Does trade affect Earth's rotation? Every country is trading with other countries around the world, some more than others.  I was wondering if there would be any change to the Earth's rotation because of the imbalance of trade between countries.  
 A: This question is impossible to answer comprehensively, but conceptually we might be able to offer some new insight.  Certain aspects of trade can affect rotation, but there are many human activities for which we have no expectation of impacting the Earth rotation.  I will try to enumerate some here.


*

*ships should not affect the Earth's rotation.  A container ship starts buoyant in the water at rest and ends at rest.  It uses its prop to push against the water to move.  While its mass was transferred, water was transferred in the opposite direction.  With no net mass transfer, the kinematics of Earth can't be affected by rule.

*trucks that move back and fourth won't have any net effect on rotation.  One could imagine that one truck traveling East is balanced by another traveling West.

*net movement of mass at constant latitude has no first-order effect on the speed of rotation.  Due to manufacturing economics, we may have increased the weight of one continent somewhat relative to others from shipping bulk materials.  But this doesn't affect the rotation of Earth if they're delivered to the same elevation.  The rotation can only be affected while the materials are actively moving, and only if it's not a ship.


So if you're not talking about these, what are you talking about?  The activities that can have an effect (at all) on rotation are limited to a extremely marginal slice of global trade.  Mainly, you can only speak about:


*

*Temporary effects, from the net velocity of vehicles moving (the ISS should count)

*Altitude effects, moving materials up or down

*Latitude effects, moving materials to and from the poles will change the moment of inertia and affect rotation similar to altitude effects


