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. If the price is less than W1, for example what happens to x1*,x2*?
WLOG (without loss of generality) we can assume that π€1 < π€2. If the price is less than W1, for example what happens to x1*,x2*?
Answers
GPT 3.5
To sketch the long-run supply curve, we need to find the optimal inputs that maximize the production function subject to some input prices. Let's assume the input prices are π€1 and π€2 for π₯1 and π₯2 respectively.
To find the optimal inputs, we need to solve the following problem:
maximize π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 1)
subject to the constraint π€1π₯1 + π€2π₯2 β€ π
where π is the price.
We can solve this problem using calculus. Taking the partial derivatives of the production function with respect to π₯1 and π₯2, we have:
βπ/βπ₯1 = 1/(π₯1 + 1)
βπ/βπ₯2 = 1/(π₯2 + 1)
Setting these derivatives equal to zero and solving, we find that π₯1* = π^(π€2 - π€1) - 1 and π₯2* = π^(π€1 - π€2) - 1.
Now let's consider the case where the price π is less than π€1. In this case, the constraint π€1π₯1 + π€2π₯2 β€ π cannot be satisfied. Since we assume π€1 < π€2, the price is less than the cost of input π₯1.
As a result, there will be no positive levels of π₯1* and π₯2* that can maximize the production function. This means that there are no corner solutions in this case.
In summary, if the price is less than π€1, the optimal levels of π₯1* and π₯2* will be zero due to input π₯1 being too expensive. There are no corner solutions in this scenario.
To find the optimal inputs, we need to solve the following problem:
maximize π(π₯1, π₯2) = ππ(π₯1 + 1) + ππ(π₯2 + 1)
subject to the constraint π€1π₯1 + π€2π₯2 β€ π
where π is the price.
We can solve this problem using calculus. Taking the partial derivatives of the production function with respect to π₯1 and π₯2, we have:
βπ/βπ₯1 = 1/(π₯1 + 1)
βπ/βπ₯2 = 1/(π₯2 + 1)
Setting these derivatives equal to zero and solving, we find that π₯1* = π^(π€2 - π€1) - 1 and π₯2* = π^(π€1 - π€2) - 1.
Now let's consider the case where the price π is less than π€1. In this case, the constraint π€1π₯1 + π€2π₯2 β€ π cannot be satisfied. Since we assume π€1 < π€2, the price is less than the cost of input π₯1.
As a result, there will be no positive levels of π₯1* and π₯2* that can maximize the production function. This means that there are no corner solutions in this case.
In summary, if the price is less than π€1, the optimal levels of π₯1* and π₯2* will be zero due to input π₯1 being too expensive. There are no corner solutions in this scenario.