Lecture 7: Deviations from PPP and LOOP explored and explained

Birmingham MSc Open Economy MacroAutumn 2015

Tony Yates

Recall lecture on DMF model

• DMF model had short run fluctuations in the real exchange rate.

• And these fluctuations caused by combination of sticky prices, money shocks, and overshooting.

• Now we dive deeper into purchasing power parity, law of one price, and long run departures from these concepts.

• More microfounded economics.• And traced back to Balassa-Samuelson.

Law of one price

P SP

P SP

LOOP holds if the price at home is the same as the nominal exchange rate times the foreign price level.

Goods are more expensive if home price level exceeds RHS, cheaper otherwise.

LOOP for Big Macs.

eBigMac SPBigMac

PBigMac

Law of one price fails miserably, on the face of it, for Big Macs.

Suggests failure of arbitrage to work across countries and borders?

Or does it?

A Big Mac is not just a hamburger. It’s food plus distribution services.

Short and long run RER movements

• Contrast between this lecture’s perspective on RER movement [ppp failure] and the lecture on the DMF model.

• Today’s lecture provides reasons to explain why PPP will fail over the very long run.

• DMF lecture provided reasons to expect damaging but short run RER movements.

LOOP and PPP

• Purchasing power parity= generalisation of the notion of the law of one price, for a basket of goods.

• Think, eg, of a ‘McDonalds meal index’, including chips and soft drink, as a first step.

• Typically/ideally, we study a representative basket of all goods, using expenditure surveys.

• Measurement issues abound, not least since there are differences in consumption patterns across countries and within countries too.

• Survey data not always available, or of the right quality.

Balassa-Samuelson

• There are traded and non-traded goods.• Productivity improvements in the traded

goods sector bid up wages in both sectors.• Labour not mobile across countries [or not as

mobile, anyway]

Plan for the analysis

• First use price index theory and the definition of the real exchange rate / PPP/ LOOP to diagnose proximate causes of failure of those laws.

• Then study a static, competitive model of the economy to reproduce the Balassa-Samuelson result.

Diagnosing proximate causes of LOOP/PPP failure

PT SPT

PN SPN

Two groups of goods, traded, ‘T’ and non-traded ‘NT’.

Think of retail distribution services as ‘non-traded’, for example.

LOOP holds for traded, say.

And suppose it does not hold for non-traded goods.

The aggregate price index in theory

P PT,PN

PT,PN PT, PN

P PT PN/2

P PT PN 1

We use a theoretical price index to aggregate across traded and non traded goods.

It’s homogenous of degree 1.

Two examples here of index functional forms that would obey this requirement to be homogenous of degree 1.

You are asked to confirm this in an exercise.

e S P

P S PT ,PN

PT,PN

Subsitute in the defn of the home and foreign price indices into the defn of the real exchange rate.

Divide and multiply numerator by p_star_T,

Do same to denominator but using p_T.

e S. 1,PN /PT .PT

PT,PN/PT .PT

PT SPT

e 1,PN /PT

1,PN/PT e S. 1,PN

/PT .PT

PT,PN/PT .PT

Substitute in our ass. that LOOP holds for traded goods.

e 1 PN /PT

PN/PT

PPP failure diagnosed as a difference in the ratio of non-traded to traded goods prices across countries.

Now we look at the BS model of why this ratio can differ across countries.

Static model of competitive production

QT TLTQN NLN

P iQi wL i

0 P iQi wL i P i iL i wL i P i i w

Production functions in the 2 sectors.

Profits = revenues less costs. Note wage common across sectors.

Free entry means zero profits. Substitute in production function to this ‘zero profit condition’.

Deriving the B-S result

PT T w PN N

PTPN

N T [2]

PT

PN N

T

e 1,

T

N

PT, TN

Equate the ‘wage equation’ across sectors since the wage is common across sectors.

Derive expression for ratio of traded to non-traded goods prices.

This relation holds for the home country, and, also, for the foreign economy too.

Substitute into our earlier equation for the RER, and we get the BS result: PPP failures, due to differences in relative sectoral productivity.

Microfoundations of price indices

• So far we said nothing about where the price index function we used should come from.

• This index should have a form that arises from a sound and empirically valid economic theory, and consistent with the theory we also use to diagnose the LOOP failure.

• So let’s go back and explore this briefly.

Microfounding our price index

U uC

C CTCNT

1

P minCT,CNPTCT PNCN

s.t. 1 CTCNT

1

Assume consumers get utility from an aggregate consumption good C,

Which is built using a Cobb-Douglas function from non traded and traded goods

Price index is a weighting function that solves a minimisation problem.

Min amount of expenditure to build 1 unit of the aggregate, subject to the aggregation ‘technology’.

Plan for algebra-analysis

• Remember we are trying to figure out a consistent microeconomic-theory-justified index function to use in our BS analysis of price indices in different countries.

• We will do static constrained optimisation now.

