Notes on Winnie Choi s Paper (Draft: November 4, 2004; Revised: November 9, 2004)

Transcription

1 Dave Backus / NYU Notes on Winnie Choi s Paper (Draft: November 4, 004; Revised: November 9, 004) The paper: Real exchange rates, international trade, and macroeconomic fundamentals, version dated October 9, 004. See: rc65/readinggroup/choi.pdf These are rough notes outlining my (probably idiosyncratic) interpretation of the paper. I work through the logic in a relatively simple setting, but most of it generalizes. As usual, no guarantees of accuracy or sense. Background Consider a world with two agents/countries (i =, ) and two goods (apples a and bananas b). Country is endowed with y apples, country with y bananas. Each consumes both, although typically we posit a preference in favor or the local good. Use the usual stochastic setting: a tree with histories s t. Utility is additive over time and across states with common discount factor β and probabilities π(s t ). The period/state utility function has two components: aggregators that define consumption as composite goods, c c = [ω a ρ + ω b ρ ] /( ρ) = [ω b ρ + ω a ρ ] /( ρ), and a period utility function, u(c i ) = c α i / α. Here ρ > 0 governs the elasticity of substitution between apples and bananas (the elasticity is /ρ), ω > ω indicates home bias for goods, and α > 0 is a curvature parameter that controls (among other things) risk aversion over the composite good. In what follows, we can let the ω s vary across countries, but ρ (and sometimes α) cannot. The resource constraints are y = a + a y = b + b. (We ll modify them later to allow transport costs.) With this setup, the equilibrium in pretty much any model that allows frictionless trade in spot markets has allocations that solve a Pareto problem for each history s t : max θ i ci α /( α) + (y a a ) + (y b b ) a i,b i i

2 for some welfare weights (θ, θ ). All of these variables (including the θ s) depend on s t. The first-order conditions are: ω a ρ ω a ρ ω b ρ ω b ρ. We solve these four equations plus the resource constraints for the unknowns (a, a, b, b,, ). It s relatively easy to convert the foc s to a relation between real exchange rates and the ratio of consumptions. Multiply the first and third foc s by, resp, a and b and sum: ( ) a + b ω a ρ + ω b ρ. The lhs is p c, where p is the (minimum) price of one unit of consumption. The rhs becomes θ c α. Dividing both sides by c gives us: p = θ c α. It should be clear that all we ve used is the power utility function and properties of homothetic functions (ie, apply Euler s theorem to the consumption aggregator). If we do the same thing for the second agent s first-order conditions, we find that the real exchange rate (relative price of consumptions) is e p /p = (θ /θ )(c /c ) α. The point is that the social planner equates weighted marginal utilities to relative prices. With complete markets, the θ s don t depend on s t. As a result, e (s t ) p (s t )/p (s t ) = (θ /θ )[c (s t )/c (s t )] α, () so we have a perfect (negative) correlation between the real exchange rate and the consumption ratio (in logs). Of course, we see nothing like this in the data. Choi s analysis Choic introduces two new wrinkles: (i) The welfare weights can vary with the history. (ii) Transport costs: if country ships x apples to country only η x arrive. The transport cost is ( η ) per unit. Similarly, let η be the analogous parameter for shipping bananas from country to. This reverts to the no-cost version if η = η =. The resource constraints become y = a + a /η y = b /η + b

3 and the first-order conditions change to /η /η ω a ρ ω a ρ ω b ρ ω b ρ. Note that the variables on the left are the prices quoted in the appropriate country: /η is the price of apples in country. With this interpretation, we get the same relation between the real exchange rate and the consumption ratio we saw earlier, namely p i = θ i c α i () and its implication, equation (). Choi takes a different approach to the same problem: use () to substitute for the θ s in the first-order conditions: /η /η = p c ρ ω a ρ = p c ρ ω a ρ = p c ρ ω b ρ = p c ρ ω b ρ. The first and second imply p /p = (/η )(ω /ω )(c /c ) ρ (a /a ) ρ. The third and fourth imply p /p = (η )(ω /ω )(c /c ) ρ (b /b ) ρ. Either of these relations could be used in place of (). Choi suggests the geometric average: p /p = (η /η ) / (c /c ) ρ (a /a ) ρ/ (b /b ) ρ/. In practice, this works better, which I regard as a major accomplishment. Free advice This is worth what you paid for it, but here s a possible plan for the paper. I think you have two really good ideas one theoretical, the other empirical but it would help to make the exposition crystal clear. The following plan is an attempt to do this: to make the logic easy to follow and use it to set up your striking evidence. My outline goes like this, organized by section: 3

