How the price mechanism allocates resource in a perfectly competitive market.?

Answer 1

Well, lets begin by saying that resources are limited but wants are unlimited, so we have to make choices.
In a capitalist system, the producers and consumers get to make those choices.

For every choice we make, we lose the opportunity to do something else with that time and money. (If I choose to spend an 2 hours and $20 going to see Bridge to Terabithia, I can't spend that 2 hours and $20 to sip 4 lattes at Starbucks).

Now, how does the price mechanism help us to allocate resources. Simple, it provides a measure of value.
If I have $20, I can go to the movie, or sip a latte, or eat a lot of Krispy Kreme doughnuts.
The price of things help me decide which item (the 36 doughnuts, or 3 lattes, or the 1 movie) is worth more to me than the others. Since price sets a uniform value (money) I can more easily compare the relative value of lattes to a movie and decide where I will spend my scarce resources.

Hope this helps,
Good luck!

Answer 2

The question of resource allocation is that of allocating inputs to create output, and outputs to provide utility to consumers.

The price mechanism assists in two levels:
1) Signals relative excesses (either of demand or supply) in the output market and input market, which aids the production decision.
2) Functions to correspond to utility in the output market.

A complete proof would take far too long. See Arrow and DeBrue's 1961 paper outlining the proof.