International Financial Management 6

Figure 8.4: PPP – Persistent deviation from Rt = 1 (2000-2022) Real FX: AUD, BRL, CAD, CHF, JPY, DKK & GBP 12/1/2021 2/1/2021 4/1/2020 6/1/2019 8/1/2018 GBP 10/1/2017 12/1/2016 2/1/2016 DKK 4/1/2015 6/1/2014 8/1/2013 JPY 10/1/2012 12/1/2011 2/1/2011 4/1/2010 CHF 6/1/2009 8/1/2008 CAD 10/1/2007 12/1/2006 BRL 4/1/2005 6/1/2004 8/1/2003 10/1/2002 12/1/2001 2/1/2001 4/1/2000 AUD 2/1/2006 1.4 1.2 1 0.8 0.6 0.4 0.2 0 ‐0.2 ‐0.4 ‐0.6 With some exceptions, the Big-Mac tends to be more expensive in developed countries (U.S.A., Euro area, Australia) than in less developed countries (Colombia, Guatemala, Pakistan).¶ Empirical Fact: Price levels in richer countries are consistently higher than in poorer ones. It is estimated that a doubling of income per capita is associated with a 48% increase in the price level. This empirical fact is called the Penn effect. Many explanations, the most popular: The Balassa-Samuelson (BS) effect. • Absolute PPP: Qualifications The big deviations from absolute PPP are usually attributed to different reasons: (1) PPP emphasizes only trade and price levels. Other financial, economic, political factors are ignored. (2) Absence of trade frictions. This is an implicit assumption: No tariffs, no quotas, no transactions costs. Realistic? It is estimated that transportation costs add 7% to the price of U.S. imports of meat and 16% to the import price of vegetables. Some products are heavily protected, even in the U.S. For example, peanut imports are subject to a tariff between 131.8% (for shelled peanuts) and 163.8% (for unshelled peanuts). (3) Perfect competition. Imperfect competition, usually related to (2), can create price discrimination. For example, U.S. pharmaceuticals sell the same drug in the U.S. and in Canada at different prices. (4) Instantaneous adjustments. Another implicit PPP assumption, related to another trade friction. Not realistic. Trade takes time and it also takes time to adjust contracts. Think of PPP as long-run model. (5) PPP assumes Pf and Pd represent the same basket, not the usual situation for CPI baskets. This is why the Big Mac is a popular basket: it is standardized around the world with an easy to get price. (6) Internationally non-traded (NT) goods (~50%-60% of GDP) –i.e., haircuts, hotels, restaurants, home & car repairs, medical services, real estate, etc. NT goods have a big weight on the CPI basket. Suppose the proportion of NT in the price basket is αN. Then, using log price notation, we can write the (log) price index as: pt = αN pNT,t + (1 – αN) pT,t where pNT,t is the log price of NT goods and pT,t is the log price of taded (T) goods. If PPP holds for IFM-LN.77 traded goods, st + (pf,T,t – pd,T,t) = 0, then, after some algebra: rt = αN (pf,NT,,t – pd,NT,t + st). Does PPP hold for NT goods? (7) The NT sector also has an effect on the price of traded goods. For example, rent, distribution and utilities costs affect the price of a Big Mac. (It is estimated that 25% of Big Mac’s cost is due to NT goods.) • Borders Matter You may look at the Big Mac Index and think: “No big deal: there is also a big dispersion in prices within the U.S., within Texas, and, even, within Houston!” It is true that prices vary within the U.S. For example, in 2015, the price of a Big Mac (and Big Mac Meal) in New York was USD 5.23 (USD 7.45), in Texas was USD 4.39 (USD 6.26) and in Mississippi was USD 3.91 (USD 5.69). Engel and Rogers (1996) computed the variance of LOOP deviations for city pairs within the U.S., within Canada, and across the border. They found that distance between cities within a country matter, but the border effect is very significant. To explain the difference between prices across the border using the estimate distance effects within a country, they estimate the U.S.