To understand the problem, it is worth starting with this video, which sets the scene nicely:
Now, the problem is that these well-meaning charities, who want to reduce the number of slaves by buying them and releasing them (referred to as slave redemption), simply increase the demand for slaves. Increased demand for slaves (as shown in the diagram below from D0 to D1) increases the price of slaves (from P0 to P1), and importantly increases the quantity of slaves traded (from Q0 to Q1). So slave redemption increases the quantity of slaves traded - the opposite of what was intended.
To make matters worse, the actions of the slave redeeming charities creates some other perverse incentives. Since slavery is now more profitable than before, and we're talking about a poor country like the Sudan where per capita income in 1990 in the middle of the Second Sudanese Civil War was about US$242 (in 2000 US$), one way that non-slavers could increase their income was to pretend to be slavers. Here's how it worked: A bunch of people would gather together, and one of them would pose as a slaver and the others as slaves. Then when the well-meaning Western slave redeemers show up, the 'slaver' pockets the money, the 'slaves' are freed, and they could meet up afterwards to split the proceeds. Rinse and repeat. So, the slave redeemers might not have even be freeing 'real' slaves at all.
Anyway, MRUniversity has just released a video on Elasticity and the Economics of Slave Redemption, which extends the analysis to consider the impact of elasticity:
As Tyler Cowan notes in the video, the number of extra slaves (including 'fake' slaves) traded will depend on the price elasticity of supply of slaves, as illustrated in the diagram below (which is not drawn to scale). If supply is relatively inelastic (the supply curve is steep, like SSR), then the number of additional slaves for a given increase in demand (from D0 to D1) will be small (quantity increases from Q0 to Q1), and the price of slaves will increase greatly (from $15 to $300). In contrast, if supply is relatively elastic (the supply curve is flat, like SLR), then the number of additional slaves for the same increase in demand (from D0 to D1) will be large (quantity increases from Q0 to Q2), and the price of slaves will not increase by much (from $15 to $50).
According to this article in the Atlantic from 1999, the actions of the slave redeemers increased the price of slaves from around US$15 to US$300 (as shown in the diagram above), so it seems likely that supply was relatively inelastic (SSR) initially. However, prices fell later (to US$100 then to US$50), showing that supply is more elastic in the long run (SLR) than the short run. This is because in the long run slavers have more time to adjust to the increased demand by increasing the intensity of their slaving activity (by enslaving more people). And non-slavers take time to recognise the opportunity and gather a group of friends to take advantage of the incentives to sell 'fake' slaves.
So there you have it - the economics of slave redemption. And amazingly, despite the unintended consequences, slave redemption continues today.