If there are two countries whose exchange rates are officially locked
(e.g., the U.S. dollar and the Saudi Arabia riyal), does inflation in
one country automatically occur in the second country at the same
exact rate?

Taking the SAR example, everybody knows that inflation in the U.S. has
been about 2-3% per year for the last few years.  But occasionally I
read that inflation in Saudi Arabia is "essentially zero," even though
that country's currency has been locked to the dollar at 1USD=3.75SAR
since 1986.

Inflation is something that's kinda hard to wrap my mind around. 
Please help clear up the clouds!

Like many things in finance/economics, it's important to focus on the
difference between REAL and NOMINAL. You can think of "real" as being
a world w/out money... everything has to do with goods and services
(think of money as just another good... the shoes you produce at work
are magically turned into paper bills, which are then turned into food
you eat). The shoes to food relationship is what matters.

The nominal world treats money as actual money, so things like
inflation creep into the picture.

There is such a thing as a nominal exchange rate and a real exchange
rate. A nominal anything can be a 'snapshot' in time... you don't need
any data besides the present to understand the numbers. This is the
case with published exchange rates; they're nominal.

A real exchange rate requires more than a 'snapshot', because you need
to know the history of how goods are exchanged in the two economies. A
real exchange rate takes inflation into account.

In the case of currency pegs, you have a nominal peg. So this doesn't
factor inflation into the picture.

***NOTE*** maintaining a currency peg often leads to
inflationary/deflationary effects (and a host of other problems)... so
the pegged currencies will theoretically converge to the same REAL
exchange rate (b/c their inflation rates will converge). When you hear
people talk about China "exporting its inflation to the world",
they're talking about China's nominal currency peg leading to
inflation in the US.

The previous comment was very interesting, but I think the question
asks something else.
Inflation rates are calculated by comparing the current price for a
"basket" of everyday, representative purchases and services with the
price for this "basket" at an earlier date.  The "basket" will include
basic foods, clothing, water and electricity, average auto costs,
etc., etc.  In some countries the comparison is between the price of
the basket 12 months before; in others, it can be the comparison with
the price of the basket in the previous month, multiplied by twelve to
give an annual rate of inflation.

Since these calculations are based on the "basket" for a single
country, the fact that the exchange rate between the two currencies is
locked is immaterial.

Thank you so much, fianceeco, for all the extensive (and unpaid)
commentary you have provided to my economics questions -- not just
with this one but with a number of others.  I study very carefully
every word you (and everybody else) write; your help is much
appreciated.

This question is particularly befuddling for me.

For anybody who might be interested, I managed to get an answer from
an economics instructor (at Yale).

THE QUESTION WAS:

"How does inflation work for two countries, when the currency of one
is pegged to that of the other (such as how Saudi Arabia's riyal) is
pegged to the US dollar?  Will the inflation rates be locked as well?

"I've searched a number of economics textbooks for a discussion of
this, but to no avail.

"If forced to come up with an answer, I would guess that the inflation
rates between the two countries will tend to correlate, but with some
variance.  They're not necessarily the same.

"My thinking is this:  it depends on what's chiefly causing the
inflation. Demand-pull forces, I reason, will tend towards parity of
the inflation rate, but this is not necessarily so with cost-push
forces.

"In the example I mention, if the price of oil shoots through the
roof, it would certainly shift the supply curve to the right and help
push up inflation.  But if inside Saudi Arabia the cost of petroleum
has not changed, that country will not be effected by this inflation.

"This is borne out by what numbers I can come up with (economics is
not my field): apparently Saudi has been under 1.0% for the last few
years, while the U.S. has been hovering at around 2-3%.

THE ANSWER WAS:

"I agree with all of what you said:

"This is ultimately an empirical question, which we have to look at case by case.

"It will ultimately depend on two things: how would the increasing
price level in the U.S. affect the exchange rate without the peg,
which will depend on the import/export structure of both countries,
for instance. And secondly, how does the monetary intervention in the
other country to defend the peg, feed into the price level abroad,
which is often and at least in the short run not one-to-one.

"But we can expect them to comove: With inflation in the U.S., foreign
commodities become more attractive, so people will want to sell dollar
and buy foreign currency, to finance imports. But that would lead to a
rise in the dollar price of foreign currency, so that the foreign
central bank will have to increase the supply of foreign currency,
which will lead - as a trend - to inflation abroad. Again, this is
what qualitatively will happen, but the magnitude of the effect
depends on a lot of things...."

Anyhow. Thanks to both of your, finaceeco and myoarin, for your help.

And thank you, gonossie-ga for pumping in some additional information..