Price levels and Exchange Rate in the Long Run

1. Price levels and Exchange Rate in the Long Run WONG Ka Fu 7th February 2001

2. Basic math review X=A/B  ln X = ln A - ln B Y=Y(x)  d ln Y / dx = d lnY / dY  dY / dx = (1/Y) (dY/dx)

3. Basic math review P=P(t)  d ln P / dt = d lnP / dP  dP / dt =(1/P) (dP/dt) Take the change of t (dt) from s to s+1.  d ln P / dt = [1/P(s)] [P(s+1) - P(s) /1] = [P(s+1) - P(s) ] / P(s) = percentage change in P at time s.

4. Law of one price • In a competitive markets • free of 1. transportation costs and 2. official barriers to trade (such as tariffs), • identical goods sold in different countries must sell for the same price when their prices are expressed • in terms of the same currency.

5. Law of one price implies exchange rate • For any good i sold in both home and foreign countries PHi = (EH/F)  (PFi) • Hence, the implied exchange rate is EH/F = PHi / PFi

6. Absolute Purchasing Power Parity (Absolute PPP) • For a given reference commodity basket sold in both the home and the foreign countries PH= (EH/F)  (PF) • Hence, the implied exchange rate is EH/F = PH/ PF • The implied exchange rate from the Economist’s Big Mac index

7. Relative PPP • Prices and exchange rates change such that the ratio of each currency’s domestic and foreign purchasing powers are preserved. • Hence, (EH/F,t - EH/F,t-1 )/ EH/F,t-1 = H,t- F,t where t = (Pt - Pt-1 ) / Pt-1

8. Relative PPP • If absolute PPP does not hold because of frictions and other factors and we have EH/F =  PH/ PF where  is a constant that measures the difference from absolute PPP. EH/F(t) =  PH(t) / PF(t) ln EH/F (t)= ln  + ln PH(t) - ln PF(t) Taking derivative with respect to t: dln EH/F (t)/dt = dln  /dt + dln PH(t)/dt - dln PF(t)/dt

9. Relative PPP • Hence, (EH/F,t - EH/F,t-1 )/ EH/F,t-1 = H,t- F,t where t = (Pt - Pt-1 ) / Pt-1 percentage change in EH/F,t = percentage change in PH,t- percentage change in PF,t

10. Long-run exchange rate based on absolute PPP • EH/F = PH/ PF • PH = MHs / L ( RH, YH ) • PF = MFs / L ( RF, YF ) • Monetary policy = money supply

11. Effect of an increase in home money supply on LR EH/F MHs PH because PH = MHs / L ( RH, YH ) because EH/F = PH/ PF EH/F

12. Effect of an increase in foreign money supply on LR EH/F MFs PF because PF = MFs / L ( RF, YF ) because EH/F = PH/ PF EH/F

13. Effect of an increase in home interest rate on LR EH/F RH LH because L ( RH, YH ) PH because PH = MHs / L ( RH, YH ) EH/F because EH/F = PH/ PF

14. Interest rate can change due to reasons other than monetary policy Factors that are not already explicit but implicit in the L(R,Y) function For example: technology advancement may improve the profitability of investment and hence the interest rate willing to pay to borrow money to invest.

15. Effect of an increase in foreign interest rate on LR EH/F RF LF because L ( RF, YF ) PF because PF = MFs / L ( RF, YF ) EH/F because EH/F = PH/ PF

16. Effect of an increase in home output on LR EH/F YH LH because L ( RH, YH ) PH because PH = MHs / L ( RH, YH ) EH/F because EH/F = PH/ PF

17. Effect of an increase in foreign output on LR EH/F YF LF because L ( RF, YF ) PF because PF = MFs / L ( RF, YF ) EH/F because EH/F = PH/ PF

18. Long-run exchange rate based on absolute PPP • EH/F = PH/ PF • PH = MHs / L ( RH, YH ) • PF = MFs / L ( RF, YF ) • EH/F = (MHs / MFs )  [L ( RF, YF ) /L ( RH, YH )]

