The production function is 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1). 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 find the optimal values of x1 and x2 at different price levels. We can differentiate the production function with respect to x1 and x2 to find the marginal products:
∂𝑓/∂𝑥1 = 1/(𝑥1 + 1)
∂𝑓/∂𝑥2 = 1/(𝑥2 + 1)
Setting these marginal products equal to the wage rates, w1 and w2 respectively, we get:
1/(𝑥1 + 1) = w1
1/(𝑥2 + 1) = w2
Solving these equations for x1 and x2, we get:
𝑥1 = 1/w1 - 1
𝑥2 = 1/w2 - 1
Now, let's consider the three cases:
1. p < w1:
In this case, the firm cannot cover its costs and will not produce anything. Therefore, there is a corner solution.
2. w1 < p < w2:
In this case, the firm will produce some positive levels of x1 and x2. The long-run supply curve will be upward sloping as the firm responds to higher prices by increasing its production.
3. w2 < p:
In this case, the firm will produce the optimal levels of x1 and x2 given the wage rates. The long-run supply curve will be horizontal at the optimal levels of x1 and x2.
In summary, the long-run supply curve will consist of a corner solution at p < w1, an upward sloping segment between w1 < p < w2, and a horizontal segment at the optimal levels of x1 and x2 for w2 < p.
To sketch the long-run supply curve, we need to analyze the conditions for optimal production. The long-run supply curve represents the quantities of output that a firm is willing to produce at different market prices in the long run when all inputs can be adjusted.
The production function 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1) suggests that the firm's output depends on two inputs, 𝑥1 and 𝑥2, where 𝑥1 represents the first input and 𝑥2 represents the second input.
To find the long-run supply curve, we need to determine the profit-maximizing quantities of inputs (𝑥1 and 𝑥2) for various price levels. We maximize profit by comparing the marginal cost of production to the market price.
Now let's analyze the three cases:
1. Case: p<w1 (where p is the market price and w1 is the cost of input 𝑥1)
In this case, the market price is lower than the cost of input 𝑥1. The firm will not produce any output because it would result in a loss. Therefore, the long-run supply curve would start at zero output.
2. Case: w1<p<w2 (where p is the market price, w1 is the cost of input 𝑥1, and w2 is the cost of input 𝑥2)
In this case, the market price is between the cost of input 𝑥1 and 𝑥2. The firm will adjust the quantities of 𝑥1 and 𝑥2 to maximize profit. To find the specific quantities, we need to set up the profit maximization problem and find the optimal production levels.
3. Case: w2<p (where p is the market price and w2 is the cost of input 𝑥2)
In this case, the market price is higher than the cost of input 𝑥2. The firm will adjust the quantities of 𝑥1 and 𝑥2, again optimizing profit. The specific quantities can be found by setting up the profit maximization problem.
Now, regarding corner solutions, they occur when certain inputs are not used in production due to high costs or inefficiency. In this case, we could have corner solutions if the cost of either 𝑥1 or 𝑥2 is too high relative to the market price. This would result in adjusting the production quantities only for the more cost-effective input and completely disregarding the other input.
To determine if there are any corner solutions, we need to compare the costs of 𝑥1 and 𝑥2 to the market price in each case. If the cost of an input is higher than the market price, then there would be a corner solution for that input.
Please note that to obtain the exact quantities and shape of the long-run supply curve, we need more specific information about the cost functions of 𝑥1 and 𝑥2 and the market price.
To sketch the long-run supply curve for the given production function, we need to understand the concept of the long-run production function and analyze the different cases of input prices.
The long-run production function represents the maximum output that can be produced given any combination of inputs, assuming that firms can change their levels of capital and labor as needed. In the case of the given production function 𝑓(𝑥1, 𝑥2) = 𝑙𝑛(𝑥1 + 1) + 𝑙𝑛(𝑥2 + 1), the inputs are represented by 𝑥1 and 𝑥2.
Now, let's analyze the three cases of input prices to determine if there are any corner solutions.
1. Case: p < w1 (where p represents the price and w1 represents the price of input 1)
In this case, input 1 is relatively expensive compared to the price of the output. The production decision would be to minimize the use of input 1 since it is costly. Thus, the firm would choose to set 𝑥1 = 0 to minimize costs. This leads to a corner solution where an input is not used.
2. Case: w1 < p < w2 (where w2 represents the price of input 2)
In this case, input 1 is relatively cheaper compared to input 2 and the price of the output. The firm would choose to increase 𝑥1 to produce more output while incorporating 𝑥2. The solution lies on the interior of the feasible region, and there are no corner solutions.
3. Case: w2 < p (where w2 represents the price of input 2)
In this case, input 2 is relatively expensive compared to the price of output. The production decision would be to minimize the use of input 2 since it is costly. Similarly to case 1, the firm would choose to set 𝑥2 = 0 to minimize costs. This leads to a corner solution where an input is not used.
Now let's sketch the long-run supply curve based on these cases:
Case 1: p < w1
In this case, the long-run supply curve has a kink at 𝑥1 = 0 and then slopes upward as 𝑥1 increases.
Case 2: w1 < p < w2
In this case, the long-run supply curve is a smooth curve with a positive slope, increasing as both 𝑥1 and 𝑥2 increase.
Case 3: w2 < p
In this case, the long-run supply curve has a kink at 𝑥2 = 0 and then slopes upward as 𝑥2 increases.
Overall, the long-run supply curve for this production function exhibits corner solutions in cases 1 and 3 and a smooth curve in case 2.