-Canada border should have a width of 75,000 miles! This huge estimate of the implied border width between the U.S. and Canada has been revised downward in subsequent studies, but a large positive border effect remains. • Balassa-Samuelson Effect Balassa (1964) and Samuelson (1964) developed a general equilibrium model of the real exchange rate (BS model). The model explains the above mentioned empirical fact: richer countries have consistently higher prices. Labor costs affect all prices. We expect average prices to be cheaper in poor countries than in rich ones because labor costs are lower. Rich countries have higher productivity, and higher wages, in the traded-goods sector than poor countries do. In the NT goods sector, productivity is similar. But, because of competition for labor, wages in NT goods and services are also higher in rich countries. Then, overall prices are lower in poor countries. For example, the productivity of McDonald’s employees around the world is likely very similar, but the wages are not. In 2000, a typical McDonald’s worker in the U.S. made USD 6.50/hour, while in China made USD 0.42/hour. This difference in NT costs may partly explain over/under-valuations when we compare currencies from developed to less developed countries. Again, standard applications of PPP, like in the Big Mac example above, will not be very informative. We need to “adjust” prices to incorporate the effect of GDP per capita in the price level. Practitioners tend to incorporate the Balassa-Samuelson effect into PPP calculations in a straightforward manner. Suppose we want to adjust Big Mac PPP-implied exchange rates. Then: IFM-LN.78 1) Estimate a regression using Big Mac Prices (in USD, PBM,t) as the dependent variable against GDP per capita (GDP_p). That is, run the following regression: PBM,t = α + β GDP_pt + 𝜀 2) Compute fitted values (GDP-adjusted Big Mac Prices). That is, 𝑃BM,GDP-adjusted = 𝛼 + β GDP_per_capitat Based on the GDP-adjusted Big Mac Prices, re-estimate the PPP implied over/under-valuation: GDP-adjusted over/under valuation: (BM Price/𝑃BM,GDP-adjusted) – 1 The regression line tells us what the “expected price” in a country is, once we take into consideration its GDP level. Using data from The Economist for July 2022, we estimated the above regression: 𝑃BM,GDP-adjusted = 3.045895 + 0.0000332 * GDP_per_capitat Figure 8.5 shows the regression line using data from in July 2022: Figure 8.5: PPP – Regression to Adjust Big Mac Prices by GDP per capita Switzerland Brazil Hong Kong Now, using the computed red line above, we calculate the “Expected BM prices, given the GDP of a given country.” For example, we compute the above value for Uruguay. Uruguay’s GDP per capita in July 2022 was USD 15,169.153. Then, 𝑃BM,GDP-adj (Uruguay) = 3.045895 + 0.0000332 * 15,169.153 = 3.549511 That is, the expected USD Big Mac price, in Uruguay, given its GDP per capita, was USD 3.55. Since the observed local BM price was UYU 255, which translates to USD 6.08 (= UYU 255 * 41.91 USD/UYU), then the GDP-adjusted over/under valuation was: IFM-LN.79 6.08 / 3.549511 – 1 = 71.29% (71.29% overvalued) On the other hand, we have Japan; which according to the adjusted index, its currency is undervalued by 35%. That is, these adjustments to PPP implied exchange rates