Answer to Question #195596 in Microeconomics for Natalie

Question

(a) Can a firm be producing efficiently when the marginal product of labor is negative?

(b) Show mathematically that the ratio of the marginal product of labor and capital equals the 

MRTS.

(c) When a firm uses different factors of production or least cost combination the optimum 

combination of factors is achieved when MPL / PL = MPL / PL. intuitively explain this 

statement. 

(d) With reference to the indifference curve, explain the convexity assumption. 

Answer 1

Solution:

a.). A firm will not be producing efficiently when the marginal product of labor is negative. Efficiency means achieving the maximum output attainable from given inputs. Producing any less means production is inefficient. A negative marginal product of labor implies that the additional unit of labor has decreased the output, rather than increasing it. The reason behind this is the diminishing marginal productivity of labor.

 

b.). The Marginal Rate of Technical Substitution (MRTS) equals the absolute value of the slope. The MRTS shows the rate at which you can substitute one input such as labor for another input such as capital, without changing the level of the resulting output.

MRTS is equal to the change in capital to change in labor, which equals the ratio of marginal product of labor to the marginal product of capital.

The mathematical form of how labor (L) can be substituted for capital (K) in production is shown below:

 

MRTS (_{L, K}) = $-\frac{\triangle K }{\triangle L} = \frac{MP_{L} }{MP_{K}}$

MRTS_LK = $\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

$\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

We can rearrange the above equation to have

$\frac{MP_{L} }{w} = \frac{MP_{K} }{r}$

 

 

c.). If the marginal benefit of additional labor, $\frac{MP_{L} }{P_{L}}$, exceeds the marginal cost, $\frac{MP_{K} }{P_{K}}$, then the firm will be better off by spending more on labor and less on capital. Similarly, if $\frac{MP_{K} }{P_{K}}$ is greater than $\frac{MP_{L} }{P_{L}}$ , the firm will be better off spending more on capital and less on labor. The equilibrium is when the firm produces the maximum amount of output at a given cost, which occurs where $\frac{MP_{L} }{P_{L}} = \frac{MP_{K} }{P_{K}}$.

d.). An indifference curve, concerning two commodities, is a graph showing the combinations of two products that leave the consumer equally well off or equally satisfied, hence indifferent in having any combination on the curve.

The convexity assumption of the indifference curves implies that the MRTS of X and Y fall as more of X is substituted for Y. Therefore, indifference curves are convex to the origin when the principle of diminishing marginal rate of substitution holds the product. Each indifference curve is convex to the origin and no two indifference curves can ever intersect. Consumers are always assumed to be more satisfied when achieving a bundle of goods on an indifference curve farther from the origin.