The production function is 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1). Assume w1<w2. Sketch the long-run supply curve. Are there corner solutions? (3 cases: p<w1; w1<p<w2; w2<p)
To sketch the long-run supply curve, we need to determine the profit-maximizing quantity of inputs for different levels of output prices.
The profit maximization problem for the firm can be written as follows:
maximize 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1) - 𝑤1𝑥1 - 𝑤2𝑥2
Where 𝑤1 and 𝑤2 are the prices of inputs x1 and x2 respectively.
Let's consider the three cases:
1. When p < w1:
In this case, the price of the output is lower than the price of input x1. This means that the firm would not produce any output because it would be more costly to produce than the revenue it generates. Therefore, there are no corner solutions, and the supply curve is zero.
2. When w1 < p < w2:
In this case, the price of the output is between the prices of inputs x1 and x2. The firm would choose to use only input x1 because it is relatively cheaper compared to input x2. The firm will choose the quantity of input x1 that maximizes its profit, which will be positive. This results in a positive slope supply curve where only input x1 is used.
3. When w2 < p:
In this case, the price of the output is higher than the price of input x2. The firm would choose to use both inputs x1 and x2 since it is now profitable to do so. The firm will choose the quantities of inputs x1 and x2 that maximize its profit. This results in a positive sloping supply curve where both inputs are used.
In summary:
- When p < w1: No corner solutions, the supply curve is zero.
- When w1 < p < w2: There is a corner solution where only input x1 is used.
- When w2 < p: No corner solutions, both inputs x1 and x2 are used.
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), given the different price levels.
Let's analyze the three cases separately:
Case 1: p < w1
In this case, the firm's revenue from selling its output is not sufficient to cover the costs of inputs, even at the lowest wage rate, w1. Therefore, the firm will not produce any output, resulting in a supply quantity of zero. Hence, the long-run supply curve will be a vertical line at quantity zero.
Case 2: w1 < p < w2
In this case, the firm's revenue from selling the output is sufficient to cover the cost of x1 but not x2. By setting the marginal cost of x1 (which is the wage rate w1) equal to the marginal revenue product of x1 (which is the marginal product of x1 multiplied by price p), we can find the optimal value of x1.
Taking the derivative of the production function with respect to x1 and setting it equal to 1/p, we have:
1/(x1 + 1) = 1/p
Therefore, x1 = p - 1. We can substitute this value back into the production function to get the optimal x2:
f(p - 1, x2) = ln(p) + ln(x2 + 1)
We can now plot the long-run supply curve by varying the price level p and determining the optimal x2 for each price level. As p increases, x2 will also increase, representing higher production quantities.
Case 3: w2 < p
In this case, the firm's revenue from selling the output is sufficient to cover the costs of both x1 and x2. By equating the marginal cost of both inputs (w1 and w2) to the marginal revenue product of each input, we can find the optimal values of x1 and x2.
Taking the derivative of the production function with respect to x1 and x2 and setting them equal to 1/p, we get:
1/(x1 + 1) = 1/p
1/(x2 + 1) = 1/p
Solving these two equations simultaneously, we find the optimal values of x1 and x2:
x1 = p - 1
x2 = p - 1
Again, we can plot the long-run supply curve by varying the price level p and determining the optimal x1 and x2 for each price level. As p increases, both x1 and x2 will increase, indicating higher production quantities.
In summary, for the three cases:
- Case 1 (p < w1): The long-run supply curve is a vertical line at quantity zero.
- Case 2 (w1 < p < w2): The long-run supply curve is an upward sloping line with increasing quantities as p increases.
- Case 3 (w2 < p): The long-run supply curve is an upward sloping line with increasing quantities as p increases.
Regarding corner solutions, there are no corner solutions in these three cases since the optimal values of x1 and x2 are positive and not zero.
To sketch the long-run supply curve, we need to analyze how changes in prices affect the optimal input quantities used in production. In this case, we have a production function defined as 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1).
Let's start by understanding the general concept of a long-run supply curve. In the long run, firms can adjust their input quantities (in this case, 𝑥1 and 𝑥2) based on the price of the inputs (wages in this case). The long-run supply curve represents the relationship between the price of the output and the quantity of the output that firms are willing to supply in the long run once they have adjusted their inputs optimally.
Now, let's consider the three cases mentioned:
1. When p < w1:
In this case, the price of the output is lower than the price of input 1 (w1). Since the firms aim to maximize profits, they will not use input 1 at all and only use input 2. Thus, the optimal input quantities will be 𝑥1 = 0 and 𝑥2 > 0. So, the long-run supply curve will start from the point (0, 𝑥2).
2. When w1 < p < w2:
Here, the price of the output is between the prices of input 1 (w1) and input 2 (w2). In this situation, firms will choose to use both inputs to maximize profits. To find the optimal quantities, we need to consider the marginal productivity of each input. Taking partial derivatives of the production function with respect to 𝑥1 and 𝑥2, we get:
∂𝑓/∂𝑥1 = 1/(𝑥1 + 1)
∂𝑓/∂𝑥2 = 1/(𝑥2 + 1)
Setting these equal to the respective input prices, we have:
1/(𝑥1 + 1) = w1
1/(𝑥2 + 1) = w2
Solving these equations will give us the optimal input quantities (𝑥1* and 𝑥2*) at the given output price. The long-run supply curve will plot these input quantities for different output prices within the given range.
3. When w2 < p:
In this case, the price of the output is higher than the price of input 2 (w2). Similar to the first case, the firms will choose to not use input 1 at all, and only input 2. Hence, the optimal input quantities will be 𝑥1 = 0 and 𝑥2 > 0, leading to a long-run supply curve starting from the point (0, 𝑥2).
Regarding the presence of corner solutions, we have corner solutions in the first and third cases (p < w1 and w2 < p), where one of the inputs is not used at all. In the second case (w1 < p < w2), there are no corner solutions, as both inputs are used optimally.
To summarize, the long-run supply curve will include three segments: one starting from the origin when p < w1, one segment connecting the optimal input quantities when w1 < p < w2, and a segment starting from the origin when w2 < p.