can be significant. See Exhibit 8.2 below from The Economist for July 2011. Exhibit 8.2: The Economist’s Adjusted-Big Mac Index (July 2011) The Balassa-Samuelson effect can explain (or partially explain) why absolute PPP does not hold between a developed country and a less developed country, for example, after correcting for the BS effect, China’s currency is no longer undervalued. But the BS effect cannot explain why PPP does not hold among developed countries (say, Switzerland and the U.S.) or among less developed countries (say, Brazil and Argentina). • Pricing-to-market Krugman (1987) offers an alternative explanation for the strong positive relationship between GDP and price levels: Pricing-to-market –i.e., price discrimination. Based on price elasticities, producers discriminate: the same exact good is sold to rich countries (lower price elasticity) at higher prices than to poorer countries (higher price elasticity). For example, Alessandria and Kaboski (2008) report that U.S. exporters, on average, charge the richest country a 48% higher price than the poorest country. That is, the price of T goods consumed in the domestic (home) market, PTH, is different from the T goods consumed in the foreign market, PTF. Using log price notation: pTH,t ≠ pTF,t. Again, pricing-to-market struggles to explain why PPP does not hold among developed countries with similar incomes. For example, Baxter and Landry (2012) report that IKEA prices deviate 16% from the LOOP in Canada, but only 1% in the U.S. Similarly, The Economist (2019) reports that Pret a Manger, a coffee-shop chain, sells the same items in Boston at higher prices than in London, for instance, an egg- IFM-LN.80 sandwich costs GBP 1.79 (USD 2.15) in the U.K. and USD 4.99 in the U.S. • Absolute PPP: Empirical Evidence Several tests of the absolute version have been performed. The absolute version of PPP, in general, fails (especially, in the short run), even when using the same basket or the same good. No surprise here, see the Big Mac example above, where Rt 1. Trade frictions, especially transportation and distribution costs, are considered a major problem for absolute PPP. 8.1.2 Relative PPP A more flexible version of PPP: The rate of change in the prices of products should be similar when measured in a common currency. (As long as trade frictions are unchanged). Thus, Relative PPP addresses the assumption of no trade frictions. (All the other qualifications still apply!) The following formula states the relative version of PPP: 𝑒 , = = 1 (Relative PPP), where 𝑒 , = percentage change in the value of FC 𝐼 = (Pf,,t+T/Pf,t) -1 = foreign inflation rate from t to t+T; 𝐼 = (Pd,,t+T/Pd,t) -1 = domestic inflation rate from t to t+T. Linear approximation (from a 1st-order Taylor series): 𝑒, 𝐼 , 𝐼, Since this relationship is not expected to hold at every time interval, it is usually rewritten in terms of conditional expectations (averages): Et[𝑒 , ]  Et[𝐼 ] - Et[ 𝐼 ]. We expect to observe a one-to-one relation between (Id-If) and ef,t. Example: Suppose that, from t=0 to t=1, prices increase 10% in Mexico relative to those in Switzerland. Then, SMXN/CHF,t should increase 10%; say, from S0=9 MXN/CHF to S1=9.9 MXN/CHF. If, at t=1, S1=11 MXN/CHF > S1PPP = 9.9 MXN/CHF, then according to Relative PPP the CHF is overvalued. ¶ Example: Forecasting St (USD/ZAR) using PPP (ZAR=South Africa). It’s 2015. You have the following information: CPIUS,2021=104.5 CPISA,2021 = 100.0 S2021 = 0.2035 USD/ZAR. IFM-LN.81 You are given the 2022 forecast for the CPI in the U.S. and South