If you have a problem where you know the amount of labor and total output and are asked to determine the efficient amount of capital used, you would need additional information to solve the problem. Specifically, you would need a production function, which describes the relationship between inputs (labor and capital) and output.
A common production function is the Cobb-Douglas function:
Q = A * L^α * K^β
Where Q is the quantity of output, L is the quantity of labor input, K is the quantity of capital input, A is the total factor productivity constant, and α and β are the output elasticities of labor and capital, respectively.
Given the quantity of labor (L) and total quantity of output (Q), you can solve for the quantity of capital input (K) using the production function. However, you would also need the values for A, α, and β, which are typically estimated using real-world data or provided in the problem set.
Once you have all the necessary information, you can substitute the values of L and Q into the Cobb-Douglas production function and solve for K. With K determined, you now have the efficient mix of labor and capital that produces the given level of output. To draw the isoquant curve for this output level, you can vary the levels of labor and capital while keeping the output constant and plot the resulting combinations. The isoquant curve represents all the combinations of labor and capital that yield the same level of output.
How to draw isoquant curves for a certain quantity? What does "increasing return to scale" and "decreasing returns to scale" mean? What is "capital input" if all I have is quantity of labor and total quqntity of product?
Typically, a production isoquant is drawn on a graph with labor on one axis, capital on the other. Because of decreasing returns to scale, isoquants have a concave shape (bow pointing towards the (0,0) point). The isoquant represents the mix of capital and labor that will produce a given, fixed level of output. (So, increasing labor and holding capital constant would move you to a higher isoquant. Substituting labor for capital could move you along the isoquant curve).
Increasing returns to scale: increasing an input by x% creates MORE than an x% increase in output. For example, two construction workers, working together, can build more than twice as fast as one worker working alone).
Decreasing returns to scale: increasing an input by x% creates LESS than an x% increase in output.
Im not sure how to answer your last question. If your only input is labor, then your optimal isoquant will be a "corner" solution -- where the isoquant crosses the labor axis. Ergo, capital input is zero.
Or do you have a problem where you know the amount of labor and you know total output and you are asked to determine the efficient amount of capital used?
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