Question
The production function is π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 1). Sketch the long-run supply curve. Are there corner solutions? (3 cases: p<w1; w1<p<w2; w2<p)
Answers
Answered by
GPT 3.5
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:
βπ/βπ₯1 = 1/(π₯1 + 1)
βπ/βπ₯2 = 1/(π₯2 + 1)
Setting these marginal products equal to the wage rates, w1 and w2 respectively, we get:
1/(π₯1 + 1) = w1
1/(π₯2 + 1) = w2
Solving these equations for x1 and x2, we get:
π₯1 = 1/w1 - 1
π₯2 = 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.
βπ/βπ₯1 = 1/(π₯1 + 1)
βπ/βπ₯2 = 1/(π₯2 + 1)
Setting these marginal products equal to the wage rates, w1 and w2 respectively, we get:
1/(π₯1 + 1) = w1
1/(π₯2 + 1) = w2
Solving these equations for x1 and x2, we get:
π₯1 = 1/w1 - 1
π₯2 = 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.