I was hoping someone could help me with this;
Suppose that a firm produces 1 output with n inputs.
The production function equals: y=x_{1}^{a}x_{2}^{a}...x_{n} With 0<a<1.
Derive the cost function of this firm.
To answer Shakarri's question first, a production function relates physical quantities of inputs and the physical quantity of output. For example, it will relate man hours of labor, tons of iron ore, tons of coke, and tons of limestone required to produce so many tons of carbon steel. A cost function relates monetary cost to physical units of output produced. What cost, you ask? There are total cost functions, average cost functions, and marginal cost functions. The theory of the firm explains what a profit maximizing firm would do given certain constraints on production, cost, and revenue functions and exact knowledge of those functional relationships.
As for the original question, it is unanswerable AS POSED. We do not know what type of cost function (total, average, or marginal) is required. Are input prices fixed regardless of quantity? I doubt we have been given the context of the problem.