Marginal Productivity Theory of Factor Pricing in Competitive Markets

Introduction

According to the traditional economic theory, by factor inputs of production, we mean broadly land, labour, capital and entrepreneurship. The prices of these factors are called rent, wage, interest and profit, respectively, and the determination of each of these factor prices is discussed under a separate body of theory.

In order to determine these factor prices, we need to understand the fact that product markets and factor markets are interdependent; that is, if anything happens in the product market (for example, an increase in product price), it will have an impact on the factor market and vice versa.

Among these four factors, labour is considered as a variable factor, and the rest of the others are known as fixed factors, which can be varied in the long run. We begin our analysis in this module with the determination of the price of the variable factor, i.e., labour.

Like product price, the determination of the equilibrium price of labour follows the basic rules of demand and supply. However, the derivation of demand and supply of this labour input differs from that of output.

Output demands are derived from consumers’ preferences, and the supply of outputs is derived from the profit maximization of firms, while the demand for labour input is derived from the output market. We will first examine the determination of wages in the competitive market framework. Subsequently, we will incorporate the assumption of market imperfection in our analysis.

In order to determine the price of labour, i.e., wage, we have to explain the so-called marginal productivity theory of distribution. This theory postulates that when perfect competition prevails in product and factor markets, factors (labour inputs) are paid according to the value of their marginal product. The equilibrium factor price, wage, is determined by its market demand and supply schedules. The market demand and supply schedules are derived by aggregating individual demands for labour and supplies of labour.

In the following sections, we will first derive demand for labour by a single firm and then, by aggregating individual demands, will reach market demands. Similarly, on the supply side, we will first derive the individual labour supply curve and then the market supply curve.

Demand Curve for Labour in a Perfectly Competitive Market

We will analyze the demand for labour in two different situations:

• (i) When labour is the only variable factor of production,
• (ii) When there are several variable factors.

Demand of a Firm for a Single Variable Factor:

In this context, we have to adopt the following assumptions to derive the labour demand curve:

• The product market is perfectly competitive, and only one good Y is produced in that market.
• The goal of the producer is to maximize profit.
• Labour is the sole variable factor whose market is perfectly competitive.
• Production technology is constant.

Since the labour market is perfectly competitive, the wage, 𝑤̅, is given to all firms. This implies that at the given market wage rate, the firm can hire any amount of labour it wants. Therefore, the supply curve of individual labour will be perfectly elastic and parallel to the horizontal axis at 𝑤̅ (figure-1).

For a given technology, the production function follows the so-called law of diminishing return to factor shown in Figure 2. The slope of the production function reflects the marginal.

Productivity of labour (𝑀𝑃_𝐿) which is decreasing as labour input is employed more and more. If we multiply the given product price ̅𝑃̅𝑌̅ with the 𝑀𝑃_𝐿 at each level of employment, we obtain the value of the marginal product curve (Figure 3).

Since the producer is a profit maximizer, he will employ labour until the total revenue generated exceeds the total cost and the gap between them is highest. That is, a producer will hire labour up to the point at which marginal revenue equals to the marginal cost of the production. The formal derivation of profit maximization with respect to labour is given below.

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

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

Note that under perfect competition, since ̅𝑃̅𝑌̅ = 𝑀𝑅 (marginal revenue), the value of marginal productivity of labour (VMP_L)and the marginal revenue productivity of (MRP_L) are identical. The above marginal condition of profit maximization is shown at point 𝑒 in Figure-4 & 5. We obtain the profit-maximizing level of labour 𝑙^∗ from the equilibrium condition. To the left of 𝑙^∗, 𝑤̅  < VMP_L. Hence, the producer has a scope to increase profit by hiring more workers. By contrast to the right of 𝑙^∗ the̅ 𝑤̅̅ > VMP_L and the entrepreneur’s profit will be reduced. Therefore, profit will be maximized when 𝑤̅  =  VMP_L at 𝑙^∗ level of employment.

