Outline & Motivatio •ooo oooooooooooooooooooooo Brüha-Podpiera model Mumerical techniques oooooooooooooooooooooooooooooooooooo evropský sociální fond V ČR EVROPSKÁ UNIE !7** MINISTERSTVO ŠKOLSTVÍ, OP Vzdelávaní MLÁDEŽE A TĚLOVÝCHOVY pro konkurenceschopnost INVESTICE DO ROZVOJE VZDELÁVANÍ Two country-models in international economics: modeling, applications, and solution Jan Bruha Lecture given at the Masaryk University, October, 2011 Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OflOO oooooooooooooooooooooo oooooooooooooooooooooooooooooooooooo Outline of the Lecture O Motivation O International Economics • International Trade • Balassa-Samuelson effect O Application to Central European Countries: Brüha-Podpiera model © Numerical Techniques International Economics Bruha-Podpiera model Mumerical techniques oooooooooooooooooooooo oooooooooooooooooooooooooooooooooooo Goal of the lecture Outline & Motivation oo«o Goal of the lecture During this lecture, I will introduce some models from international economics, which may be useful for understanding real convergence, trade flows, or external balance of open economies. One can investigate these phenomena from different perspectives, such as: • business-cycle dynamics, • trends, Outline & Motivatio ooo» oooooooooooooooooooooo Bmha-Podpiera model Mumerical techniques oooooooooooooooooooooooooooooooooooo I will concentrate on modeling trends. Hence, most models will be casted in a perfect-foresight framework with no aggregate uncertainty. This is distinct from DSGE models in: • Goal: understanding of trends rather than business cycle fluctuations • Approach: perfect foresight rather than rational expectations; • Solution: • most DSGE — dynamics around BGP, where trends are exogenous (sometimes even around steady state) • this kind of models — dynamics of trends 1 The main issues: O Why there is trade? O What is traded? 0 Who trade with whom? O At which price? Selected frameworks: • Comparative advantages (David Ricardo) » Intra-industry trade (Paul Krugman) • Intra-industry trade + heterogenous firms (Jacques Melitz) Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO 0*00000000000000000000 oooooooooooooooooooooooooooooooooooo Ricardian theory of trade • Countries differ in their technology. a Key assumption: it is easier to move goods than technologies. • Motive for trade - it is statically efficient to trade if technologies are different (so-called comparative advantages.) This theory predicts that: Most trade will occur between countries with different technologies (North-South trade should dominate) • As countries converge, motives for trade fall Modern version of the model: Eaton and Kortum (2002) Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOÄOOOOOOOOOOOOOOOOOOO oooooooooooooooooooooooooooooooooooo Heckscher-Ohlin model of trade (1933) • Countries differ in their factor endowments. • Key assumption: it is easier to trade goods than factors of production. • Key finding: trade alone may equalize factor prices. • Motive for trade: endogenous differences in technology. Countries must differ in order to trade: • Ricardo model - technologies differ; • HO model - factor endowments differ. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo ooo«oooooooooooooooooo oooooooooooooooooooooooooooooooooooo Empirical challenges to Ricardo and Heckscher-Ohlin • Countries with similar technologies trade. • Countries with similar factor endowments trade. • =4> North-North trade dominates trade flows (technologically advanced countries, capital abundance) • A large fraction of trade is two-way intra-industry trade. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOO0OOOOOOOOOOOOOOOOO oooooooooooooooooooooooooooooooooooo Krugman model of trade (1980) Very elegant model, which can explain why countries with identical technology and preferences trade. Key ingredients • monopolistic competition; • increasing-returns-to scale (product specialization); • love-for-variety (consumers want to consume all possible goods). The model relied by the then advances in modeling of imperfect competition (Dixit-Stiglitz approach). Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques oooo ooooo«oooooooooooooooo oooooooooooooooooooooooooooooooooooo Krugman model - stylized exposition /l Consumers: utility maximization: s.t. PjXj = Income. Parameter 9 > 1 measures the elasticity of substitution (if 9 —> oo), goods are perfect substitutes (perfect competition). Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques OOOO OOOOOO0OOOOOOOOOOOOOOO oooooooooooooooooooooooooooooooooooo Krugman model - stylized exposition /2 Demand function: 'Pi P Note: O P does not depend on x,-; i 0 If pi = p, then P = pn1-0 - this is called love-for-variety. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo ooooooo«oooooooooooooo oooooooooooooooooooooooooooooooooooo Krugman model - stylized exposition /3 Firms:Total costs = marginal cost (constant for simplicity) + fixed costs of production: w TQ = qi— + f, a (a is technology, f is fixed costs). Resulting optimal supply: 9 w Without trade: w Profit/ = TRf- - TQ = p;q; - q-,--f, a Profit,-_/ a y-11 (o^-iy-1 _ and the zero-profit condition yields the equilibrium real factor price w/P oc Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOO0OOOOOOOOOOOOO oooooooooooooooooooooooooooooooooooo Krugman model - stylized exposition /5 Trade: iceberg costs - a fraction of goods sent is lost during transportation t. Domestic price: pf- = ^pj; Foreign price: pf = (l+t)^f Results: O all goods are traded even if countries are perfectly symmetric (love-for-variety effect); O specialization (each country produces a subset of goods); Q trade gains: increase the number of products (increase of profits); O decrease in t: effect of P, but not on average of p,. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo ooooooooo«oooooooooooo oooooooooooooooooooooooooooooooooooo Krugman model - stylized exposition /6 Asymmetric countries (n a large market (or in a country with better technology, i.e., lower marginal costs): • lower price index P, but higher average price P\ • consumers are less willing to import additional unit of foreign varieties (due to constant elasticity of the demand); • relative factor price increases (aka currency appreciation) • higher nominal income, lower price index P - higher real income. Interesting implications in the economic geography. Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques OOOO 0000000000*00000000000 oooooooooooooooooooooooooooooooooooo Krugman model - empirical problems • There is a lot of heterogeneity across firms, within any sector. • Very few firms export (or engage in FDI). • Exporters are very different from non exporters (usually bigger and more productive). Heterogeneity: • Firms differ in productivity Trade barriers: • Iceberg costs • Fixed entry cost to export market Extensions • In the original Melitz model, countries are symmetric • In the original Melitz model, firms differ only by productivity All these assumptions can be relaxed Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO 000000000000*000000000 oooooooooooooooooooooooooooooooooooo Melitz model - implications Implications: Three sets of firms: • non-producers; a those who produce only for the domestic market, • exporters. Sorting is based on productivity. Original model has labor only, but if capital is added, then exporters would be larger than non-exporters. Trade liberalization: • Aggregate productivity is increasing; • Reallocation to more productive firms; • The effect of the liberalization can be seen even before the liberalization actually happens. CES preferences are used in most international-trade models: • Simplicity • Constant-elasticity of the demand • No choke prices (even with very large price, there is some demand) Alternative: linear-quadratic utility: tf = a£/<7/-/3£/<7?