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?
Answers
GPT 3.5
To sketch the long-run supply curve, we need to determine the relationship between the input variables π₯1 and π₯2 and the level of output. In this case, the production function is given by:
π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 1)
Let's analyze the function to see if it exhibits any corner solutions. A corner solution occurs when one of the variables becomes zero while the other remains positive.
Taking the derivative of π(π₯1, π₯2) with respect to π₯1, we get:
βπ/βπ₯1 = 1/(π₯1 + 1)
Taking the derivative of π(π₯1, π₯2) with respect to π₯2, we get:
βπ/βπ₯2 = 1/(π₯2 + 1)
The first-order conditions for corner solutions occur when either π₯1 or π₯2 is zero. However, in this case, neither π₯1 nor π₯2 can be zero since we have ππ(π₯1 + 1) and ππ(π₯2 + 1) in the production function. Thus, there are no corner solutions in this case.
To sketch the long-run supply curve, we would need information on the prices of inputs and the level of output. However, without this information, we cannot directly determine the long-run supply curve.
π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 1)
Let's analyze the function to see if it exhibits any corner solutions. A corner solution occurs when one of the variables becomes zero while the other remains positive.
Taking the derivative of π(π₯1, π₯2) with respect to π₯1, we get:
βπ/βπ₯1 = 1/(π₯1 + 1)
Taking the derivative of π(π₯1, π₯2) with respect to π₯2, we get:
βπ/βπ₯2 = 1/(π₯2 + 1)
The first-order conditions for corner solutions occur when either π₯1 or π₯2 is zero. However, in this case, neither π₯1 nor π₯2 can be zero since we have ππ(π₯1 + 1) and ππ(π₯2 + 1) in the production function. Thus, there are no corner solutions in this case.
To sketch the long-run supply curve, we would need information on the prices of inputs and the level of output. However, without this information, we cannot directly determine the long-run supply curve.