If the supply of a good decreases and it causes total revenue to increase, this shows that the good has an a) inelastic demand b) inelastic supply...

We are shifting the supply curve, which means we're moving along the demand curve--so the elasticity we're interested in is the elasticity of demand, which narrows it down to (a), (d), or (e).By decreasing supply, we increase revenue. This means that as quantity sold Q goes down, the price P rises so much that the revenue P*Q goes up.In terms of math, this means that the rate of change in [P*Q]...

We are shifting the supply curve, which means we're moving along the demand curve--so the elasticity we're interested in is the elasticity of demand, which narrows it down to (a), (d), or (e).

By decreasing supply, we increase revenue. This means that as quantity sold Q goes down, the price P rises so much that the revenue P*Q goes up.

In terms of math, this means that the rate of change in [P*Q] with respect to Q is negative:

d[PQ]/dQ < 0

Using the Chain Rule we can separate this into two parts:

P + dP/dQ * Q < 0

With some simple algebra, we can turn this into an expression of the inverse elasticity:

dP/dQ * Q < - P

dP/dQ * Q/P < -1

The elasticity is just the inverse of this, and yes, indeed, dP/dQ = 1/(dQ/dP), though this is actually a calculus theorem and not nearly as obvious as it looks at first.

Also, don't forget to reverse the inequality when you take the reciprocal of both sides:

dQ/dP * P/Q > -1

That is, demand is inelastic, because an elasticity greater than -1 means an elasticity closer to 0. Therefore the answer is (a), inelastic demand.

In words, what we're saying is that we only have to decrease the quantity sold a little bit to get the price to rise a lot, and that means quantity is falling slower than price is rising, so demand must be inelastic.