-7(£;<7/)2 • Demand: q, = a — b * p, + c * P, with P = £,-p/. • There is a choke price: pf- = a+c*p • Elasticity of demand increases with price • Complicated Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOO0OOOOOOO oooooooooooooooooooooooooooooooooooo Comparison of IT models - based on Baldwin and Harrigan (2007) Model Pr (export= =0) importer distance size remoteness Eaton-Kortum + + + Mon. comp. (CES) 0 0 0 Mon. comp. (linear demand) + 0 + Hetero. firms (CES) + - + Hetero. firms (linear demand) + + + Hetero. firms (CES + quality) + - + Model Export price importer distance size remoteness Eaton-Kortum 0 + Mon. comp. (CES) Mon. comp. (linear demand) 0 0 0 0 + Hetero. firms (CES) Hetero. firms (linear demand) Hetero. firms (CES + quality) + + + Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooo«ooooo oooooooooooooooooooooooooooooooooooo Open issues in international trade Open issues: • Why trade has increased faster than the GDP? • The Interplay between FDI and trade? • Why did trade collapse during the recent recession. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo ooooooooooooooooo«oooo oooooooooooooooooooooooooooooooooooo Real exchange rates - some definitions: Real exchange rate = nominal FX + foreign price level - domestic price level in logs: q = e + p* — p, Two sectors: tradable and non-tradable. Domestic price level: p = a * pT + (1 — a) * pNT. Hence: q = e + {p*T-pT) + [(1 - a){pNT - pT) - (1 - a){p*NT - p*T)], If PPP holds in the tradable sector, then e + (p*r - pT) = 1, i.e., real terms-of-trade: qT = e + (p*r — pT) Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooo«ooo oooooooooooooooooooooooooooooooooooo Supply side with two sectors: YT = ATF{KT, LT) and YNT = ANTG{KNT, LNT). If F and G are constant-return-to-scale, then in per capita terms (yr = Yj/Lt = f{kT) = 1/LT * F{KT/LT, 1) and so on): yT = ATf(kT) and yNT = ANTf(kNT). The F.O.C. are given as: PjAjf'{kj) = r, Pa/t^a/t^X^a/t) = r> and hence: kT = ^TQ^r;^_,), kNT = kNT(Aj^r,^r^) PTAT[f{kT) - f'{kT)kT} = w, PNTANT[f{kNT) ~ f'{kNT)kNT] = w. Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOO0OO oooooooooooooooooooooooooooooooooooo Factor price equalization If the interest rate r is exogenous (world price) and both factors can freely move across sectors, then: w = pjwj{ Aj >v_>^,) and + w = Pntwnt{A.nt',^J^,) and nence: + Pnt = wT(AT, r) PT wNT(ANT,rY i.e., just the relative productivity in both sectors determines the relative price ^p^r- This result does not depend on the demand side of the model. Log-linearization implies: MT y Labor share in NT T MT Labor share in T Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOÄO oooooooooooooooooooooooooooooooooooo BS effect: Recall: q = e + (p*T - pT) + [(1 - a)(pNT - pT) - (1 - a)(p*NT - p*T)], and plug in NT T Labor share in NT T NT Labor share in T If the technological progress is relatively biased towards tradable sector, then the real FX rate will appreciate. Pitfalls: • Why should be technological progress biased towards the tradable sector? • The RER is explained by the movements in the non-tradable prices: implications for Terms-of-Trade. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo ooooooooooooooooooooo» oooooooooooooooooooooooooooooooooooo BS effect - evidence for CEE countries The upper estimates suggest that about 1/3 of the observed RER appreciation is explained by the BS effect. Explanations: • Administrative and regulated prices • Initial undervaluation • Appreciation in the tradable sector Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO »00000000000000000000000000000000000 Motivation Bruha-Podpiera two-country models Motivation: • to mimic a strong pace of the real exchange rate appreciation observed in transition countries, • to inquire about the necessary model ingredients, The model aims at long-run trends, not medium frequency deviations, so it is formulated as a perfect-foresight DGE model. International Economics Bruha-Podpiera model Mumerical techniques oooooooooooooooooooooo o»oooooooooooooooooooooooooooooooooo Stylized Facts Outline & Motivatior oooo Stylized facts related to V4 countries: Economic convergence towards the EU average the convergence in GDP per capita towards the EU average about 1 p.p. a year Trade integration an increase in the export/GDP ratio about 2 p.p. a year Real exchange rate appreciation about 2% a year (also in the subindex of manufacturing). High-tech production share has gained from 1.5 - 2 p.p. a year International Economics Brüha-Podpiera model Mumerical techniques oooooooooooooooooooooo oo»ooooooooooooooooooooooooooooooooo 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Outline & Motivatior