Africa: CPIUS,2022 = 110.8 CPISA,2022 = 102.5. You want to forecast S2022 using the relative (linearized) version of PPP. Then, E[IUS,2022] = (110.8/104.5) – 1 = .06029 E[ISA,2022] = (102.5/100) – 1 = .025 SF2022 = S2021 * [1 + IFUS, 2022 – IFSA, 2022] = .2035 USD/ZAR * [1 + .06029 -.025] = .2107 USD/ZAR. You forecast an appreciation of the ZAR against the USD. ¶ • Relative PPP: Implications (1) Under relative PPP, Rt remains constant (it can be different from 1!). (2) Relative PPP does not imply that St is easy to forecast. (3) Without relative price changes, a multinational corporation faces no real operating exchange risk (as long as the firm avoids fixed contracts denominated in foreign currency). • Relative PPP: Absolute versus Relative Absolute PPP compares price levels, while Relative PPP compares price changes (or movements). Under Absolute PPP prices are equalized across countries, but under Relative PPP exchange rates move by the same amount as the inflation rate differential (original prices can be different). Relative PPP is a weaker condition than the absolute one: Rt can be different from 1. Example: Absolute vs Relative Absolute PPP: "A mattress costs GBP 200 (= USD 320) in the U.K. and BRL 800 (=USD 320) in Brazil –i.e., same cost in both countries." (St =1.6 USD/GBP & St =0.4 USD/BRL) Relative PPP: "U.K. inflation was 2% while Brazilian inflation was 8%. Meanwhile, the BRL depreciated 6% against the GBP. Then, relative cost comparison remains the same." ¶ • Relative PPP: General Evidence Key: On average, what we expect to happen, 𝑒 , , should happen, ef,T.  Q: Is, on average, 𝑒 ,  Id,T – If,T = ef,T? Under PPP, we should see ef,T and (Id – If)T aligned around a 45º line, like in Graph 8.1. Figure 8.6: PPP Line IFM-LN.82 PPP Line Id,T - If,T ef,T 1. Visual Evidence Figure 8.7 plots (Id-If) between Japan and the U.S. against ef(JPY/USD), using 1970-2019 monthly data. Figure 8.7: PPP Line for the JPY/USD (1970-2019)? Relative PPP: JPY/USD 0.15 ef,t 0.1 0.05 IJP ‐ IUS ‐0.04 0 ‐0.03 ‐0.02 ‐0.01 0 0.01 0.02 0.03 0.04 ‐0.05 ‐0.1 ‐0.15 There is no clear 45º line in the plot, like in Graph 8.6! (See also Graphs in book)  PPP does not track short-term movements. Figure 8.8-A plots Rt between Japan and the U.S. It can be used to check if Rt is constant (ideally, under absolute PPP, close to 1, but we do not have prices, but indices. Rt is arbitrary set to 1 in Jan 1971): IFM-LN.83 Figure 8.8-A: Real Exchange Rate between Japan and USA (1970-2019) 3.5 Real FX Rate: JPY/USD 3 2.5 2 1.5 1 0 1/1/1971 10/1/1… 7/1/1974 4/1/1976 1/1/1978 10/1/1… 7/1/1981 4/1/1983 1/1/1985 10/1/1… 7/1/1988 4/1/1990 1/1/1992 10/1/1… 7/1/1995 4/1/1997 1/1/1999 10/1/2… 7/1/2002 4/1/2004 1/1/2006 10/1/2… 7/1/2009 4/1/2011 1/1/2013 10/1/2… 7/1/2016 4/1/2018 0.5 Clearly, Rt is not constant! In general, we have some evidence for mean reversion for Rt in the long run. Loosely speaking, Rt moves around some mean number, which we associate with a long-run PPP parity (for the JPY/USD the average Rt is 1.93). But, the deviations from the long-run PPP parity are very persistent –i.e., very slow to adjust. Note that the deviations from long-run PPP parity can be big (up to 66% from the mean) and happen in every decade. Economists usually report the number of years that a PPP deviation is expected to decay by 50% (the half-life) is in the range of 3 to 5 years for developed currencies. Very slow! Note that even for currencies with a fixed exchange rate, like Belgium and Germany, Rt is not constant, as illustrated in Figures 8.8-B. Figure 