As you do the calculations, you should quickly find that mankind's hydroelectric dams will dwarf any contribution from material movement through trade.  Then open-pit mining will probably contribute more than trade too.  Melting of Antarctic ice caps almost certainly will.  Even ocean and atmosphere expansion from global warming will probably have a larger effect than simple trade.  Very few of the resources that go into trade participate in activities that can affect Earth's rotation.
A: If you consider only trade imbalance, I would think that the trade imbalance between penguins (or white bears) and elephants will have by far the greater effect per unit of mass,
much more, for example, than the trade imbalance between yetis and whales.
What I actually mean is that the most effective way to modify the moment of inertia of the planet so as to change its rotation speed is by moving mass between the equator and the poles. That gives you  6,371 km of distance change between the mass and the axis of rotation. This is more than 1000 times what you can achieve feasibly by moving mass between a mine and a high altitude city.
Furthermore, altitude change is effective only when not to far from the equator, where it is effectively fully outward. In Montreal (latitude about 45) the change in altitude is only 70% effective ($\sqrt{2}/2$) because it is angled at 45 degrees with respect to the axis. And it is totally useless at the pole.
But I am not yet asserting there is anything measurable that actually happens as a direct consequence of trade imbalance.
However, human activity may have a significant effect on Earth rotation through the climate warming that may cause a massive melting of the ice at the south pole (Archimedes saw to it that the north pole would not matter, but Greenland may do its 10% bit). The water from the melt with spread over the oceans so that a good part of it will end up much further away from the planet axis, slowing the rotation of the planet.
Exact computation is not easy as it must take into account a lot of factors, including the shape of the oceans coastline. I just did a quick search on the Internet to see whether other people had reach that conclusion. One site has a back of the envelope calculation that gives a slowdown of 8 seconds per day, which they think accurate up to a factor of 2. I would not vouch for it, as my own envelope gives 0.6 seconds per day slowdowm, but I think they made a factor 10 mistake in assessing the mass of the ice (unless I did). Actually. this is quite considerable.  But we will have other things to worry about, since that much fresh (unsalted) water in the ocean may have some drastic effects on currents, hence on climate. It may also considerably harm the ecological balance of marine life. And what else ...
The effect on Earth rotation should be slowly compensated for a part as the Antarctic continent will surge upward when the ice is gone (it is sort of heavy), thus taking mass from the rest of the planet.  Possibly we may have some earthquakes to help us forget about oceanic problems, but it should be very slow.
Geophysicists and climatologists would know all that better than I, including whether the warming will actually melt the polar ice. But observational evidence is not too reassuring.
Some figures, from a well sized envelope
The back of envelope computation. The continental ice sheet at the
southpole is between 25 and 30 millions cubic kilometers, to which we
can add 2.6 millions for the Greeland ice sheet. Considering the
0.92kg/dm3 density of ice, this gives a very conservative estimate of
$25\ 10^{18}$kg, which we consider as sitting at the pole, thus not
contributing to the moment of inertia. When this ice melts, it is
spread as a sperical shell on the earth surface (actually only the
oceans), producing a moment of inertia $2/3\ mR^2$ where m is the ice
mass and R the Earth radius. The Earth mass is $M=6\ 10^{24}$kg,
producing a moment of inertia $2/5\ MR^2$ for a full sphere. So the
actual contribution of the melted ice to the moment of inertia is $5/3$
of the mass ratio, as if the mass off the earth were increased by $5/3$
of the ice mass, i.e. approximately $40\ 10^{18}$kg. The ratio of the
two masses is $6\ 10^{24}/4\ 19^{19} = 1.5\ 10^5$.  Applying the same
ratio to the duration of a day, i.e., 86400 seconds, we get
approximately 0.6 seconds, which is, up to much approximation and
hopefully no goofing, the increase of day duration.
Now another possibibility to attempt to slow the planet, suggested by 
@AlanSE, is to keep
large tropical reservoirs so that there is more water near the equator,
and less on the surface of the oceans. However, for a given mass $m$ of
water, the effect on the moment of inertia is $mR^2$ instead of $2/3\
mR^2$ for a spherical shell. Hence the gain is only $1/3\ mR^2$, as
opposed to $2/3\ mR^2$ for ice melted at the pole. It is only half as
effective.  But what is much worse is that there is no chance that the
mass involved will be anywhere near the mass of the polar ice.
If we look differently at the first result on ice melting, we se that
the effect can be read as a change of a bit more than 1 nanosecond a
day for 50 million tons of ice spread over the ocean. The effect is
50% more if it is kept at the equator, rather than spread on the
planet surface. Hence trade imbalance between elephants and penguins
can result in a variation of the duration of the day of about 1.5
nanoseconds for 50 millions tons of goods.
On the accuracy of the result.
The calculation contains lots of approximations. First the ice is not
really sitting at the pole (this changes the result for ice melting,
but not for elephant-penguin trade if penguins live at the pole). Also
the water does not spread uniformly on the surface.  But these are
probably minor approximations.  One easily overlooked approximation
concerns the radial density distribution of the planet (see Wikipedia). The inner
core is about twice as dense on a bit more than half the radius. That
means that the inner 1/8th of the plannet, close to the axis, contains
1/4 of the mass. Hence the formula for spherical moment of inertia is
not quite correct. Furthermore, though minor, Earth is an oblate
spheroid, which changes slightly the moment of inertia (0,3% effect).
This moment of inertia for the polar axis is actually $8.04\
10^{37}\ kgm^2$ according to current estimates.
Let us correct this aspect of the calculation so as to have a better
feeling for the precision of our result.
The moment of intertia of the ice as a spherical shell is $2/3\ mR^2$,
i.e. $25\ 10^{18}\ \times\ 2/3\ \times\ 6,371^2\ 10^6\ kg m^2$, approximately
$25\ 10^{18}\ \times\ 2/3\ \times\ 40\ 10^{12}\ kg m^2$, or $2000/3\
10^{30}\ kg m^2$, finally giving: $6.7\ 10^{32}\ kg m^2$.
Compared to Earth, this gives a ratio of $1.2\ 10^5$ rather than $1.5\ 10^5$ in the
previous calculation, i.e. a 20% difference, giving then a slowdown of
0,72 second per day, for $25\ 10^{15}$ tons of melted ice.
Given that the effect is 50% more important when the mass is on the
equator rather than spread as a spherical shell, that gives approximately an angular speed
change of 1 nanosecond per day for a displaced mass of 25 millions
tons between pole and equator (trade imbalance between penguins and elephants).
