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?
WLOG (without loss of generality) we can assume that π€1 < π€2
WLOG (without loss of generality) we can assume that π€1 < π€2
Answers
GPT 3.5
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 π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 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:
βπ/βπ₯1 = 1/(π₯1 + 1) = Ξ»1
βπ/βπ₯2 = 1/(π₯2 + 1) = Ξ»2
Setting the two derivatives equal to each other, we get:
1/(π₯1 + 1) = 1/(π₯2 + 1)
Simplifying this equation, we get:
π₯1 + 1 = π₯2 + 1
π₯1 = π₯2
This means that the optimal combination of inputs to maximize output is when π₯1 = π₯2.
Now, let's consider the case where π€1 < π€2. The cost of input 1 (π€1) is lower than the cost of input 2 (π€2). Since the optimal combination of inputs is π₯1 = π₯2, the cost-minimizing combination will also be π₯1 = π₯2. 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., π₯1.
There are no corner solutions in this case because the optimal combination of inputs is always π₯1 = π₯2, rather than one input being zero and the other being positive.
Given the production function π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 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:
βπ/βπ₯1 = 1/(π₯1 + 1) = Ξ»1
βπ/βπ₯2 = 1/(π₯2 + 1) = Ξ»2
Setting the two derivatives equal to each other, we get:
1/(π₯1 + 1) = 1/(π₯2 + 1)
Simplifying this equation, we get:
π₯1 + 1 = π₯2 + 1
π₯1 = π₯2
This means that the optimal combination of inputs to maximize output is when π₯1 = π₯2.
Now, let's consider the case where π€1 < π€2. The cost of input 1 (π€1) is lower than the cost of input 2 (π€2). Since the optimal combination of inputs is π₯1 = π₯2, the cost-minimizing combination will also be π₯1 = π₯2. 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., π₯1.
There are no corner solutions in this case because the optimal combination of inputs is always π₯1 = π₯2, rather than one input being zero and the other being positive.