Answer to Question #152955 in Microeconomics for sara

Question

Assuming that labor is the only variable input and the wage rate is constant, explain the reason for the U-shape of the MC curve.

Answer 1

ANSWER

The first step. Total product of labour.

According to the question, the total product (denoted by TP) depends on the Direct Labour (denoted by L) only (it can be services with no material and amortization cocts). It is easy to show that growth in L is the reason for the growth in TP. However, it is easy to imagine that when increasing in L from 1 to 2 workers, for example, the growth in TP is greater than in the case of the growth in L from 100 to 101. In this example the marginal product of labour (denoted by MP(L) ) in the case of L=2 is greater than that of in the case of L=101. Hence, $MP(2)>MP(101)$ . When the TP is low, TP might increase at an increasing rate. After that the TP can be equal to zero (inflaction point of the TP function with respect to the L). After that TP is big and TP might increase at an decreasing rate. This example is consistent with the the law of diminishing marginal productivity (it is a famous economic principle). This Law states that advantages gained from slight improvement on the input factor will only advance marginally per unit and may level off or even decrease after a specific value. 

Let's consider the three areas: labour (denoted by L) range from 0 to L1. In this area MR(L) increases (or $dMP(L)/dL>0$ ). The inflaction point (the point at which $dMP(L)/dL=0$ ) is denoted by L1. In the range that can be described by means of inequality $L>L1$ we can write that $dMP(L)/dL<0$ .

The second step. Marginal costs and labour.

According to the question the total cost (denoted by TC) is impacted by the L. Let's infer this function.

$TC=VC+FC$ , where VC os variable costs and FC is fixed costs.

According to the question any changes is the VC are consistent with the changes in the L by means of constant wage rate (denoted by W).

Hence, $VC=w*L(q)$ .

Hence, $TC=w*L(q)+FC$

According to the marginal cost definition, the MC function can be calculated as the first order derivative of TC with respect to q.

Hence,

$MC(q)=dTC(q)/dq=d(w*L(q)+FC)/dq=w*dL/dQ=w*1/MP(L)$

The third step. The U-shape of the MC curve.

From 0 to L1 (or from q=0 to q=q1) the MP increases, and hence $1/MP(L)$ decreases. Hence, MC decreases from L=0 to L = L1.

When L=L1 (or q=q1) there is inflaction point of MC.

From L>L1 (or q>q1) the MP decreases, and hence $1/MP(L)$ increases. Hence, MC increases after L1.

Conclusion: the shape of MC curve is U-shape. The necessary function diagrams are depicted in the picture below.