The production function is 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1). Sketch the long-run supply curve. Are there corner solutions?
WLOG (without loss of generality) we can assume that 𝑤1 < 𝑤2
To sketch the long-run supply curve, we need to consider the maximum quantity of output that can be produced for different combinations of inputs in the long run.
Given the production function 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1), we can maximize output by choosing the optimal combination of inputs.
To find this optimal combination, we can use the first-order conditions for maximization:
∂𝑓/∂𝑥1 = 1/(𝑥1 + 1) = λ1
∂𝑓/∂𝑥2 = 1/(𝑥2 + 1) = λ2
Setting the two derivatives equal to each other, we get:
1/(𝑥1 + 1) = 1/(𝑥2 + 1)
Simplifying this equation, we get:
𝑥1 + 1 = 𝑥2 + 1
𝑥1 = 𝑥2
This means that the optimal combination of inputs to maximize output is when 𝑥1 = 𝑥2.
Now, let's consider the case where 𝑤1 < 𝑤2. The cost of input 1 (𝑤1) is lower than the cost of input 2 (𝑤2). Since the optimal combination of inputs is 𝑥1 = 𝑥2, the cost-minimizing combination will also be 𝑥1 = 𝑥2. This means that firms will only choose to produce output using the lower-cost input (input 1) until its cost equals the cost of input 2.
Therefore, the long-run supply curve will be a straight vertical line at the quantity of output that can be produced using only input 1, i.e., 𝑥1.
There are no corner solutions in this case because the optimal combination of inputs is always 𝑥1 = 𝑥2, rather than one input being zero and the other being positive.
To sketch the long-run supply curve from the given production function, we need to determine the optimal values for x1 and x2 that maximize output.
Step 1: Solve for the partial derivatives of the production function.
Taking the partial derivative of f with respect to x1:
∂f/∂x1 = 1 / (x1 + 1)
Taking the partial derivative of f with respect to x2:
∂f/∂x2 = 1 / (x2 + 1)
Step 2: Set the partial derivatives equal to the wage ratio.
Since we assume w1 < w2, we have:
w1 / (x1 + 1) = w2 / (x2 + 1)
Step 3: Rearrange the equation to solve for x2.
Cross multiplying, we get:
w1 * (x2 + 1) = w2 * (x1 + 1)
Expanding, we have:
w1 * x2 + w1 = w2 * x1 + w2
Step 4: Simplify the equation.
Rearranging, we get:
w1 * x2 - w2 * x1 = w2 - w1
Step 5: Solve for x2 in terms of x1.
Dividing the equation by (w2 - w1), we get:
x2 = (w2 - w1) / (w1 * x1 - w2)
Step 6: Determine the conditions for corner solutions.
To check for corner solutions, we need to find the values of x1 and x2 that produce an undefined or negative output. In this case, since f(x1, x2) = ln(x1 + 1) + ln(x2 + 1), we cannot have values of x1 or x2 that are less than or equal to -1.
Thus, if x1 or x2 becomes -1 or less, the production function is undefined, and we have corner solutions.
Step 7: Sketch the long-run supply curve.
Using the derived equation x2 = (w2 - w1) / (w1 * x1 - w2), we can plot the points on a graph that satisfy the equation. However, without specific values for w1 and w2, we cannot determine the exact shape of the long-run supply curve.
In summary, to sketch the long-run supply curve for the given production function, we determine the equation x2 = (w2 - w1) / (w1 * x1 - w2) and plot the points that satisfy the equation. Additionally, the presence of corner solutions depends on specific wage values and can occur if x1 or x2 becomes less than or equal to -1.
To sketch the long-run supply curve, we need to find the optimal values of x1 and x2 that maximize the production function f(x1, x2) while also considering the assumption that w1 < w2.
The first step is to find the derivative of f(x1, x2) with respect to x1 and x2 to determine the marginal products.
∂f/∂x1 = 1/(x1 + 1)
∂f/∂x2 = 1/(x2 + 1)
Setting them equal to the wage ratio (w1/w2), we get:
1/(x1 + 1) = w1/w2
1/(x2 + 1) = 1
Simplifying the equations, we have:
x1 = w2/w1 - 1
x2 = 0
This means that the optimal values for x1 and x2 are x1 = w2/w1 - 1 and x2 = 0.
Now, we can plot the long-run supply curve:
- On the x-axis, we can have x1 (input 1).
- On the y-axis, we can have the output level f(x1,x2).
Since x2 = 0, the production function becomes f(x1, 0) = ln(x1 + 1) + ln(1) = ln(x1 + 1).
Plotting ln(x1 + 1) on the graph, we can observe the long-run supply curve.
Regarding the presence of corner solutions, in this case, since x2 is fixed at 0 and x1 is determined by the wage ratio, we do not have any corner solutions. The optimal solution is always x1 = w2/w1 - 1 and x2 = 0.