Therefore, it can be said that a profit-maximizing firm will hire units of variable factor input until the point is reached at which the value of a marginal product of the input is exactly equal to its price. For a given wage rate or supply of labour, the competitive firm determines the quantity of labour demand by equating the VMP_L to the wage rate. When the wage rate is 𝑂𝑤₁ (Figure-5), the firm will hire 𝑂𝑙₁ units of labour to equate the VMP_L to the given wage rate (at point𝑒₁). Similarly, if the wage rate falls to 𝑂𝑤₁, the firm will employ 𝑂𝑙₂ units of labour at the new equilibrium point 𝑒₂. In this way, we have the individual demand curve for labour (when labour is the single variable factor) established by the VMP_L curve.

Demand of a Firm for Several Variable Factors:

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 VMP_L curve. This is 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, 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 Figure 6.

Assume that the 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^/₁. This will rotate out the iso-cost line from 𝑎𝑏 to 𝑎𝑐 by pivoting around 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 Figure 6. 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 to 𝑎𝑐 and also tangent to the initial iso-quant 𝑌₁ so that it shows the new factor price ratio and at the same time maintain the initial level of production. The new iso-cost line 𝑑𝑒 is tangent to the isoquant 𝑌₁ at 𝑒^/₁. The movement from 𝑒₁ to 𝑒^/₁ 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 Figure 6, the firm increases the level of employment from 𝑂𝑙₁ to 𝑂𝑙^/₁. Next, the movement from 𝑒^/₁ 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 and is able to produce large output. This is shown by the rise of labour from 𝑂𝑙^/₁ to 𝑂𝑙₂ and the increase in capital from 𝑂𝑘^/₁ to 𝑂𝑘₂.

However, the story does not end here. There is another prominent effect, i.e., the profit-maximizing effect arising as a firm increases its expenditure in order to maximize profit. More intuitively, when the wage rate falls, the marginal cost of production declines. In Figure 6, it is shown by the downward shift of marginal cost curve 𝑀𝐶₁ to 𝑀𝐶₂ for a given output price̅ 𝑃̅̅𝑌̅. The rectangle 𝑓 𝑌₁𝑌₂ 𝑔 in Figure 7 is the additional expenditure made by the firm to attain the profit-maximizing level output 𝑌₃.

Going back to Figure 6 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 show how these three effects help us to derive the labour demand curve. 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 Figure 3.2.3 this is shown by the rightward shift of 𝑉𝑀𝑃_𝐿 curve when the wage rate declines from 𝑤 to 𝑤^/ and to 𝑤^// for a given level of̅ 𝑃̅𝑌̅. For different wage rates, we have corresponding levels of employment by equating the wage lines with the separate VMP_L curves. Suppose for three different wage rates 𝑤, 𝑤^/, and 𝑤//, we get three different equilibrium points 𝑒₁, 𝑒₂ and 𝑒₃ on the three separate VMP_L curves. Thus, by joining 𝑒₁, 𝑒₂ and 𝑒₃, we have the new demand curve for labour in case of several variable factors.

The Market Demand for a Factor

Unlike the market demand for output, the market demand for factors like labour is not the simple horizontal summation of the demand curves of individual firms. This happens due to the fact that when the price of input, say wage, declines, it is not only for a single firm that wants to hire more labour, but all other firms also seek to employ more workers. This resulting increase in the employment of labour will lead to an expansion of aggregate output. Thus, the market supply curve of output will be shifted to the right, and consequently, the price of output 𝑃_𝑌 will fall.

This reduction of output price will depress the value of marginal productivity of labour at all levels of employment and also for all individual firms. All these facts related to individual as well as market demand for labour are clearly depicted in Figures 9(a) & 9(b) respectively. The straight line dl in Figire-9 (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 behave in the same manner, the market supply of output increases and product price 𝑃_𝑌 falls. This will result in a downward shift of 𝑉𝑀𝑃_𝐿 curve or 𝑑_𝐿 curve to the left. The firm is now at point 𝑒₂ on the new demand curve for labour 𝑑^/₂ and hires 𝑂𝑙₂ units of labour at wage rate 𝑤₂ in Figure-9 (a).

Next, we see what happens in the market demand curve for labour, which is shown in Figure-9 (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 9 (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^/₂, and at 𝑤₂, the market demand for labour will be 𝑂𝐿₂ units.