• Instead of Lagrangian, use the constraint to substitute out for one of the variables….

Optimising wrt C_T

CN CT 1

P PTCT PNCT 1

CT PNPT

1

1

Use aggregation technology constraint to substitute out for C_N

This is the new objective function that we will minimise, now with just one ‘choice variable’.

Now differentiate wrt C_T, and set equal to zero.

CN CT 1

CN PTPN

1

Now we have an expression for C_T, we can use it to find C_N, substituting in here…

To get C_N in terms of things other than C_T.

And once we do that, we can go back to the expression for the unconstrained objective function and substitute in both C_T and C_N….

Substituting in C_T and C_N expressions into the unconstrained objetive fn…

P PTCT PNCN

P PT PNPT

1

1 PN PT

PN1

P PT PN1 A

A 1 1

Substitute into this unconstrained ‘objective fn’ expressions for C_T, C_N

Yields this expression for the price index.

We can simplify to get this, with the constant A defined in terms of the utility parameter alpha.

Remarks about apples and oranges

• Often encounter the informal comment that you can’t compare apples and oranges.

• Well, with utility theory and optimisation, you can!• Only problem is, how do we find alpha?• Neuro-economics attempts to do things like this by

experiments and brain monitoring.• But we can do it by following through with the same

theory.• Obviously, the answer we get is only as good as the

theory we use.

Measurement and theory

• Before we go on….• Note how we are required to use theory even to create

the data…• …that we will use in empirical work….• …to test other theories.• …a theory that we ought to hope to test with other

data….• …constructed how…• Intimate two way relationship between theory and

data.

Finding an empirical, theory consistent measure of the alpha

CN CT 1

PNPTCT

CN PNPTCT

CT 1

PNPTCT

CN PNPTCT

11

One of the equations we worked out earlier….

* Both sides by P_N/(P_T*C_T)

Collect terms on the RHS in C_T to get this.

Substitute for C_T on the RHS only, to get this equation…..

PNPTCT

CN 1

Finding the alpha as f(expenditure weights)

PTCTPTCT PNCN

P PT PN 1 A

e 1,

T

N

PT, TN

We find that alpha=share of spending (consumption*price) of one good in total spending.

So, no need to look inside brain. Just get data on spending patterns across goods.

Where we use alpha in this optimal price index formula.

And the price indexes are used to model and diagnose the cause of RER movements [PPP failure] in this expression.

NB the index is the phi function here.

Stock take on what we have done

• We wanted a model to help diagnose deviations from PPP, or, equivalently, fluctuations in the real exchange rate.

• To do this we compared price indices that combine goods from 2 different sectors in each country.

• We then realised we needed a theory-consistent way of doing this.

• Static consumer optimisation gave us the answer, in terms of the utility parameter alpha.

Recap on DMF

• Single good model. Not microfounded. Small country assumption.

• One period sticky prices combined with exogenous money stock movements causes nominal and therefore real exchange rates to move.

Recap on today

• Microfounded model. 2 large countries. 2 goods in each country. Arbitrage in traded goods prices. No labour movement permitted.

• Theory consistent construction of aggregate price index.

• RER moves due to productivity differences in the traded goods sectors.

Failure of ‘absolute PPP’

Source: used by SGU, W textbook, p263

Success of ‘relative PPP’

Price indices should move in tandem [foreign price expressed in home currency] to make sure real exchange rate does not continually grow or fall.

Seems to hold for US/UK over 2 centuries.

Source: taken from SGUW textbook, p265

Borders, distance and LOOP failures

• Famous paper ‘How wide is the border?’• Study of US/Canada prices of different goods

in different cities.• Price differences for the same good=f(distance

between cities)• Yet price difference much greater, for same

distance if cities have the US-Canada border between them.

Real exchange rate persistence

• Regress RER on a lag of itself.• High coefficient means high ‘persistence’:– Shock hits, takes a long time to die out, since

tomorrow’s RER depends heavily on today.• Many interesting themes in this work.• Let’s mention briefly 3 of them.– Chari Kehoe, McGrattan– Benigno– Giraitis, Kapetanios and Yates [yes, that’s me].

Chari, Kehoe, McGrattan

• Part of RER persistence explanation rested on price stickiness.

• Recall DMF model with 1 period sticky prices.• But CKM simulate modern, microfounded

sticky price, RE model, and conclude that RER in that model much less persistent than data.

Benigno

• Takes CKM as his starting point.• Simulates a modern, microfounded, sticky prie

RE model with different central bank interest rules.

• Notices that the higher the coefficient on lagged interest rates in those rules, the more persistent is the RER.

• So explanation is sticky prices+interest rate inertia in policy rules.

Giraitis, Kapetanios, Yates

• Developing techniques to detect structural change in time series processes.

• Investigate whether RER persistence changes over time in different countries.

• Finding: it changes a lot.• Conclusion: suggests contribution of policy

rules, since these are things mostly likely to be changing.