19. How is long-run exchange rate determined? • Anything that raises (lowers) LH lowers (raises) EH/F • Anything that lowers (raises) LF lowers (raises) EH/F • An increase (A decrease) in MHs raises (lowers) EH/F • An increase (A decrease) in MFs lowers (raises) EH/F

20. Growth rate of money supply: a mathematical derivation • Money supply level : MHs (t) • Growth rate : (MHs (t+1) - MHs (t) ) / MHs (t) • Define y(t) = ln( MHs (t) ) • dy(t)/d(t) = d ln( MHs (t) )/dt = dy(t)/d MHs (t)  d MHs (t) /dt = 1/ MHs (t)  d MHs (t) /dt • dt = t+1 - t = 1

21. Fisher effect • Uncovered interest parity: RH,t = [EH/F,t+1e - EH/F,t] / EH/F,t+ RF,t • let t+1 e = (Pt+1e - Pt ) / Pt and • t+1 = (Pt+1 - Pt ) / Pt • Relative PPP : (EH/F,t+1 - EH/F,t )/ EH/F,t = H,t+1- F,t+1 • (EH/F,t+1e - EH/F,t )/ EH/F,t = H,t+1e - F,t+1e • RH,t - RF,t= H,t+1e - F,t+1e

22. If MHS is growing at a rate of • PH grows at a rate of because PH = MHs / L ( RH, YH ) • I.e., expect H,t+1=  • or , H,t+1 e =  • Hence, RH,t - RF,t= H,t+1e - F,t+1e=  if F,t+1e = 0

23. If MHS is growing at a rate of Log(MHS) Slope =  t0

24. If MHS is growing at a rate of RH RH1 t0

25. If MHS is growing at a rate of Log (PH) Slope =  t0

26. If MHS is growing at a rate of Log(EH/F) Slope =  t0

27. If MHS is growing at a rate of( + ) • PH grows at a rate of( + ) because PH = MHs / L ( RH, YH ) • I.e., expect H,t+1= ( + ); • or , H,t+1 e = ( + ) • Hence, RH,t - RF,t= H,t+1e - F,t+1e= ( + ) if F,t+1e = 0

28. If the rate of MHS growth increases from  to ( + ) • Suppose RF,tfixed and F,t+1e = 0 because a stable monetary policy, for example. • RH,t increases by  because H,t+1e is expected to increase by . • Note that, however, MHS does not change at time t0 -- only the future growth rate • Hence, PH has to jump from PH1= MHs / L ( RH1, YH ) to PH2= MHs / L ( RH2, YH )

29. Effect of an increase in the growth rate of MHS Log(MHS) Slope =  +  Slope =  t0

30. Effect of an increase in the growth rate of MHS RH RH2 = RH1 +  RH1 t0

31. Effect of an increase in the growth rate of MHS Log (PH) Slope =  +  Slope =  t0

32. Effect of an increase in the growth rate of MHS Log(EH/F) Slope =  +  Slope =  t0

33. The lesson learnt is much more general • The story was: • A change in money supply growth leads to change in expected inflation. • A change in expected inflation leads to a jump in interest rate. (Through Fisher) • A jump in interest rate leads to a jump in exchange rate. • More generally, • Any thing that cause a change in expected inflation will lead to a jump in interest rate. • A jump in interest rate leads to a jump in exchange rate.

34. The lesson learnt is much more general • What will cause a change in expected inflation? • The release of economic indicators (say, unemployment, GDP, interest rate, confidence index, etc.) may change our expectation of inflation. • Any release of indicators that cause a change in expected inflation will lead to a jump in exchange rate.

35. Empirical test PH= (EH/F)  (PF)  ln PH= ln EH/F + ln PF • Regression: ln PH,t= 0 + 1ln EH/F,t + 2ln PF,t + t or ln PH,t= 0 + 1ln EH/F,t + 2ln PF,t + 3Xt + t where Xt serves as a control variable.