4 . Intro. There are lots of puzzles about exchange rates, but one of the most striking is the lack of any systematic relation between high-frequency movements in exchange rates and anything that might be thought of as a fundamental. See Frankel and Rose (995) among many others. One example is the lack of any relation between real exchange rates and the kinds of quantities that would show up in first-order conditions for agents choosing between domestic and foreign goods. These conditions are one of the backbones of trade theory, and play a central role in most international macro models as well. bla bla bla. Cite Backus-Smith, Chari-Kehoe-McGrattan, Obstfeld- Rogoff, anyone else who fits (the idea being not to give credit, but to convince the reader this is a mainstream issue in the field). The question is what to do. Propose frictions in intertemporal trade. Somewhat surprisingly, this affects the relation between prices and quantities at a point in time and leads to an empirical relation that works substantially better than most competitors.... Consumption and exchange rates. Work through a model like the first section above. Use simpler notation than either you or me. One thought: let utility be u(x ij ) = [ i ω ijx ρ ij ] /( ρ) for country i. In this section, do a two-good version to keep things simple. Derive (). Use (say) US/Canada data to show it doesn t work. Next ask: would it help to allow the θ s to vary? Graph time series for US/Canada of θ /θ = (p /p )(c /c ) α for α = 0.5,,. (Put real exchange rate on same graph?) The idea: the evidence suggests lots of variation in the weights. Note that this is exactly what you get from lots of models with capital market frictions: incomplete markets, participation constraints, etc. [But don t write down the equations for these models!] Looking ahead, note that the ratio of US exports to Canada to Canadian exports to the US is highly correlated with the real exchange rate (I m guessing, but something must be). 3. Capital market frictions. Do the theory outlined in the second section, again in the two-country case, without transport costs. (They re a red herring, and you can deal with them later when they re less of a distraction.) Redo the evidence for US/Canada, show how well it works. Be specific about the source of the data, make sure you convince people that the export quantities didn t slip in the exchange rate somehow (ie, they re pure quantities). 4. A general theory. Add many countries, nontraded goods. You may be able to generalize the utility function, too (nested CES? general hd?). Rather than the square root, I d show that the model implies several relations for the real exchange rate, and propose a geometric average as a summary. But in the evidence, the separate relations should all give you the same answer: if they don t, it s a problem for the theory. Perhaps some discussion of this is called for. [And ignore my comment about some goods not being traded in equilibrium not worth the effort.] 5. Evidence for OECD countries. Follow with evidence for many countries, as in the paper. The major challenge here is expositional: taking huge tables and making them 4

5 easy to follow. Personally, the counting (50 out of 55 or whatever are higher) isn;t as striking as a picture if you can come up with one. I d suggest one that Mario Crucini uses. You have lots of correlations and want to say yours are higher than the traditional ones. Mario graphs a smoothed density estimate of each and compares them. You should see that yours is shifted to the right. See how it works. The hope is that a single picture will make your point in a visually effective way. Then put the tables in the appendix (or skip). 6. Trade frictions. Add transport costs, show that they enter your theory in a natural way. Talk about the impact of letting the η s be state-dependent. Perhaps talk about how they might complement the capital market frictions you proposed above. [Notation: I like the Alvarez-Lucas use of κ for η, but it s a tiny point.] 7. Open issues. What kind of model leads to this variation in θ s? Do we see more variation of θ in emerging markets corresponding to greater variation in exchange rate? (Ie, can we attribute greater exchange rate volatility to greater frictions in capital markets?) Etc. Your call, but that s what I d do. It sets up the next paper (some kind of explicit model of imperfect capital markets). Let me know if you d like to discuss further. 5