oooo Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooo«oooooooooooooooooooooooooooooooo Stylized facts 00.O Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo oooo«ooooooooooooooooooooooooooooooo How to generate the RER appreciation? It is not trivial to generate the RER appreciation after an uniform increase in productivity. Why? Because of the dowr sloping demand curve Possible approaches: O Horizontal investment (expansion in new varieties) 0 Harrod-Balassa-Samualson story O Vertical investment (quality) Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooooooooooooooooooooo ooooo«oooooooooooooooooooooooooooooo Horizontal investments Love-for-variety The horizontal investment explanation is based on a dichotomy between welfare-theoretical price indexes and 'average' observable price indexes. A more productive country has ceteris paribus higher average prices, but welfare-theoretical price index is lower because of expansion in varieties. Krugman (1980), Melitz (2003) International Economics Bruha-Podpiera model Mumerical techniques OOOOOOOOOOOOOOOOOOOOOO OOOOOO0OOOOOOOOOOOOOOOOOOOOOOOOOOOOO Export Eligibility Outline & Motivatior oooo The productivity increase may be biased towards tradable goods, then the usual HBS effect causes the RER appreciation. Why should be productivity biased towards tradables? The self-selection mechanism, Bergin, Glick, Taylor (2006). Data - very limited scope for the HBS in the V4 countries: Podpiera, Cincibuch (2006), Egert (2007). International Economics Bruha-Podpiera model Mumerical techniques oooooooooooooooooooooo ooooooo»oooooooooooooooooooooooooooo Vertical Investment Outline & Motivatior oooo The productivity increase vertical margin (quality investment), which implies that more goods can be sell for higher prices. The RER appreciation after a productivity increase is based on dichotomy between quality- adjusted and quality- unadjusted prices. Price indexes are rarely adjusted for quality: Ahnert, Kenny (2004). Task is to integrate the vertical margin in a two-country DGE model and to inquire whether implications are consistent with the facts outlined above. International Economics Brüha-Podpiera model Mumerical techniques OOOOOOOOOOOOOOOOOOOOOO OOOOOOOO0OOOOOOOOOOOOOOOOOOOOOOOOOOO Framework Outline & Motivatior oooo • Two countries in discrete time • Each country endowed with a representative consumer and heterogeneous firms • Foreign country - big and advanced • Domestic country - small and converging • A metaphor for a transition country (domestic country) versus the Euro area (foreign country) Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooo«oooooooooooooooooooooooooo Vertical Investment Margin We consider the following production function: qjt = Atzjk*!1-*, where At is the TFP, zj is the idiosyncratic productivity, k is the quality input, / is labor and a £ [0 1). If a = 0, the production function is linear and all types goods have the same quality (as is standard e.g. in Ghironi, Melitz 2005). If a > 0, then it is optimal to choose k > 0. The optimal amount of invested capital k = k{ At , zj ). International Economics Brüha-Podpiera model Mumerical techniques OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOO0OOOOOOOOOOOOOOOOOOOOOOOOO Firms Outline & Motivatior oooo Firms are NPV optimizers and choose: • labor input (variable); • export eligibility (fixed at entry, sunk costs); 0 quality level (fixed at entry). Think of firms as of projects! Backward induction used for solution of firms' problem: O labor is chosen as to equalize MPL with real wage; O the quality level is increasing in zj and is higher for exporters; O there is a cut-off of zj, which determines the exporter status. Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooooo«oooooooooooooooooooooooo Market structure - Dixit-Stiglitz The aggregate good is defined as: nT / q?T; dG(j) + n*T / l£q£t' dG(J) where nT is the number of entrants. The market structure implies the aggregate price index: Pt = ( £(1 - *)t_T \"r J Pjrt° dG(J) + n*Tj\fTpf;B dG(j) l 1-8 Today, I would experiment with the linear-quadratic utility. International Economics Brüha-Podpiera model Mumerical techniques OOOOOOOOOOOOOOOOOOOOOO 000000000000*00000000000000000000000 Households Outline & Motivatior oooo The household maximizes oo maxU = J2ßtu(Ct), t=o subject to -1 1 VJ/d 0 Bt = (l+rt*_1)ßt_1+— (Ct - Wt£)+- {Et - ctnt)-^-B2t+Tt, Vt Vt z Et = J2^-S)t-5ns so(l-«J)VrvPt,t International Economics Bruha-Podpiera model Mumerical techniques oooooooooooooooooooooo ooooooooooooo«oooooooooooooooooooooo General Equilibrium Outline & Motivatior oooo General Equilibrium is a sequence of prices and quantities such that all agents maximize and all market clears. • Labor Markets clear • Goods Markets clear (GDP identity in the two countries) Consistency of Portfolios Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO 00000000000000*000000000000000000000 Computational experiments We use a computer-intensive sampling scheme to understand the implications of the various modeling assumptions. Parameter Lower bound Upper bound exit shock 5 0.050 0.750 CES parameter 9 3.500 7.500 icebergs t 0.025 0.150 investment cost c" 2.000 10.00 export-eligibility costs ce 1.050 5.000 International Economics Brfjha-Podpiera model Mumerical techniques oooooooooooooooooooooo ooooooooooooooo»oooooooooooooooooooo Implications Outline & Motivatior oooo Is there a combination of parameters which could generate the reasonable REER appreciation? No under the standard assumptions (i.e. a = 0) Yes if the model framework is extended by the quality investments. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO 0000000000000000*0000000000000000000 Implications of Different Investment Margins Export self-selection and horizontal margin helps Export self-selectiveness can explain why more productive economies have higher price levels and help to explain why the 'observed' real FX rate of a converging economy is expected to appreciate. ... but they are alone insufficient Quality investment needed to explain the observed pace. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOÄOOOOOOOOOOOOOOOOOO The convergent trajectory International Economics oooooooooooooooooooooo Brüha-Podpiera model Mumerical techniques oooooooooooooooooo«ooooooooooooooooo Applications The modeling framework has been applied in a different context: The assessment of the EMU inflation criterion by Bruha and Podpiera (2007), ECB WP 740 The calibration of the Czech economy by Bruha, Podpiera and Polak (2010), The Convergence Dynamics of a Transition Economy: The Case of the Czech Republic, Economic Modelling 27, January 2010, pp. 116-124. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOO0OOOOOOOOOOOOOOOO The assessment of the EMU inflation criterion RER decomposition: rft = St + 7Tj - 7Tt, Conditional on stable nominal exchange rate st = 0, and the price stability of the EA, -k*t = 0.02, we evaluate the dynamic path for the trend inflation of the converging country as f0ll0WS:7Tt = 7T( — rj^. The path can be in turn compared against the benchmark inflation (average inflation in the three best performing EU Member states plus 1.5 percentage points), i.e., 7r" = 7r£ + 0.015 Probability of fulfillment of the criterion: Prob(7Tf* > 7rt|o",st = 0,rff). Historical evaluation using detrended (Hodrick-Prescott filter A = 100) inflation (CPI index) over period 1995-2010. Outline & Motivatio oooo International Economics oooooooooooooooooooooo Brüha-Podpiera model Mumerical techniques oooooooooooooooooooo«ooooooooooooooo Table: Parameters of the model Parameter CZ HU PO SK Elasticity of intra, subst. 9 6.32 Utility function e 0.50 Production function a 0.20 Exit shock 5 0.05 Iceberg costs t 0.27 Sunk cost of exporting cx 0.50 Portfolio adj. costs 10.0 Productivity m 1.72 1.79 2.31 1.18 Productivity n 6.28 7.37 8.97 6.58 Productivity A* 1.35 1.35 1.23 1.43 Productivity T 9.33 9.33 11.70 9.33 Relative country size £*/£ 30 30 10 60 Domestic productivity: At l+mexp(-(t-1995)/r) M l+nexp(-(t-1995)/r) " Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOO0OOOOOOOOOOOOOO Figure: Czech Republic si exchange rate [Index] 2ÜÜÜ 2ÜÜ5 2Ü1Ü 2015 2Ü2Ü 2Ü25 2Ü3Ü 2035 2ÜAÜ 2ÜA5 Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo oooooooooooooooooooooo«ooooooooooooo Figure: Hungary GDP convergence [% of EA] 