Supply of Variable Factor Input (Labour)

In the previous sections, we concentrated our efforts on deriving the demand curve for variable factor labour at the individual firm level as well as the market level. Now, we finally focus on the derivation of the market supply curve of the variable factor in order to determine the equilibrium price of the variable factor. The nature and characteristics of the supply curve of variable factors like raw materials and various intermediate goods are the same as the supply curve of any commodities, while the supply curve of the variable factor like labour is derived by following somewhat different principles. In this section, we first derive the individual labour supply curve and then the market supply curve of labour.

The decision to supply labour depends on a worker’s choice between the amount of work effort and the time he will spend for leisure with the changes in the wage rate. Therefore, we can derive the supply of labour by an individual with the help of the indifference curve approach.

In Figure 10, along the horizontal axis, we measure the hours available for leisure and work over a given period of time, say 𝑂𝑇. Along the vertical axis, we measure money income. Say, for instance, if an individual spends entire 𝑂𝑇 time for work and zero hours for leisure, she will earn a maximum of 𝑂𝑀₀ level of total money income. The slope of the straight line joining the points 𝑀₀ and T shows the wage rate, i.e.,

𝑤 = 𝑂𝑀₀ / 𝑂𝑇

Now we draw indifference curve 𝐼₀, which is tangent to the 𝑂𝑀₀ line at point 𝑎. Assume that the initial wage rate is 𝑤₀ at which the individual enjoys 𝑂𝑙₀ hours of leisure and puts 𝑙₀𝑇 hours of work effort. Suppose that the wage rate increases to 𝑤₁. The individual will move to the higher indifference curve 𝐼₁, which is tangent to the new income line 𝑀₁𝑇 at 𝑎^/. When money income increases due to an increase in wage rate, the decision to choose more work effort or leisure depends on income effect and substitution effect. The substitution effect tells us that when the wage rate increases, leisure will become relatively costlier as compared to earning money income by putting in more work effort.

On the other hand, the income effect happens because the increase in the wage rate will raise the purchasing power of the individual, and with a higher income, she can buy many goods, including leisure. Initially, when the wage rate increases, the substitution effect dominates the income effect and leads individuals to give more work effort. This is shown in Figure-4.1, in which at point 𝑎^/ individual earns 𝑂𝑀₁ level of income corresponding to 𝑂𝑙₁ hours of leisure. Up to the wage rate 𝑤₂, the substitution effect continues to be higher than the income effect, and the individual also continues to add more effort to earn more income.

However, at a very high wage rate, say 𝑤₃, the individual moves again to a higher indifference curve 𝐼₃, and the equilibrium takes place at point 𝑎^/// and at this tangency point, the income effect starts to dampen the substitution effect. This results in a reduction of work effort from 𝑙₂𝑇 hours to 𝑙₃𝑇 hours with the increase in money income to 𝑂𝑀₄ level. If we join all the tangency points 𝑎, 𝑎^/, 𝑎^//, and 𝑎^///, we can obtain a curve denoted as 𝐿𝐿 which mirrors the labour supply curve 𝑆_𝐿. From the wage rate 𝑤₀ to 𝑤₂, the labour supply curve will be upward rising. After that, it starts to bend in a backward direction, as shown in Figure 11.

Although both individual and market labour supply curves may have a backward bending shape, in general, the market supply curve of labour appears as upward rising in the long run. This occurs due to the fact that with the rise in wages in the long run, young workers may be attracted, or old workers may increase their effort in enhancing their skills for sustaining at their present job. Therefore, in our analysis, we consider the upward-sloping market supply curve shown in Figure-12.

Determination of Equilibrium Factor Price (Wage) in Perfectly Competitive Markets

We are now able to determine the equilibrium factor price in a perfectly competitive market by combining the market demand curve and supply curve for variable factor labour derived in previous sections. Figure-5.1 shows that the interaction between the market demand for labour 𝐷𝐿 and the market supply of labour 𝑆_𝐿 determine the equilibrium factor price 𝑤^∗ corresponding to 𝑂𝐿^∗ level of employment of labour.

Read More in: Advanced Microeconomics

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3. Roles of Labour Union in Factor Market