Graph the isoquant for which Q=100 given the production function Q = L0.5K0.5. Can we produce 100 units using only capital?

Question

Graph the isoquant for which Q=100 given the production function  Q = L0.5K0.5. Can we produce 100 units using only capital?

BlackTom AI · Accepted Answer

To approach this question, we first restate what is being asked: can we achieve Q = 100 using only capital given the production function Q = L^0.5 K^0.5?
Option A (the only option provided): 'No, because this is a Cobb-Douglas production function.'
First, evaluate what happens if we set one input to zero. If L = 0, then Q = 0.5 portion of L makes Q = 0 regardless of K, so output cannot reach 100 with no labor. Conversely, if K = 0, Q = 0 as well, regardless of L. This shows that with a Cobb-Douglas form like Q = sqrt(LK), any positive output requires both inputs to be positive; an isoquant for Q = 100 cannot be played with one input equal to zero.
However, digging deeper into the rationale: saying 'because this is a Cobb-Douglas production function' labels the functional form but does not explicitly justify the inability to produce with only capital. The true reason is the multiplicative structure Q = L^0.5 K^0.5 implies output is proportional to the geometric mean of L and K; if either L or K is zero, the product is zero, so Q cannot reach 100 without both inputs positive.
Another way to see it is to consider the isoquant for Q = 100: it requires L and K such that sqrt(LK) = 100, or LK = 10000. This equation cannot be satisfied if either L = 0 or K = 0, because it would force LK = 0, not 10000. Therefore, producing 100 units with only capital would require L > 0 as well, so the 'only capital' scenario is infeasible.
In summary, the key concept is the multiplicative Cobb-Douglas form causing output to be zero if any input is zero, not the label of the function alone. The provided justification mentions the Cobb-Douglas form but omits the more precise reason tied to the requirement that both inputs must be positive to achieve a positive output level like 100.

ECON3200001.1251 Midterm Exam II

Graph the isoquant for which Q=100 given the production function Q = L0.5K0.5. Can we produce 100 units using only capital?

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