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 si exchange rate [Index] 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOO0OOOOOOOOOOOO Figure: Poland GDP convergence [% of EA] 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 si exchange rate [Index] 2000 2005 2010 2015 2020 2025 2030 2035 2040 2045 Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo oooooooooooooooooooooooo«ooooooooooo Figure: Slovakia GDP convergence [% of EA] Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooooooooooooooooooo»oooooooooo Figure: Probability of fulfillment of the inflation criterium Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooooooooooooooooooo»oooooooooo How to solve perfect-foresight models This part of the lecture will overview selected solution techniques for perfect-foresight discrete-time economic models. Problem statement Two-point boundary value problem (with infinite Two difficult points: • perfect-foresight: what agents do today depends on the current state (what they did yesterday) and their expectations on what would happen tomorrow (what they will do in future); • infinite-horizon: equilibrium is an infinite-dimensional system (policy function is of no help, if the model is not autonomous). International Economics Bruha-Podpiera model Mumerical techniques OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOO0OOOOOOOOO Problem statement Outline & Motivatior oooo General problem statement: O Initial condition for state variables (e.g., capital and technology): ki, A\ given; Q Law of motion for exogenous states (e.g. productivity): {/4t}^1 - agents know this; O Law of motion for endogenous states (such as capital accumulation: kt+i = (1 — 5)kt + /t); O Equilibrium conditions (agents' decisions, market clearing) F(kt,ct,At) = 0 for all t e Z+; @ Transversality conditions (usually in the form of limt_).00/3tu(ct,/ct) = 0). The goal is to find {/ct}^1 and {ct}^1 consistent with conditions above. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooooooooooooooooooooo«oooooooo Simple example - a growth in an open economy: model • Two countries in discrete time; • One country big and advanced, the other country small and converging; • In each country, there is a representative consumer with recursive utilities: Ut = Yl^LtPT~tu{ct), • Budget constraint: Ct = (1 + rt)Wt - Wt+1 - T{AWt+1) +Yt- it • Production technology Yt = f(kt,At), the market clearing Yt = ct + k+xu • Capital accumulation kt+\ = (1 — 5)kt + it; • Balance-of-payments Wt+i = (1 + rt)Wt +xt; • Initial conditions ki, W\. • Terminal conditions \'\mt^,oo filu'{ct)kt = 0, Wvrit^oo fru'^wt = 0. Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO 0000000000000000000000000000*0000000 Simple example - a growth in an open economy: equilibrium equations o Optimal investments(t/(ct) = (3u'(ct+1)[fk(kt+1, At+1) + (1 - 5)], (l + T'(Al/l/t+1) = /3(l + rt+1)^ • Production technology Yt = f(kt,At), the market clearing Yt = ct + it + xt; • Market clearings xt = —x^ and Wt = -l/l/t* • Capital accumulation /ct+i = (1 — 5)kt + it; • Balance-of-payments Wt+i = (1 + rt)Wt +xt; • Initial conditions klt Wlt k\, • Terminal conditions Wmt^oo (3lu'(ct)kt = 0, Wvrit^oo fru'^wt = 0. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooooooooooooooooooooooo«oooooo Three possible approaches O Domain-truncation techniques O First-order iterations (Fair-Taylor) Q Quasi-Newton techniques (L-B-J) Q Projection techniques Domain truncation techniques solve the model for T periods with the hope that for t > T, endogenous variables will be at the constant levels (hence the infinite dimensionality is approximated by the dynamics with finite horizon). Projection techniques approximate the equilibrium dynamics by a (linear) combination of few elements (basic functions). International Economics Brüha-Podpiera model Mumerical techniques OOOOOOOOOOOOOOOOOOOOOO 000000000000000000000000000000*00000 Fair-Taylor approach Outline & Motivatior oooo Fair-Taylor: O choose T and guess {Af, c°}J-i O set / = 1 and for t = 1,... 