Useful Notes on Short-Run Demand Curve for Labour

Let us assume that labour and capital are the only two factors of production, of which one factor is fixed. The production function is then defined by Q = TP_L = f(L)K, where ‘Q’ denotes total product of labour, ‘L’ is the total amount of labour used and K shows the amount of capital used which is fixed in the short run. The function shows a direct relation between the number of workers and total output produced till some point after which the output declines with increase in number of workers.

The above TP_L schedule would generate a downward sloping marginal physical product of labour curve indicating diminishing returns to the variable factor (labour in this case). Ultimately, MPP_L becomes negative showing a decline in TP_L, when ‘L’ is increased.

The marginal physical product curve of labour or MPP_L is the change in TP_L that occurs when one extra unit of labour is employed. Mathematically, it is denoted by

dQ/dL=MPP_L

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We further assume that the factor as well as commodity markets are perfectly competitive. Since there is perfect competition in the commodity market, demand for the commodity is perfectly elastic (firm is a price taker) and P = AR = MR.

Similarly, in the labour market, the supply curve for labour is perfectly elastic (again the firm is a price taker). It shows that the firm can hire any amount of labour at the going wage rate (Fig. 17.1).

Now, before proceeding further let us define two new concepts

(i) The value of the marginal product of labour or factor demand curve is given by the price or the value of its marginal product. Mathematically, VMP_L= MPP_L x P or MPP_L x AR (since P = AR under perfect competition).

(ii) The marginal revenue product of labour is the addition to total revenue from the sale of output produced by one additional nit of labour. Mathematically,.

dTR_x

MRP_L=dTR_x/dQ= dTR_x/ dQ . dQ /dL= MPP_L × MR_x

Under perfect competition, AR or P= MR. Therefore, VMP_L = MRP_L. But, under monopoly or imperfect competition, AR_x > MR, So, VMP_L > MRP_L (Fig. 17.2).

In our model, given that P = MR and given that MPP_L is downward sloping, we get the following relation between MPP_L and VMP_L or MRP_L

Since the firm is a profit maximise, an additional worker yields MRP_L amount of benefit. But, an additional worker costs w amount of wage. As long as marginal benefit > marginal factor cost, it is profitable to hire an additional worker and stop when marginal benefit = marginal factor cost.

Or, MRP_L = w or VMP_L = w (since MR = P under perfect competition) mathematically, the same argument can be shown as follows:

π = TR – TC

Starting from an initial equilibrium condition, if wage rate falls, but ‘P’ remains constant, then MPP_L > w. To restore equilibrium, MPP_L has to fall which will only occur if more labour is hired. Therefore, as wage rate falls, employment increases and the VMP_L curve turns out to be the demand curve of labour for the firm in the short run, when labour is the only variable factor of production.