Cap and trade schemes often use a policy of adjustable allowance supply with the intention to stabilize the market for allowances. We investigate whether these policies deliver. Our focus is on the sensitivity of allowance prices to the interest rate. We restrict attention to policies that rely on either the allowance price (price measures) or the surplus of unused allowances (quantity measures) to adjust supply in a dynamic cap and trade market. These policies are modelled after the existing policy landscape. Compared to a situation with fixed supply, we find that price measures stabilize allowance prices. Quantity measures do the opposite. Though phrased in the context of changing interest rates, our results warn more generally against the belief that quantity measures are a suitable instrument to promote a stable cap and trade market.