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?
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.
To set up the firm's profit maximization problem in the long run, we make the following assumptions:
1. The firm operates in a perfectly competitive market.
2. The firm's objective is to maximize its long-run profits.
In the long run, the firm can adjust its inputs (𝑥1 and 𝑥2) to maximize its profits. To do this, the firm chooses the levels of inputs that maximize the difference between its revenue and its costs.
Step 1: Calculate the revenue function, 𝑅.
In a perfectly competitive market, the firm's revenue is equal to the price of the output multiplied by the quantity of output sold. Let 𝑤1 and 𝑤2 be the prices of the inputs 𝑥1 and 𝑥2, respectively. The quantity of output sold can be expressed as 𝑞 = 𝑓(𝑥1, 𝑥2).
So, 𝑅 = 𝑝𝑞 = 𝑝(𝑓(𝑥1, 𝑥2))
Step 2: Calculate the cost function, 𝐶.
The cost function is given by 𝐶 = 𝑤1𝑥1 + 𝑤2𝑥2, which represents the total cost of producing the given output levels.
Step 3: Formulate the profit maximization problem.
The firm's profit is given by 𝜋 = 𝑅 − 𝐶.
Maximize 𝜋 = 𝑝(𝑓(𝑥1, 𝑥2)) − (𝑤1𝑥1 + 𝑤2𝑥2).
Step 4: Find the long-run supply curve.
To find the long-run supply curve, the firm must determine the optimal input levels (𝑥1* and 𝑥2*) that maximize its profits. We can find these by taking the first-order partial derivatives of the profit function with respect to 𝑥1 and 𝑥2 and setting them equal to zero:
∂𝜋/∂𝑥1 = 𝑝∂𝑓/∂𝑥1 − 𝑤1 = 0
∂𝜋/∂𝑥2 = 𝑝∂𝑓/∂𝑥2 − 𝑤2 = 0
Solving these equations will give us the optimal input levels 𝑥1* and 𝑥2*.
Step 5: Sketch the long-run supply curve.
The long-run supply curve can be graphically represented by plotting the optimal input levels 𝑥1* and 𝑥2* against the corresponding prices 𝑝.
To determine if there are corner solutions (i.e., situations where one or both inputs are zero), we need to consider the feasibility of the production function. Since the production function is 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1), it is not defined for values of 𝑥1 or 𝑥2 less than -1. Therefore, the firm cannot have corner solutions where 𝑥1 or 𝑥2 are zero.
Note: Additional information on the prices 𝑤1 and 𝑤2 is needed to derive a specific functional form for the long-run supply curve.
To set up the firm's profit maximization problem in the long run, we need to take the derivative of the production function with respect to inputs 𝑥₁ and 𝑥₂, and set it equal to zero. This will help us find the optimal combination of inputs that maximize profit.
The production function is 𝑓(𝑥₁, 𝑥₂) = 𝑙𝑛(𝑥₁ + 1) + 𝑙𝑛(𝑥₂ + 1).
To maximize profit in the long run, we need to solve the following problem:
max 𝜋 = 𝑝𝑦 - 𝑤₁𝑥₁ - 𝑤₂𝑥₂,
where 𝜋 represents profit, 𝑝 is the market price of output, 𝑦 is the level of output, 𝑤₁ and 𝑤₂ are the input prices for 𝑥₁ and 𝑥₂ respectively.
In this case, 𝑦 is the total production given by 𝑦 = 𝑙𝑛(𝑥₁ + 1) + 𝑙𝑛(𝑥₂ + 1).
To find the long-run supply curve, we need to solve for the optimal values of 𝑥₁ and 𝑥₂ that maximize profit by taking the derivative of 𝜋 with respect to 𝑥₁ and 𝑥₂:
∂𝜋/∂𝑥₁ = 𝑝(1/(𝑥₁ + 1)) - 𝑤₁ = 0, (1)
∂𝜋/∂𝑥₂ = 𝑝(1/(𝑥₂ + 1)) - 𝑤₂ = 0. (2)
Solving equations (1) and (2) will give us the optimal values of 𝑥₁ and 𝑥₂.
To find the long-run supply curve, we express 𝑥₁ and 𝑥₂ as functions of 𝑝, 𝑤₁, and 𝑤₂. Once we have the optimal values of 𝑥₁ and 𝑥₂, we can substitute them back into the production function to find the corresponding level of output 𝑦 as a function of 𝑝, 𝑤₁, and 𝑤₂. This will give us the long-run supply curve.
To determine if there are corner solutions (cases where 𝑥₁ or 𝑥₂ are zero), we need to check if the optimal solutions for 𝑥₁ and 𝑥₂ are indeed positive. If they are positive, it means there are no corner solutions. If either 𝑥₁ or 𝑥₂ is zero, it indicates a corner solution.
Unfortunately, without knowing the specific values of 𝑝, 𝑤₁, and 𝑤₂, we cannot determine the exact long-run supply curve or whether there are any corner solutions. However, by solving equations (1) and (2) using the specified values of 𝑝, 𝑤₁, and 𝑤₂, you can obtain the long-run supply curve and identify if there are corner solutions.