8.8-B: Real Exchange Rate between Belgium and Germany (1970-2019) 1.15 Real FX Rate: Belgium/Germany 1.1 1.05 1 0.95 0.9 1/1/1971 10/1/1972 7/1/1974 4/1/1976 1/1/1978 10/1/1979 7/1/1981 4/1/1983 1/1/1985 10/1/1986 7/1/1988 4/1/1990 1/1/1992 10/1/1993 7/1/1995 4/1/1997 1/1/1999 10/1/2000 7/1/2002 4/1/2004 1/1/2006 10/1/2007 7/1/2009 4/1/2011 1/1/2013 10/1/2014 7/1/2016 4/1/2018 0.85 IFM-LN.84 2. Statistical Evidence Let’s look at the usual descriptive statistics for (Id – If)t and ef,t, using the 1971-2019 monthly data used above. For the JPY/USD, they have similar means, but quite different standard deviations (look at the very different minimum and maximum stats). A simple t-test for equality of means (t-test=-0.976) cannot reject the null hypothesis of equal means, which is expected given the large SDs, especially for ef,t. Rt IJP IUS IJP-IUS ef,T (JPY/USD) Mean 0.0020 0.0032 -0.0012 0.0016 1.9485 SD 0.0062 0.0038 0.0059 0.0313 0.4261 Min -0.0107 -0.0192 -0.0198 -0.1474 0.9977 Median 0.0010 0.0029 -0.0018 -0.0004 1.9318 Max 0.0423 0.0182 0.0335 0.1029 3.3218 If we think of the average over the whole sample, as a long-run value, we cannot reject PPP in the longrun! But, the average relation over the whole sample is not that informative, especially with such a big SD. We are more interested in the short-run, in the contemporaneous relation between ef;T and (Id – If)t. That is, what happens to ef;T when (Id – If)t changes? To test the contemporaneous relation we have a more formal test, a regression: 𝑒 , = (St+T – St)/St = α + β (Id – If)T + εT, (where εt is the regression error, E[εt]=0). The null hypothesis is: H0 (Relative PPP holds): α = 0 & β = 1 The alternative hypothesis is: H1 (Relative PPP does not hold): α ≠ 0 and/or β ≠ 1 Tests: t-test (individual tests on the estimated α and β) and F-test (joint test): (1) t-test = tθ = θ0 = [𝜽 – θ0] / S.E.( 𝜽) ~ tv (v = N – K = degrees of freedom). (2) F= [RSS(H0) – RSS(H1)]/J RSS(H1)/(N K) ~ FJ,N-K (J = # of restrictions in H0). Notation for tests: θ = (α, β) 𝜽 = Estimated θ et residuals ef;t – [ 𝜶 + 𝜷 (Id – If)t ] H0 (theory is true): θ = θ0 N = # of observations K = # of parameters in our model, in the PPP case 2: (α, β) RSS = Residual Sum of Squares = Σt (et)2. J = # of restrictions in H0, in the PPP case 2: α=0 & β=1 . α = significance level –most popular, α =.05 (5 %). IFM-LN.85 tv = t-distribution with v degrees of freedom (df). (When v > 30, it follows a normal). FJ,N-K = F-distribution with J df in the numerator and (N – K) df in the denominator. Rules for tests: If |t-test| > |tv,α/2|, reject H0 at the α level. (When α = .05 & v > 30, t.025 = 1.96.) If F-test > FJ,N-K,α, reject H0 at the α level. (When α = .05 & (N – K) > 300, F2,300+,.05 3.) Example: We want to test relative PPP for the JPY/USD exchange rate (we use α = .05). We use the monthly Japanese and U.S. data from the graph (1/1975 - 12/2022). We fit the following regression: = α + β 𝐼 𝐼 + 𝜀. ef,t (JPY/USD) = (𝑆 – 𝑆 )/𝑆 R2 = 0.005621 Standard Error (σ) = .02617 F-stat (slopes=0 –i.e., β=0) = 3.244 (p-value = 0.07219) F-test (H0: α=0 and β=1) = 19.185 (p-value: lower than 0.0001) => reject at 5% level (F2,574,.05= 3.01) Observations = 576 Intercept (𝛼 ) 𝐼 𝐼 (𝛽 ) Coefficients -0.00209 -0.39148 Stand Error 0.001157 0.217343 t Stat -1.804 -1.801 P-value 0.0717 0.0722 Let’s test H0, using t-tests (t477.05=1.96 –when N-K > 30, t.05 = 1.96): tα=0 (t-test for α = 0): ( -0.00209 – 0)/0.001157 = -1.804 (p-value = .07)  cannot reject H0 at the 5% level tβ=1 (t-test for β = 1): (-0.39148 – 