Marginal Productivity Theory of Factor Pricing in Imperfect Commodity Market

Introduction

The demand curve for a variable factor, such as labour, is no longer the value of the marginal product of labour (𝑉𝑀𝑃𝐿) when there is imperfect competition in the commodity market and the factor market is perfectly competitive.

Under the above assumption in this module, first, we will derive demand for labour in two situations: (i) when labour is the sole variable factor and (ii) when there exist several factors and secondly, we determine equilibrium level factor input and factor price through the interaction between market demand and supply of factor input.

Demand for Factor Input by Monopolistic Firm

1. Demand for Factor Input by Monopolistic Firm in Case of Single Variable Factor (Labour)

In an imperfectly competitive product market (i.e. the existence of monopoly, monopolistic competition or oligopoly), an individual firm will hire labour up to the level at which its profit is maximized, i.e. marginal revenue productivity of labour (𝑀𝑅𝑃_𝐿) equals to the marginal cost of labour (wage rate) provided that there is perfect competition in the labour market. The derivation of a firm’s profit maximization is given as:

Let us consider a production function Y = f (L, 𝐾̅)

The total cost of production consists of the total variable cost 𝑤̅. 𝐿 and fixed cost F

i.e., total cost  𝑇𝐶 = 𝑤̅. 𝐿 + 𝐹

Assume that the market demand function faced by the firm is given as 𝑃_𝑌 = 𝑃(𝑌)

The total revenue of the firm is 𝑇𝑅 = 𝑃_𝑌. 𝑓(𝐿, 𝐾̅). Hence, the profit function can be written as

Π = 𝑇𝑅 − 𝑇𝐶

Π = 𝑃(𝑌). 𝑓(𝐿, 𝐾̅) −  𝑤̅. 𝐿 − 𝐹

In order to satisfy the first-order condition of profit maximization with respect to the variable labour, we set

That is, profit is maximized at the level of labour at which the marginal revenue productivity of labour equals the marginal cost of labour or wage rate. Thus, we obtain the demand curve for variable factor input or labour from the above marginal condition. The demand curve for labour is given by the 𝑀𝑅𝑃_𝐿 curve in the figure.

Since we assume that the labour market is perfectly competitive. The supply of labour to the individual firm is perfectly elastic, i.e. the supply curve of labour 𝑆_𝐿 faced by the firm is parallel to the horizontal axis (figure). Now, for a given wage 𝑤̅, the 𝑀𝑅𝑃_𝐿 intersects the supply curve 𝑆_𝐿 at point 𝑒, and the corresponding equilibrium level of labour is 𝑙^∗.

In an imperfectly competitive commodity market, the 𝑀𝑅𝑃_𝐿 is not the same as 𝑉𝑀𝑃_𝐿 (value of marginal productivity of labour) as price doesn’t equal to marginal revenue (𝑃_𝑌 > 𝑀𝑅). Since 𝑃_𝑌 > 𝑀𝑅, the 𝑉𝑀𝑃_𝐿 is greater than 𝑀𝑅𝑃_𝐿 shown in the figure.

2. Demand for Factor Input by Monopolistic Firm in Case of Several Variable Factors (Labour & Capital)

In case of more than one variable factor, such as capital (in the long run, capital will also vary along with labour), the demand curve for labour will no longer be the simple MRP_L curve. This is as because when several factors are employed in the production simultaneously, a change in the price of one factor can alter the demand for the other factor and which in turn shifts the marginal productivity curve of the factor whose price is changed initially.

To explain this, we apply the concepts of the substitution effect, output effect and profit-maximizing effect. These three effects are analyzed graphically in the figure.

Assume that a firm produces output 𝑌₁ by using capital 𝐾₁ and labour 𝐿₁, when factor prices are 𝑟₁ and 𝑤₁ respectively. In the diagram, this is shown by the equality of the slope of the isoquant 𝑌₁ and the iso-cost line ab at point 𝑒₁.

Suppose that the wage rate declines from the initial rate 𝑤₁ to 𝑤^/. This will rotate out the iso-cost line from 𝑎𝑏 to 𝑎𝑐 by pivoting the point 𝑎, and the firm will produce more output 𝑌₂ at the level of capital and labour 𝐾₂ and 𝐿₂ respectively. This is shown by the new equilibrium point 𝑒₂ in the figure.

The journey from initial equilibrium point 𝑒₁ to 𝑒₂ can be divided into two effects: substitution effect and output effect. In order to explain these two effects more explicitly, we draw a new iso-cost line 𝑑𝑒, which is parallel and also tangent to the initial iso-quant 𝑌₁ so that it shows the new factor price ratio and, at the same time, maintains an initial level of production. The new iso-cost line 𝑑𝑒 is tangent to the isoquant 𝑌₁ at 𝑒₁^/.

The movement from 𝑒₁ to 𝑒₁^/ happens due to the substitution effect. This implies that when the wage rate falls, the firm substitutes more labour in place of relatively costlier capital. In the figure, the firm increases the level of employment from 𝑂𝑙₁ to 𝑂𝑙₁^/. Next, the movement from 𝑒₁^/ to 𝑒₂ is explained by the output effect.

The output effect arises because of the parallel shift of the iso-cost line 𝑑𝑓 to the iso-cost line 𝑎𝑐. It reflects the fact that at the same expenditure, the firm hires more of both capital and labour. This is shown by the rise of labour from 𝑂𝑙₁^/ to 𝑂𝑙₂ and the increase in capital from 𝑂𝑘₁^/ to 𝑂𝑘₂.

