Final answer:
1. The production function for x₂=A=1 is y=ln(x₁+1). 2. The cost minimization problem involves finding the combination of inputs that minimizes the cost. 3. The variable cost (cᵥ), fixed cost (F), average cost (AC), average variable cost (AVC), average fixed cost (AFC), and marginal cost (MC) can be determined for the cost functions.
Step-by-step explanation:
1. To illustrate the production function, let's substitute A=1 and x₂=1 into y=f(x₁,x₂)=Aln(x₁+1)x₂. We have y=ln(x₁+1).
2. The cost minimization problem involves finding the combination of inputs that minimizes the cost of producing a given output. In this case, we want to minimize the cost function C(w₁,w₂,p,x), subject to the constraint y=f(x₁,x₂), where C represents the cost, w₁ and w₂ are the prices of the inputs, p is the price of the output, and x represents the quantities of the inputs. By solving the cost minimization problem, we can find the cost function C(w₁,w₂,p,x).
3. The variable cost (cᵥ) is the cost of the variable inputs, which includes the prices of the inputs (w₁ and w₂) multiplied by the quantities of the inputs (x₁ and x₂). The fixed cost (F) is the cost associated with the fixed inputs, which in this case is x₂. The average cost (AC) is the total cost divided by the quantity of output (y). The average variable cost (AVC) is the variable cost divided by the quantity of output. The average fixed cost (AFC) is the fixed cost divided by the quantity of output. The marginal cost (MC) is the change in total cost resulting from a one-unit increase in output.
4. To find the supply function of the firm, we need to determine the quantity of output the firm is willing to supply at different prices. In this case, the supply function is given by the equation p=MC, where p is the price and MC is the marginal cost. By substituting the cost function C(w₁,w₂,p,x) into MC, we can find the supply function.