1)/0.217343 = -6.402 (p-value: < .00001) => reject H0 at the 5% level Let’s test H0, using the F-test (F2,574,.05 = 3.01): RSS(H0) = 0.4211 RSS(H1) = 0.3930 J = 2; (N – K) = 574 F-test = {[0.4211 – 0.3930]/2}/{0.3930/579} = 19.185 > 3.015 ⇒ reject at the 5% level. Regression Notes: ⋄ If we look at the R2, the variability of monthly (IJAP – IUS) explain very little, 0.01%, of the variability of monthly ef,t. ⋄ We can modify the regression to incorporate the Balassa-Samuelson effect, by incorporating GDP differentials. Say, 𝑒 , (JPY/USD) = α + β (IJAP – IUS) t + δ (GDP_capJAP – GDP_capUS) t + εt. ¶ Relative PPP tends to be rejected in the short-run (like in the example above). In the long-run, there is a debate about its validity. As mentioned above there is some evidence of (slow) mean reversion. In the long-run, inflation differential matter: Currencies with high inflation rate differentials tend to depreciate. IFM-LN.86 • PPP: Rt and St Research shows that Rt is much more variable when St is allowed to float. Rt‘s variability tends to be highly correlated with St‘s variability. This finding comes from Mussa (1986). Figure 8.9: Nominal and Real Exchange Rate for the USD/GBP (1970-2017) USD/GBP: Real and Nominal FX Rate (1971-2017) 3 S(t) 2.5 R(t) R(t), S(t) 2 1.5 1 0.5 1/1/2015 11/1/2016 3/1/2013 5/1/2011 7/1/2009 9/1/2007 11/1/2005 1/1/2004 3/1/2002 5/1/2000 7/1/1998 9/1/1996 11/1/1994 1/1/1993 3/1/1991 5/1/1989 7/1/1987 9/1/1985 1/1/1982 11/1/1983 3/1/1980 5/1/1978 7/1/1976 9/1/1974 11/1/1972 1/1/1971 0 In the Figure 8.9 above, we see the finding of Mussa (1986) for the USD/GBP exchange rate: After 1973, when floating exchange rates were adopted, Rt moves like St. As a check to the visual evidence: the monthly volatility of changes in Rt is 2.94% and the monthly volatility of changes in St is 2.91%, with a correlation coefficient of .979. Almost the same! In the graph above, we see the finding of Mussa (1986) for the USD/GBP exchange rate: After 1973, when floating exchange rates were adopted, Rt moves like St. As a check to the visual evidence: the monthly volatility of changes in Rt is 2.94% and the monthly volatility of changes in St is 2.91%, with a correlation coefficient of .979. Almost the same! Recall that economists tend to think that nominal variables can affect nominal variables, but not real variables. The above graph shows that St moves like Rt, which we think is affected by real factors. We can easily incorporate this idea into St (using the definition of Rt, we solve St): St = Rt Pd / Pf. Now, we have St affected by real factors (through Rt) and nominal factors (through Pd / Pf). • PPP: Sticky Prices From the above USD/GBP graph, which is representative of the usual behavior of Rt and St, we infer that price levels play a minor role in explaining the movements of Rt (&St). Prices are sticky/rigid –i.e., IFM-LN.87 they take a while to adjust to shocks/disequilibria. A potential justification for the implied price rigidity: NT goods. Price levels include traded and NT goods; traded-goods should obey the LOOP. But, Engel (1999) and others report that prices are sticky also for traded-goods (measured by disaggregated producer price indexes). A strange result for many of us that observe gas prices change frequently! Possible explanations: (a) Contracts Prices cannot be continuously adjusted due to contracts. In a stable economy, with low inflation, contracts may be longer. We find that economies with high inflation (contracts with very short duration) PPP deviations are not very persistent. (b) Mark-up