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Lecture 7: Deviations from PPP and
LOOP explored and explained
Birmingham MSc Open Economy Macro
Autumn 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.
P
BigMac
e BigMac S P
BigMac
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
P T SP T
P N SP N
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 nontraded goods.
The aggregate price index in theory
P 
P T , P N
We use a theoretical price index to
aggregate across traded and non
traded goods.

P T , P N 
P T , P N 
It’s homogenous of degree 1.
P 
P T P N 
/2
Two examples here of index functional
forms that would obey this
requirement to be homogenous of
degree 1.
P 
P T 
P N 1
You are asked to confirm this in an
exercise.



P
T,P N
e S PP S P
e S.
T,P N


1,P 
.P 
N/P T
T

1,P N/P T 
.P T
Substitute 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, the foreign traded goods
price
Do same to denominator but using
p_T, the home traded goods price.
PPP failure due to ratio of traded/non
traded prices at home compared to
abroad
P T SP T
e S.


1,P 
.P 
N/P T
T

1,P N/P T 
.P T
e 1  P N /P T P N /P T
Substitute in our ass. that LOOP holds for
traded goods.
e


1,P 
N/P T

1,P N/P T 
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 T L T
Production functions in the 2 sectors.
QN N L N
 P i Qi wL i
Profits = revenues less costs. Note wage
common across sectors.
0  P i Qi wL i
Free entry means zero profits. Substitute
in production function to this ‘zero profit
condition’.
 P i i L i wL i
 P i i w
Deriving the B-S result
P T T  w P N N

P
 T  N [2]
T
PN
Equate the ‘wage equation’ across sectors
since the wage is common across sectors.
Derive expression for ratio of traded to
non-traded goods prices.
P
T
P
N
N
 
T
e
1,
1,
This relation holds for the home country,
and, also, for the foreign economy too.

T

N
T
N


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 u
C
1
C C 
T C NT
P minCT,CN P T C T P N C N 
1
s.t. 1 C 
T C NT
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’.
Note this is NOT the alpha that
appears in production fn studied
earlier.
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
Use aggregation technology
constraint to substitute out for C_N.
This leaves just 1 choice variable left
to minimise wrt.

1
C N C T

1
P P T C T P N C T
CT 
1
PN 
P T 1
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.

1
C N C T
CN 
P T 1 
PN 
We can use expression for C_T derived on last
slide to substitute in here….
…To get C_N in terms of things other than C_T.
And once that’s done…
We go back to the expression for the
unconstrained objective function [the
expression for P we are minimising]…
and substitute in both C_T and C_N….
Substituting in C_T and C_N
expressions into the unconstrained
objective fn…
Substitute into this
unconstrained ‘objective fn’
expressions for C_T, C_N
P P T C T P N C N
P P T
1
PN 
P N
P T 1
P 
P T 
P N 1A
A 
1 1
P T 1 
PN 
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

1
One of the equations we worked out
earlier….
C N C T
PN
C
P TC T N
PN
C
P TC T N
PN
C
P TC T N

1
 PPCN C T
* Both sides by P_N/(P_T*C_T)
T T

1
PN
1
C
T
PT

 1

Collect terms on the RHS in C_T to get
this.
Substitute for C_T on the RHS only, to
get this equation…..
Finding the alpha as f(expenditure
weights)
  P TCPTTCPTNCN

1
P 
P T 
PN  A
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.
e
1,
P T,

T

N
T
N


And recall whole point is to use this price
index 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.