7", compute /c/ and c[ using /c£_ and Cj_1 and /c^J and c^J; O check the convergence, if the convergence is not achieved, increase /<—/' + 1 and go to 2. Advantages: • economic intuition - learning; Disadvantages: • it may not converge - Gauss-Seidel method; • sometimes a dampening factor is helpful (*/ = /i*r + (i-/i)*r1); • even if it converges, it is slow (linear convergence). International Economics Bruha-Podpiera model Mumerical techniques oooooooooooooooooooooo ooooooooooooooooooooooooooooooo»oooo L-B-J approach Outline & Motivatior oooo L-B-J (due to Lafargue, 1990, Boucekkine, 1995, and Juillard et al., 1998): O choose T and form a huge (really huge) system H(ki, ci,..., kt, Q,..., kj, cj) = 0 (and set kj+\ equal to kj when appropriate. 0 apply a (quasi-) Newton techniques. O if you are clever, you can make this approach efficient (the Jacobian is usually tri-diagonal, clever ways of updating of the Jacobian, ...) Advantages: • if it converges, it is fast (quadratic convergence); Disadvantages: • it is really a huge system: a system of equations with TM unknowns (M being the number of endogenous variables); • How to choose Tl T should be much larger than the horizon of projection. Outline & Motivation International Economics Brfjha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO 00000000000000000000000000000000*000 Projection techniques /l Projection techniques (due to Judd, 2002): • Approximate the path of endogenous variables by a (linear) combination of basis functions: kt = sffi{t). • Choose so that equilibrium conditions are satisfied. • The infinite dimensional problem is reduced to find coefficients • Basis functions can be: (orthogonal) polynomials, splines, radial basis functions, finite elements, ..... Judd (2002) recommends: kt = e-xt + £ a?/Kt)) + (1 " e-Xt)kSS, where f;[t) = Z_/(2A£)e~At and L-, are Laguerre polynomials, A governs the speed of convergence to the new steady state k$s and could (actually should) be computed based on the linearization of the model. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques OOOO OOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOÄOO Projection techniques /2 How to choose coefficients a? • Set residual function R(t,a). • Brut force: solve the optimization problem mm, J2Li \\R{t,a)\\p for suitable p. • If p = 2, then you solve a non-linear least-square problem. • you still have to truncate the time to compute the sum, but instead of T coefficients, you need only /. It is possible to combine L-B-J with projection techniques: • If the trajectory of endogenous variables is not smooth (abrupt, unexpected changes), then it is hard to approximate it with smooth basis functions (such as polynomials) - you would need a large /. • The idea is to approximate for first t by L-B-J and then use projection. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo oooooooooooooooooooooooooooooooooo«o Projection techniques /3 There are better ways to chose the coefficients a: Galerkin method • consider the integral /0°° R(t, a)tpj(t)dt, where ipj(t) are test functions. • if you choose ip(t) = R(t, a) you are back to non-linear least-square problem. • Hope is that if you chose test functions ipj(t) cleverly, then /0°° R(t, a)ipj(t)dt will be zero if R(t, a) is. 0 use a quadrature to approximate /0°° R(t, a)iPj(t)dt * J2k R(tk, a)iPj(tk)wk. • Therefore, you need not to compute the residual function R(t, a) for all t = 1,..., 7~, but only for (rounded) values tk. Not always applicable. Outline & Motivation International Economics Brüha-Podpiera model Mumerical techniques oooo oooooooooooooooooooooo ooooooooooooooooooooooooooooooooooo» Application to Bruha-Podpiera model The model is rewritten into the first-order form and the idea is to rewrite all variables in term of 7 endogenous variables - a great reduction in the dimensionality of the problem. It has its costs as the Jacobian for L-B-J is no longer tridiagonal and all 1 < t < T should be computed even for the Galerkin method. • Fair-Taylor: the method failed; » L-B-J: in general it works, but it is relatively slow during first iterations ; • Projections: safe and method, but sometimes difficult to obtain precise results (slow last iterations); • The best way seems to use projections to get relatively accurate results (error about 10~6) and then use L-B-J if further accuracy is required.