adjustments There is a tendency of manufacturers and retailers to moderate any increase in their prices in order to preserve their market share. For example, changes in St are only partially transmitted or pass-through to import/export prices. The average ERPT (exchange rate pass-through) is around 50% over one quarter and 64% over the long run for OECD countries (for the U.S., 25% in the short-run and 40% over the long run). The average ERPT seems to be declining since the 1990s. Income matters: ERPT tends to be bigger in low income countries (2-3 times bigger) than in high countries. (c) Repricing costs (menu costs) It is expensive to adjust continuously prices; in a restaurant, the repricing cost is re-doing the menu. For example, Goldberg and Hallerstein (2007) estimate that the cost of repricing in the imported beer market is 0.4% of firm revenue for manufacturers and 0.1% of firm revenue for retailers. (d) Aggregation Q: Is price rigidity a result of aggregation –i.e., the use of price index? Empirical work using detailed micro level data –say, same good (exact UPC barcode!) in Canadian and U.S. grocery stores– show that on average product-level Rt –i.e., constructed using the same traded goods– move closely with St. But, individual micro level prices show a lot of idiosyncratic movements, mainly unrelated to St: Only 10% of the deviations from PPP are accounted by St. • PPP: Puzzle The fact that no single model of exchange rate determination can accommodate both the high persistent of PPP deviations and the high correlation between Rt and St has been called the “PPP puzzle.” See Rogoff (1996). • PPP: Summary of Empirical Evidence ⋄ Rt and St are highly correlated, domestic prices (even for traded-goods) tend to be sticky. ⋄ In the short run, PPP is a very poor model to explain short-term exchange rate movements. ⋄ PPP deviation are very persistent. It takes a long time (years!) to disappear. ⋄ In the long run, there is some evidence of mean reversion, though very slow, for Rt. That is, StPPP has long-run information: Currencies that consistently have high inflation rate differentials –i.e., (Id-If) IFM-LN.88 positive- tend to depreciate. The long-run interpretation for PPP is the one that economist like and use. PPP is seen as a benchmark, a figure towards which the current exchange rate should move. • Calculating StPPP (Long-Run FX Rate) We want to calculate 𝐒𝐭𝐏𝐏𝐏 = Pd,t / Pf,t over time. Steps: (i) Divide and multiply 𝐒𝐭𝐏𝐏𝐏 by S = Pd,o/Pf,o (where t=0 is our starting point or base year). (ii) After some algebra, 𝐒𝐭𝐏𝐏𝐏 = S * [Pd,t / Pd,o] * [Pf,o/Pf,t] By assuming S = So, we can plot 𝐒𝐭𝐏𝐏𝐏 over time. (Note: S = So assumes that at time 0, the economy was in equilibrium. This may not be true. That is, be careful when selecting a base year.) Figure 8.10 plots 𝐒𝐭𝐏𝐏𝐏 and St for the MXN/USD exchange rate during the 1987-2019 period. During the sample, Mexican inflation rates were consistently higher than U.S. inflation rates –actually, 322% higher during the 1987-2015 sample period). Relative PPP predicts a consistent appreciation of the USD against the MXM. Figure 8.10: StPPP and St for the MXN/USD (1987-2019) Actual vs Long Run PPP: MXN/USD 30 25 20 15 10 5 PPP MX/US MXN/USD 4/1/2018 3/1/2017 2/1/2016 1/1/2015 12/1/2013 11/1/2012 9/1/2010 10/1/2011 8/1/2009 7/1/2008 6/1/2007 5/1/2006 4/1/2005 3/1/2004 2/1/2003 1/1/2002 12/1/2000 11/1/1999 10/1/1998 9/1/1997 8/1/1996 7/1/1995 6/1/1994 5/1/1993 4/1/1992 3/1/1991 2/1/1990 1/1/1989 12/1/1987 0 In the short-run, Relative PPP is missing the