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?
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.
If you know the quantity of labor and the total quantity of the product, but not the capital input, you can use the production function to determine the capital input.
The production function relates the inputs of labor and capital to the quantity of output produced. It is typically written as Q = f(L, K), where Q represents the output, L represents the quantity of labor, and K represents the quantity of capital input.
To determine the capital input, you can rearrange the production function to solve for K. For example, if the production function is Q = 10L^0.5K^0.5, and you know the quantity of labor (L) is 100 and the total quantity of the product (Q) is 1000, you can substitute these values into the production function to solve for K.
1000 = 10(100)^0.5K^0.5
Simplifying the equation, you have:
100 = (100)^0.5K^0.5
Taking the square of both sides:
10 = K^0.5
Now, you can solve for K by taking the square of both sides again:
K = 10^2
K = 100
Therefore, the optimal capital input in this case would be 100.
Note that this is just a simplified example to illustrate the process. In practice, production functions can vary in complexity and may require additional information or calculations to determine the capital input accurately.
To draw isoquant curves for a certain quantity, start by selecting a fixed level of output that you want to represent on the graph. Then, plot various combinations of labor and capital that produce that fixed level of output. Labor is typically plotted on the x-axis, and capital on the y-axis. Connect the points on the graph to form a curve, which represents all the possible combinations of labor and capital that yield the desired output.
Now, let's move on to the concepts of "increasing returns to scale" and "decreasing returns to scale."
Increasing returns to scale occur when increasing the inputs by a certain percentage leads to more than a proportional increase in output. For example, if you double the amount of labor and capital used in the production process, and the output more than doubles, then we have increasing returns to scale. This suggests that economies of scale exist, and as production increases, the average cost per unit decreases.
On the other hand, decreasing returns to scale occur when increasing the inputs by a certain percentage results in less than a proportional increase in output. For instance, if you double the inputs, but the output increases by less than double, then we have decreasing returns to scale. This implies that the production process becomes less efficient as it grows, leading to higher average costs per unit.
Regarding your question about "capital input," if all you have is the quantity of labor and the total quantity of the product, it means you are dealing with a production function that only depends on labor. In this case, the production function only takes labor as an input and does not require any capital input. Therefore, the optimal isoquant will intersect the labor axis, indicating that the labor input is sufficient for producing the desired output.