All told, there are two ways this could go down. Both have been touched upon in the answers of @Burki and @VilleNiemi respectively, but just to recap:
A) The quantity your company can mine is insignificant next to the size of the global market
In this case, the demand will not change significantly, so there's no need to worry about a market collapse. Some of the earth-bound mines might stop being profitable and go out of business, but since your space company cannot provide enough material to satisfy global demand on its own, the rest will keep going. Since the demand for most materials is increasing long-term, perhaps not even that will happen.
Your profit margin is directly equivalent to proportion of your unit cost and the market price, still mostly dictated by the earth-bound methods. You will be able to make out like thieves, until everyone else catches wind of what you're doing and starts competing with you.
B) Your company is mining enough materials to significantly affect the global market
This is the Ville Niemi's scenario; the price will drop, but new applications will increase demand. Expect competition to spring up shortly.
There is one huge advantage that your space mining company has; the material they mine and send to Earth is already in space. This instantly slaps another $4k-14k per kilogram (in current economic conditions) to the value versus Earth-mined materials, if there is any use for said material in space technologies or if it can be advantageously processed in zero-g.
A company that mines the material on Earth would have to pay for the mining and then to launch it into orbit just to make it available for the same applications. This is in fact the principal reason why deep space mining would be considered on an industrial scale; if you're already in space, you can use it to get much, much more stuff into space than if you were to launch it.