I don't think their value is arbitrary. Just really really small. A medium of exchange or currency needs to have a "stable value" or at least a value that can exist within a certain range for an extended period of time. This builds trust. When there are multiple exchanges trading Bitcoins and each exchange is tracking a similiar exchange rate it will build trust. Eventually it will reach a critical mass.
Really now? Do you know why people used gold as money? How about unbacked paper?
Trust is not built by corresponding blips on the internet, it is built by being able to use regularly and gain an expected result...
If no one can purchase anything but $ with bitcoins then bitcoin is nothing more than a gamble on anticipating other people's expectations on the utility of bitcoin.
IF one can only purchase consumer goods OR capital goods with bitcoin, an exchange is necessary to fuel the economy with transaction material, for example:
If I sell widgets and I accept bitcoin for them, if I cannot buy widget supplies to generate widgets in bitcoin, I will require an exchange for the purpose of attaining the capital goods necessary to resupply my production cycle. Conversely, if I can buy my capital goods in bitcoin, but no one will buy my consumer good in bitcoin, I will require an exchange to resupply, using bitcoin. However if both transactions are taking place using the bitcoin medium of exchange, I may only need an exchange if there are not enough goods to exchange my profits in a (subjectively) worthwhile manner...
Is it that you and another exchange collude that give bitcoin value, or is it that I can expect to use bitcoin in exchange for goods? If an exchange rate is stable from one exchange to the next, it builds trust that the exchanges are legit, but does nothing to secure a value to bitcoin.
I'm not trying to create value out of nothing or facilitate a Ponzi scheme. Just a market for buyers and sellers and a way to track demand versus the US Dollar. Every buyer may have a different reason for buying and every seller may have a different reason for selling. Resupplying a production cycle is just one reason of many. The exchangers are not going to create trust in bitcoin all by themselves, but they are a critical piece that must be in place.