invts; gfs = log(mus1) - eta*pfs + gs; ghs = log(mus2) - eta*(ph - rer) + gs; rer - rer(-1) = delS + infls - infl; tot = pfs + rer - ph; %********************************** PRICE INDICES *************************************** 1 = mu1 * exp((1-eta)*ph) + mu2 * exp((1-eta)*(pfs+rer)); 1 = mus1 * exp((1-eta)*pfs) + mus2 * exp((1-eta)*(ph-rer)); exp((1-sig)*ph) = (1-alpha) * exp((1-sig)*phn) + alpha * exp((1-sig)*(ph(-1) - infl)); exp((1-sig)*pfs) = (1-alphas) * exp((1-sig)*pfsn) + alphas * exp((1-sig)*(pfs(-1) - infls)); (1-omega) * exp((1-phi)*wn) + omega * exp((phi-1)*winfl) = 1; (1-omegas) * exp((1-phi)*wsn) + omegas * exp((phi-1)*winfls) = 1; w - w(-1) = winfl - infl; ws - ws(-1) = winfls - infls; infh = infl + ph - ph(-1); inffs = infls + pfs - pfs(-1); infdif = infls - infl; %********************************** CONSUMER FOCs *************************************** theta*c - thetas*cs = rer; lam = -theta * c; 1/(1 + int) = beta * exp(lam(+1) - lam - infl(+1)); ikr = exp(invt - k(-1)) - delta; exp(lam) = ksi - ksi * psi * ikr; ksi/beta = exp(lam(+1) + r(+1)) + ksi(+1) * ( (1-delta) - 0.5 * psi * ikr(+1)^2 + psi * ikr(+1) * (ikr(+1) + delta) ); exp(k) = (1-delta) * exp(k(-1)) + exp(invt) - 0.5 * psi * exp(k(-1)) * ikr^2; lams = -thetas * cs; 1/(1 + ints) = beta * exp(lams(+1) - lams - infls(+1)); ikrs = exp(invts - ks(-1)) - delta; exp(lams) = ksis - ksis * psis * ikrs; ksis/beta = exp(lams(+1) + rs(+1)) + ksis(+1) * ( (1-delta) - 0.5 * psis * ikrs(+1)^2 + psis * ikrs(+1) * (ikrs(+1) + delta) ); exp(ks) = (1-delta) * exp(ks(-1)) + exp(invts) - 0.5 * psis * exp(ks(-1)) * ikrs^2; %*********************************** WAGE SETTING *************************************** wmark = phi / (phi - 1); exp((1 + phi*chi) * wn) = wmark * wb / wa; wb = exp( (1+chi)*l ) + omega*beta * wb(+1) * exp(phi*(1+chi)*winfl(+1)); wa = exp(lam + l + w) + omega*beta * wa(+1) * exp((phi-1)*winfl(+1)); exp((1+phi*chi) * wsn) = wmark * wbs / was; wbs = exp( (1+chi)*ls ) + omegas*beta * wbs(+1) * exp(phi*(1+chi)*winfls(+1)); was = exp(lams + ls + ws) + omegas*beta * was(+1) * exp((phi-1)*winfls(+1)); %************************************* PRODUCTION *************************************** z = A11 * z(-1) + A12 * zs(-1) + hepsz; zs = A21 * z(-1) + A22 * zs(-1) + fepsz; r - w = log(nu/(1-nu)) + l - k(-1); exp(y) * DPh = M * exp(z + nu * k(-1) + (1-nu) * l); mc = nu * r + (1-nu) * w - nu * log(nu) - (1-nu) * log(1-nu) - z; rs - ws = log(nu/(1-nu)) + ls - ks(-1); exp(ys) * DPfs = Ms * exp(zs + nu * ks(-1) + (1-nu) * ls); mcs = nu * rs + (1-nu) * ws - nu * log(nu) - (1-nu) * log(1-nu) - zs; %*********************************** PRICE SETTING ************************************** pmark = sig / (sig - 1); exp(phn) = pmark * pbh / pah; pbh = exp( lam + sig*ph + mc + yh) + alpha*beta * pbh(+1) * exp(sig * infl(+1)); pah = exp( lam + sig*ph + yh ) + alpha*beta * pah(+1) * exp((sig-1) * infl(+1)); exp(pfsn) = pmark * pbfs / pafs; pbfs = exp( lams + sig*pfs + mcs + yfs) + alphas*beta * pbfs(+1) * exp(sig * infls(+1)); pafs = exp( lams + sig*pfs + yfs) + alphas*beta * pafs(+1) * exp((sig-1) * infls(+1)); DPh = (1-alpha) * exp(sig * (ph - phn)) + alpha * exp(sig * (ph - ph(-1) + infl)) * DPh(-1); DPfs = (1-alphas) * exp(sig * (pfs - pfsn)) + alphas * exp(sig * (pfs - pfs(-1) + infls)) * DPfs(-1); %********************************** MONETARY POLICY ************************************ ishock = rhoi * ishock(-1) + hepsi; ishocks = rhoi * ishocks(-1) + fepsi; % Taylor Rules without output gap % int = (1-rhoint)*(1/beta - 1) + rhoint*int(-1) + (1-rhoint) * rhopi * infl + hepsi; % ints = (1-rhoint)*(1/beta - 1) + rhoint*ints(-1) + (1-rhoint) * rhopi * infls + fepsi; % Taylor Rules with output gap int = (1-rhoint)*(1/beta - 1) + rhoint*int(-1) + (1-rhoint) * rhopi * infl(+1) + (1-rhoint) * rhogap * (y - ssy) + hepsi; ints = (1-rhoint)*(1/beta - 1) + rhoint*ints(-1) + (1-rhoint) * rhopi * infls(+1) + (1-rhoint) * rhogap * (ys - ssys) + fepsi; % Foreign country stabilizes NER % delS = 0; %********************************** FISCAL POLICY ************************************** g = 0.03 * (log(gshare) + ssy - log(M)) + 0.97 * g(-1) + hepsg; gs = 0.03 * (log(gshares) + ssys - log(Ms)) + 0.97 * gs(-1) + fepsg; exp(g) = -tr; exp(gs) = -trs; %*********************************** USEFUL RATIOS ************************************* gr = M * exp(g - y); grs = Ms * exp(gs - ys); cr = M * exp(c - y); crs = Ms * exp(cs - ys); ir = M * exp(invt - y); irs = Ms * exp(invts - ys); kr = M * exp(k - y)/4; %Since K is a stock variable and Y is a flow variable krs = Ms * exp(ks - ys)/4; trr = -M * tr / exp(y); trrs = -Ms * trs / exp(ys); wrate = exp(w - ws - rer); crate = exp(c - cs - rer); relc = c - cs; rely = y - ys; %************************** STEADY-STATE AUXILIARY VARIABLES **************************** klr = k - l; klrs = ks - ls; ilr = invt - l; ilrs = invts - ls; ylr = yh - l; ylrs = yfs - ls; clr = c - l; clrs = cs - ls; end; initval; % DISTURBANCES AND AUXILIARY VARIABLES prefshock = 0; mu1 = ssmu; mu2 = 1 - ssmu; mus2 = 1 - ssmus; mus1 = 1 - mus2; pmark = sig / (sig - 1); wmark = phi / (phi - 1); z = 0; zs = 0; ishock = 0; ishocks = 0; % PRICES AND INFLATION rer = 0; delS = 0; tot = 0; ph = 0; pfs = 0; infl = 0; infls = 0; infdif = infls - infl; infh = 0; inffs = 0; winfl = 0; winfls = 0; DPh = 1; DPfs = 1; % STEADY STATE REDUCED FORM EQUATIONS int = (1/beta) - 1; ints = int; r = log( delta + int ); rs = log( delta + ints ); mc = -log(pmark); mcs = -log(pmark); w = (1/(1-nu)) * ( mc + nu * log(nu) + (1-nu) * log (1-nu) + z - nu * r ); ws = (1/(1-nu)) * ( mcs + nu * log(nu) + (1-nu) * log (1-nu) + zs - nu * rs ); wn = 0; wsn = 0; klr = w - r + log(nu) - log(1 - nu); klrs = ws - rs + log(nu) - log(1 - nu); ilr = log(delta) + klr; ilrs = log(delta) + klrs; ylr = nu * klr + log(M) + z; ylrs = nu * klrs + log(Ms) + zs; clr = log( (1/M) * ( exp(ylr)*(1 - ssmu*gshare) - exp(ylrs)*(1-ssmus)*gshares ) - exp(ilr) ); clrs = log( (1/Ms) * ( exp(ylrs)*(1 - ssmus*gshares) - exp(ylr)*(1-ssmu)*gshare ) - exp(ilrs) ); l = (w - log(wmark) - theta * clr)/(theta + chi); ls = (ws - log(wmark) - thetas * clrs)/(thetas + chi); k = klr + l; ks = klrs + ls; invt = ilr + l; invts = ilrs + ls; invh = log(ssmu) + invt; invf = log(1 - ssmu) + invt; invfs = log(ssmus) + invts; invhs = log(1 - ssmus) + invts; y = ylr + l; ys = ylrs + ls; yh = log(ssmu) + y; yhs = log(1 - ssmu) + y; yfs = log(ssmus) + ys; yf = log(1 - ssmus) + ys; c = clr + l; cs = clrs + ls; ch = log(ssmu) + c; cf = log(1 - ssmu) + c; cfs = log(ssmus) + cs; chs = log(1 - ssmus) + cs; lam = -theta * c; lams = -thetas * cs; ksi = exp(lam); ksis = exp(lams); g = log(gshare) + y - log(M); gs = log(gshares) + ys - log(Ms); gh = log(ssmu) + g; gf = log(1 - ssmu) + g; gfs = log(ssmus) + gs; ghs = log(1 - ssmus) + gs; wb = exp(l * (1 + chi)) / (1 - omega * beta); wbs = exp(ls * (1 + chi)) / (1 - omegas * beta); wa = exp(lam + w + l) / (1 - omega * beta); was = exp(lams + ws + ls) / (1 - omegas * beta); pbh = exp(lam + mc + yh) / (1 - alpha * beta); pah = exp(lam + yh) / (1 - alpha * beta); pbfs = exp(lams + mcs + yfs) / (1 - alphas * beta); pafs = exp(lams + yfs) / (1 - alphas * beta); phn = ph; pfsn = pfs; wrate = exp(w - ws - rer); crate = exp(c - cs - rer); relc = c - cs; rely = y - ys; gr = M * exp(g - y); grs = Ms * exp(gs - ys); cr = M * exp(c - y); crs = Ms * exp(cs - ys); ir = M * exp(invt - y); irs = Ms * exp(invts - ys); kr = M * exp(k - y)/4; krs = Ms * exp(ks - ys)/4; tr = -exp(g); trs = -exp(gs); trr = -M * tr / exp(y); trrs = -Ms * trs / exp(ys); ikr = exp(invt - k) - delta; ikrs = exp(invts - ks) - delta; end; steady; check; Sigma_e = [0.000049 0.000012 0 0 0 0 0; 0.000049 0 0 0 0 0; 0.0001 0 0 0 0; 0.0001 0 0 0; 0.0001 0 0; 0.000089 0.0000889; 0.000089]; % hepsz fepsz epsp hepsg fepsg hepsi fepsi % Business cycle properties, theoretical moments stoch_simul(IRF=0,ORDER=1) y, ys, c, cs, invt, rer, delS; %stoch_simul(IRF=0,hp_filter=1600,ORDER=1) y, ys, c, cs, invt, rer, delS; %stoch_simul(IRF=0,ORDER=1) y, ys, c, invt, ints, infdif, tot, rer, delS; % Recover time series for NER (note: unfiltered) % Output: a (per + 3) x (trials x 2) matrix % First (trials) columns contain RER data % Last (trials) columns contain NER data % "trials" is defined above in the Parameters section % This portion of the code generates several simulations of non-filtered RER and NER time series EXR = zeros(per+3,trials*2); YS = zeros(per+3,trials*2); Temp2 = zeros(per+3,1); for j = 1:trials Temp1 = zeros(per+3,1); stoch_simul(IRF=0,ORDER=1,noprint,nocorr,nofunctions,nomoments) y, rer, delS; EXR(:,j) = [rer]; YS(:,j) = [y]; Temp2 = [delS]; for i = 2:(per+3) Temp1(i,1) = Temp1(i-1,1) + Temp2(i,1); end EXR(:,j+trials) = Temp1; YS(:,j+trials) = Temp1; end save exr_sim_data.out EXR -ASCII %xlswrite('exr_sim_data.xls', EXR, 'Exchange Rates', 'A1'); xlswrite('HP Filters.xls', YS, 'SimData', 'B2'); % This portion of the code is for computing business cycle properties of the model SimData = zeros(per+3,7,trials); Temp2 = zeros(per+3,1); for j = 1:trials Temp1 = zeros(per+3,1); stoch_simul(IRF=0,hp_filter=1600,ORDER=1,noprint,nocorr,nofunctions,nomoments) y, c, invt, rer, delS, ys, cs; SimData(:,1,j) = [y]; SimData(:,2,j) = [c]; SimData(:,3,j) = [invt]; SimData(:,4,j) = [rer]; SimData(:,6,j) = [ys]; SimData(:,7,j) = [cs]; Temp2 = [delS]; for i = 2:(per+3) Temp1(i,1) = Temp1(i-1,1) + Temp2(i,1); end SimData(:,5,j) = Temp1; end save BCProp.mat SimData;