To set up the firm's profit maximization problem in the long run, we need to determine the firm's cost function and then maximize its profit.
Step 1: Determining the cost function
In the long run, the firm can adjust both inputs, x1 and x2, to minimize its cost and maximize profit. The cost function can be derived by minimizing the total cost (TC) subject to the production function and input prices.
We can use the concept of cost minimization to determine the cost function. According to cost minimization, the firm will choose the input combination that minimizes the cost of producing a given output level.
The firm's cost minimization problem is:
Minimize 𝑤1𝑥1 + 𝑤2𝑥2 such that 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1) = 𝑞
Where:
𝑤1 and 𝑤2 are the prices of inputs x1 and x2, respectively.
𝑞 is the desired output level.
To obtain the cost function, we need to solve this minimization problem.
Step 2: Maximizing profit
Once we have the cost function, we can determine the firm's profit function by subtracting the total cost from the firm's revenue. In this case, we are not given a revenue function explicitly, so we will assume a linear demand function for simplicity:
𝑅 = 𝑝𝑞,
Where:
𝑝 is the price of the final product (output).
𝑞 is the quantity of output.
The profit function is then given by:
𝜋 = 𝑅 − 𝑇𝐶
Where:
𝜋 is profit.
𝑇𝐶 is total cost.
Step 3: Long-run supply curve
To find the long-run supply curve, we need to determine the input choices that maximize profit for different price levels. The firm will choose its inputs to maximize profit by adjusting 𝑥1 and 𝑥2.
To simplify the analysis, let's assume that 𝑞 = 1. Therefore, the production function becomes:
𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1)
𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(1 + 1) (because 𝑞 = 1)
The firm's profit maximization problem becomes:
Maximize 𝑝 − (𝑤1𝑥1 + 𝑤2𝑥2) such that 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1) = 1
To find the long-run supply curve, we consider different price levels and find the optimal input choices. The supply curve will show the quantity that the firm is willing to supply at different price levels.
To sketch the long-run supply curve, we can plot the price levels on the y-axis and the quantity supplied on the x-axis.
Corner solutions occur when the firm optimally chooses one input to be zero. In this case, corner solutions occur when either x1 = 0 or x2 = 0. To determine if there are corner solutions in the long run, we need to examine the conditions for corner solutions by taking the derivative of the production function with respect to each input and assessing the sign.
Overall, we need additional information on price levels (p), input prices (w1 and w2), and the desired output level (q) to find the long-run supply curve and determine the presence of corner solutions. Without these specific values, we cannot proceed to find the long-run supply curve or assess the presence of corner solutions.
The production function is 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1). Set up the firm’s profit maximization problem in the long run and find the long-run supply curve (as a function of p, 𝑤1, and 𝑤2). Then, sketch the long-run supply curve. Are there corner solutions?
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