However, the story does not end here. There is another prominent effect, i.e., the profit-maximizing effect, arises as a firm increases it‘s expenditure in order to maximize profit. More intuitively, when the wage rate falls, the marginal cost of production declines. This induces the firm to make an additional expenditure for attaining the profit-maximizing level output 𝑌₃.

Going back to the figure (above) where this additional expenditure due to profit-maximization is shown by the parallel upward shift of the iso-cost line 𝑎𝑐 to 𝑎^/𝑐^/. The final equilibrium point is shown by the point 𝑒₃ at which the new equilibrium level of output 𝑌₃ is obtained for the corresponding 𝑂𝑘₃ level of capital and 𝑂𝑙₃ level of labour.

Now, we will explain how these three effects help us to derive the demand curve for labour when there are several variable factors. The substitution effect leads to a reduction in the marginal productivity of labour as it substitutes more workers in place of capital, while the output effect and profit-maximizing effect both have a positive impact on the marginal productivity of labour. This is because the employment of both the factor inputs capital and labour are raised by these two effects.

In the figure (above), this is shown by the rightward shift of 𝑀𝑅𝑃_𝐿 curve when the wage rate declines from 𝑤^/ to 𝑤^//. For different wage rates, we have corresponding levels of employment by equating the wage lines with the separate 𝑀𝑅𝑃_𝐿 curves. Suppose for three different wages rates 𝑤, 𝑤^/ and 𝑤^//, we get three different equilibrium points 𝑒₁, 𝑒₂ and 𝑒₃ on the three separate MRP_L curves. Thus, by joining 𝑒₁, 𝑒₂ and 𝑒₃, we have the new demand curve for labour in case of several variable factors.

3. Market Demand for a Factor

The market demand for variable factors by a group of monopolists is simply the sum of the individual demands of those monopolists. Unlike a perfectly competitive situation, there is no external effect of output expansion on price in an imperfect market. There is an internal effect of output expansion on price for each monopolist. When various monopolists use the variable factor, the market demand is the addition of the different component industry demand curves, where the industries may be formed by any number of firms. The derivation of market demand is the same for monopolistic competition and oligopoly.

Here, the derivation of market demand in case of monopolistic competition is considered. The individual demand curve for variable factor derived in the case of monopolistic competition is the same as the individual monopolist’s demand for factor, but when all firms in the product group expand output, there will be a fall in market price, like perfect competition. The derivation of market demand is shown in the figure (A) & (B).

The straight line 𝑑_𝐿 in Figure-(A) shows an individual firm’s demand curve for labour. Suppose at the wage rate 𝑤₁, the firm is at point 𝑒₁ and hires 𝑙₁ units of labour. Assume that the wage rate falls to 𝑤₂. The firm moves from point 𝑒₁ to 𝑒^/ along the 𝑑_𝐿 curve and increases its employment of labour from 𝑙₁ to 𝑙^/ units.

Since all other firms of the product group behave in the same manner, there will be an expansion of output and product price 𝑃_𝑌 will diminish. This will result in a downward shift of 𝑀𝑅𝑃_𝐿 curve or 𝑑_𝐿 curve to the left. The firm is now at point 𝑒₂ on the new demand curve for labour 𝑑_𝐿^/ and hires 𝑂𝑙₂ units of labour at wage rate 𝑤₂ in Figure-(A).

Next, we see what happens in the market demand curve for labour, which is shown in Figure-(B). At the initial wage rate 𝑤₁ aggregating all individual firms’ demand for labour, we get a point 𝐸 on the market demand curve for labour 𝐷_𝐿 and corresponding 𝑂𝐿₁ units of aggregate level of employment of labour. At wage rate 𝑤₂, each firm demands 𝑂𝐿₂ units of labour and aggregating over all firms, the market demand for labour will be 𝑂𝐿₂ in Figure-(B), by joining the points, 𝐸₁ and 𝐸₂ we derive the market demand curve for labour as 𝐷_𝐿.

We will not be at the point 𝐸^/ on the same demand curve 𝐷_𝐿, rather we obtain a point, say 𝐸^//, which is left to the point 𝐸^/. By adding the points 𝐸 and 𝐸^//, we finally obtain the market demand curve 𝐷_𝐿^/ and at 𝑤₂, the market demand for labour will be 𝑂𝐿₂ units.

Determination of Equilibrium Factor Price

We are now in a position to determine the equilibrium factor price in the perfectly competitive factor market, while the product market is imperfectly competitive by combining the market demand curve and supply curve for variable factor labour derived in previous sections. In this model, the individual labour supply curve is derived through the labour leisure choice of individual workers. The market supply curve is a horizontal summation of individual labour supply curves.

The figure-1 below shows that the interaction between the market demand for labour 𝐷_𝐿 and the market supply of labour 𝑆_𝐿 determine the equilibrium factor price 𝑤^∗ corresponding to 𝑂𝐿^∗ level of employment of labour. In comparison with perfectly competitive firms, imperfectly competitive firms or monopolist firms always pay a lower wage. This is shown in the figure below. The difference between 𝑉𝑀𝑃_𝐿 and 𝑀𝑅𝑃_𝐿 for a given level of employment is called monopolistic exploitation.

In the figure-2, the gap between 𝑊_𝐶 and 𝑊_𝑀 represents monopolistic exploitation. According to Joan Robinson, monopolistic exploitation is a situation in which a factor price is paid to any factor lower than its value of marginal productivity.