36. Hypotheses: • Absolute PPP implies • 0 = 0, 1 = 1, 2 = 1 • Relative PPP implies • 0 = ?, 1 = 1, 2 = 1

37. Empirical evidence on Absolute PPP • Way off the mark: The prices of identical commodity baskets, when converted to a single currency, differ substantially across countries.

38. Empirical evidence on Relative PPP • Usually performs poorly although it sometimes is a reasonable approximation to the data. • More reliable in the 1960s as a guide to the relationship among inflation and national price levels but less so since 1970s.

39. Why PPP fails? • Transport costs and restriction on trade • Monopolistic or oligopolistic practices in goods markets • Measure sof inflation differ across countries.

40. Exchange rate pass-through (ERPT) • The percentage change in local currency import prices resulting from a one percent change in the exchange rate between the exporting and importing countries. • Full or complete ERPT if the following two conditions are met: • constant markups of price over cost (e.g., when industries are perfectly competitive, and markups are constant at zero) and • constant marginal cost.

41. Exchange rate pass-through (ERPT) • Empirical: ln( pt ) = a + b Xt +c ln( Et ) + d Zt + et pt : local currency import price Xt : a measure of exporter’s cost Zt : import demand shifters Et : the exchange rate (importer’s currency per unit of exporter’s currency)

42. The interpretation of c C = d [ ln P ] / d [ ln E ] = { d [ ln P ] / dt } / { d [ ln E ] / dt } = % change in P / % change in E • ERPT is “full” or “complete” if c=1 and is “incomplete” if c<1.

43. Exchange rate pass-through (ERPT) • Empirical: ln( pt ) = a + b Xt +c ln( Et ) + d Zt + et • Estimate of c is around 60%. This implies that 40% of the exchange rate change was offset by changes in the markup.

44. Pricing to Market • Consider a monopolistic firm that sells its product in n countries (I.e., n segmented markets) • Its objective is to maximize profit (p1,…,pn) = pi qi(Eipi,vi) - C(qi(Eipi,vi),w)

45. Pricing to Market (p1,…,pn) = pi qi(Eipi,vi) - C(qi(Eipi,vi),w) pi is the price charged in i-th market, in the firm’s domestic currency qi(Eipi,vi) is the demand in i-th market, a function of Eipi, price in i-th foreign currency and vi, some demand shifters (say, income). Thus,  pi qi(Eipi,vi) is the total revenue in domestic currency. C(qi(Eipi,vi),w) is the total cost of producing qi(Eipi,vi) and w is the factors that may shift production cost.

46. Pricing to Market (p1,…,pn) = pi qi(Eipi,vi) - C(qi(Eipi,vi),w) Note that without exchange rate, Ei, the problem is the same as the standard problem of a monopoly maximizing profits in n segmented markets. We should all know its solution from basic microeconomics.

47. Pricing to Market • The optimal export price is the product of the common marginal cost and a destination-specific markup: • pi= Cq [-i /(-i +1)] • where Cq is the marginal cost, • i is the absolute value of the elasticity of demand in the foreign market with respect to changes in price, pi.

48. Pricing to Market • Thus, prices are different across markets and are related to a destination-specific markup which is a function of demand elasticity. • If pricing to market behavior dominates, PPP is unlikely to hold. Further readings: • Goldberg, Pinelopi Koujianou and Michael M. Knetter (1997): “Goods Prices and Exchange Rates: What Have we Learned?” Journal of Economic Literature, Vol. XXXV (September, 1997), pp. 1243-1272.

49. Empirical test of Fisher’s Equation • RH,t - RF,t= H,t+1e - F,t+1e • H,t+1e - F,t+1e = RH,t - RF,t • (H,t+1- F,t+1) e = RH,t - RF,t • (H,t+1- F,t+1) = RH,t - RF,t + t where (H,t+1- F,t+1) = (H,t+1- F,t+1) e + t • Run the regression • (H,t+1- F,t+1) = +  (RH,t - RF,t ) + t • should get   0 and  1

50. Evidence • Cumby and Obstfeld (1984) and Mishkin (1984) both rejected the hypothesis.