Demand is key
First of all, the bitcoin price is always determined by the net demand for holding bitcoin. With a given amount of bitcoin available at any point in time, its value must adjust until investors realize their desired allocations, denominated in e.g. USD.- To make a simplified example: If there was only one bitcoin and two investors wanted to hold 1000 USD worth of bitcoin each, that would only be possible with bitcoin valued at 2000 USD a coin.
Hodlers are fully invested
In other words, it looks like the supply side effect is irrelevant. But that is not 100% true. The reason is that a lot of bitcoin hodlers are fully invested. They will keep holding if the price goes up, but they do not have more USD to buy BTC for. The price is therefore to some degree determined by the marginal buyer and the marginal seller and not the hodlers alone.- To make a simplified example: Imagine that all existing coins are held by strong hands not selling. Miners have to sell to cover costs, but no one has to buy. A halving in the supply of new bitcoin would, for a given rate of inflow of new USD, lead to a doubling of price. Once the price has doubled, half the number of coins will be enough to absorb the incoming USD.