target, St. But, in the long-run, PPP gets the trend right. (As predicted by PPP, the high Mexican inflation rates differentials against the U.S depreciate the MXN against the USD.) Similar behavior is observed for the JPY/USD, as shown in Figure 8.11. The inflation rates in the U.S. have been consistently higher than in Japan (57% higher during the 1971-2015 period), then, according to Relative PPP, the USD should depreciate against the JPY. PPP gets the long term trend right, but IFM-LN.89 misses St in the short-run. Figure 8.11: StPPP and St for the JPY/USD (1971-2019) Actual vs Long Run PPP: JPY/USD 500 450 400 350 300 250 200 150 100 4/1/2018 7/1/2016 10/1/20… 1/1/2013 4/1/2011 10/1/20… 7/1/2009 1/1/2006 4/1/2004 7/1/2002 10/1/20… 1/1/1999 4/1/1997 10/1/19… 7/1/1995 1/1/1992 4/1/1990 10/1/19… JPY/USD 7/1/1988 1/1/1985 4/1/1983 7/1/1981 10/1/19… 1/1/1978 4/1/1976 PPP JPY/USD 7/1/1974 1/1/1971 0 10/1/19… 50 Note that in both graphs, 𝐒𝐭𝐏𝐏𝐏 is smoother than St. • PPP: Summary of Applications ⋄ Equilibrium (“long-run”) exchange rates. A CB can use 𝐒𝐭𝐏𝐏𝐏 to determine intervention bands. ⋄ Explanation of St movements (“currencies with high inflation rate differentials tend to depreciate”). ⋄ Indicator of competitiveness or under/over-valuation: Rt > 1  FC is overvalued (& Foreign prices are not competitive). ⋄ International GDP comparisons: Instead of using St, 𝐒𝐭𝐏𝐏𝐏 is used. (An additional advantage: since 𝐒𝐭𝐏𝐏𝐏 is smoother, GDP comparisons will not be subjected to big swings.) For example, per capita GDP (World Bank figures, in 2017) are reported below in Table 8.1: Table 8.1: GDP per capita in Nominal and PPP Prices (in USD) - 2017 Country Luxembourg USA Japan Costa Rica Brazil Lebanon China GDP per capita (in USD) - 2017 Nominal PPP 104,103 103,745 59,532 59,532 38,428 43,279 11,631 17,044 9,821 15,483 8,524 14,676 8,827 16,807 IFM-LN.90 India Ethiopia 1,937 767 7,056 1,899 Example: Nominal vs PPP - Calculations for China Data: Nominal GDP per capita: CNY 59,670.52 St = 0.14792 USD/CNY; 𝐒𝐭𝐏𝐏𝐏 = 0.2817 USD/CNY  “goods in the U.S. are 51.58% more expensive than in China.” - Nominal GDP_cap (USD) = CNY 59,670.52 * 0.1479 USD/CNY = USD 8,827 - PPP GDP_cap (USD) = CNY 59,670.52 * 0.2817 USD/CNY = USD 16,807. ¶ Chapter 8 - Appendix – Taylor Series Definition: Taylor Series Suppose f is an infinitely often differentiable function on a set D and c ∈D. Then, the series Tf(x, c) = Σn [ f(n)(c)/n!] (x – c)n is called the (formal) Taylor series of f centered at, or around, c. Note: If c=0, the series is also called MacLaurin Series. Taylor Series Theorem Suppose f ∈ Cn+1([a, b]) -i.e., f is (n+1)-times continuously differentiable on [a, b]. Then, for c ∈ [a,b] we have: f (n)  x 0  f c  f / c  f // c  0 1 2 x  c   x  c   x  c      x  x 0 n  R f ( x )  T ( x, c )  R  0! 1! 2! n! 1 n where Rn 1 ( x)   f (n1 )  p x  p  dp n! In particular, the Tf(x, c) for an infinitely often differentiable function f converges to f iff the remainder R(n+1)(x) →converges to 0 as n → ∞. Example: 1st-order Taylor series expansion, around c=1, of f(x)=5+2x + x2 f(x) = 5+2x + x2 f’ (x0=1) = 8 f’ (x) = 2 + 2x f ’(x0=1) = 4 f’’ (x) = 2 f ‘’(x0=1) = 2 f’’’(x) = 0 f ’’’(x0=1) = 0  1st-order Taylor’s series formula (n=1): f(x) T(x; c) = 8 + 4 (x-1) = 4 + 4x • Now, for the Relative PPP approximation, we use a Taylor series expansion, Tf(x, c), for a bivariate series: f c , d    f c , d  x  c 0  y  d 0  f x c , d  x  c 1  y  y  d 1  T ( x, y, c, d )  0! 1! 1! 1 2 2  [ f xx c , d  x  c   f xy c , d  x  c  y  d   f yy c , d  y  d  ]   2! IFM-LN.91