Chapter 8: Problem 3

When capital and labour are perfect complements, the isoquant is linear.

Short Answer

Expert verified

In conclusion, when capital and labor are perfect complements, the isoquant is linear because capital and labor are used together in fixed proportions, and neither can be replaced by the other without affecting output levels. This fixed proportionality is represented by straight-line equations, resulting in a linear isoquant.

Step by step solution

Understanding Perfect Complements

Perfect complements are goods or factors of production that are always consumed or used together in fixed proportions. In the context of production, capital and labor are considered perfect complements if the firm uses both in a fixed ratio to produce goods or services. This implies that neither capital nor labor can be substituted for the other without affecting the level of output.

Understanding Isoquants

An isoquant is a curve that represents all the combinations of inputs (capital and labor) that produce the same level of output. In other words, it shows the different capital-labor combinations that result in the same output level, assuming that the production function exhibits constant returns to scale.

The Production Function for Perfect Complements

In the case of perfect complements, the production function takes the form of \(Q = min(AK, BL)\), where \(Q\) is the output, \(K\) is the capital, \(L\) is the labor, and \(A\) and \(B\) are constants representing the fixed proportion of capital and labor used in production. Here, the output \(Q\) is produced by taking the minimum of \(AK\) and \(BL\).

Isoquant for Perfect Complements

To show that the isoquant is linear when capital and labor are perfect complements, we need to find the combinations of capital and labor that result in the same output level \(Q\). From the production function, we know that \( Q = min(AK, BL) \). If \(AK < BL\), we have \(Q = AK\). Rearranging, we get \(K = (1/A)Q - (B/A)L\). This is the equation for a straight line with slope \(-(B/A)\) and intercept \((1/A)Q\). Similarly, if \(AK > BL\), we have \(Q = BL\). Rearranging, we get \(L = (1/B)Q - (A/B)K\). This is the equation for a straight line with a slope \(-(A/B)\) and intercept \((1/B)Q\).

Conclusion

When capital and labor are perfect complements, the isoquant is linear because their relationship follows a straight-line equation, as shown in Step 4. This reflects the fact that capital and labor are used in fixed proportions and cannot be substituted for each other without affecting the output level.