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# The marginal productivity theory of distribution

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Neo classical theory of distribution, MP, Product exhaustion

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### The marginal productivity theory of distribution

1. 1. Prof. Prabha Panth, Osmania University, Hyderabad
2. 2.  The Neo-classical theory of distribution is based on Functional distribution of Income.  The distribution of income shows how total output in the market, is divided among owners of factors of production.  E.g. workers own labour, capitalists own capital, rentiers own land.  Total Product or Output has to be completely distributed to all factors.  No residues after all factors are paid.  So what should be the basis to achieve such factor payments? 2Prabha Panth
3. 3. Assumptions: 1. Derived Demand for factors, i.e. Factor D based on demand of what they produce, 2. Law of Variable Proportions applies. 3. Perfect competition in both product and factor markets. 4. Factors are independent of each other. 1. All factors are homogenous, and divisible. 2. All factors are natural factors, including capital. Capital is not manmade capital, (this gave rise to many debates on how realistic this assumption is). 5. Technology is given. 6. Ceteris paribus. 7. MP of factor = Factor price, is given by individual factor markets. (e.g. w = MPL determined in the labour market, r = MPK, determined in the capital market). D and S of factors determines factor price. 3Prabha Panth
4. 4. Demand for factors by firms: Prabha Panth 4 Taking a factor input, say labour, as more L is employed: 1) Total cost increases, as more wages are to be paid. 2) Output increases. In the figure, assuming ceteris paribus, as L increases, TP increases, because MP is still positive. TC also increases. Multiplying MP with price of x gives VMPL. VMPL = MPL × Px The firm will employ L up to the point where VMPL = MC i.e. w. Point A. Same applies to capital. VMPL w 0 MPL A Qx Labour MC Q Figure 1
5. 5. Firm’s Equilibrium 5 For a firm, the Supply curve of L is perfectly elastic. Also factor price is given. VMPL w 0 A Qx Labour w Figure 2 B L2L1 Profit Conditions for Firm’s Equilibrium 1. First order condition or Necessary Condition: VMPL = PL (i.e. VMPL = w) 2. Second order condition or Sufficient Condition: VMPL should be declining. So equilibrium of the firm is at B, and the firm employs 0L2 of labour. It earns abnormal profits of AB. Same applies to capital.
6. 6.  The downward sloping part of the VMPL is the firm’s demand curve for the factor (labour).  If the factor price (w) falls, the demand for the factor increases, up to the point where the VMPL = Price of the product.  If w falls to w2 from w1, due to external reasons, then the firm increases employment of labour, from 0L1 to 0L2.  This applies to all factors of production. Prabha Panth 6 VMPL w2 0 A Qx Labour w1 Figure 3 B L2L1
7. 7. The Adding up Problem or Product Exhaustion theorem  As assumed, if each factor is paid individually, would the sum of total factor payments be equal to the total product?  In other words, would the value of total product (TR) be equal to total factor payments?  or will there be excess remaining, (surplus then who gets it?)  or will there be insufficient funds to pay the total factor cost (i.e. deficit, then how to pay the factors?)  Wicksteed applied Euler’s theorem to show Product Exhaustion. According to him:  If each factor is paid up to the point were its VMP=Pf, then TR will be equal to TC, where TC = VMP of all factors in long run perfect competition. Prabha Panth 7
8. 8. 8 MPPL w 0 Qx Labour Total profit Figure 4a B L Total wages A MPPK r 0 Qx Capital Total Wages Figure 4b B K Total profits A In Fig. 4a, w = MPP of L, and total wages paid to Labour = 0wBL. The surplus wAB is paid to capital = total profit, In Fig 4b, r = MPP of K, and total profits = 0rBK. The residue rAB is paid to labour = total wages. The question is will owBL of fig. 4a = rAB of fig 4b? (total wages) Similarly will wAB of fig 4a = 0rBK of fig 4b? (total profits), since both factor prices are determined independently.
9. 9.  Wicksteed assumed that: 1. The production function is linear, and homogeneous, such as the Cobb-Douglas production function. This denotes that there are constant returns to scale throughout the production function, 2. Each factor price = VMP of that factor. VMPL = w, VMPK = r, etc. w = Q/L, r = Q/K, etc. 3. Applying Euler’s theorem, then the sum of the partial derivatives, multiplied by the quantity = the original amount. Then TR = wL + rK + …+…. = (Q/L)L + (Q/K)K + …. + = VTP or TR Prabha Panth 9
10. 10.  Assume a linear, homogeneous production function, such as Cobb Douglas production function. Q = F(K,L) = AKL PK = r = MPK = Q/K = AK -1 L Similarly, PL = w = MPL = Q/L = AK L-1 Prabha Panth 10
11. 11. Applying Euler’s theorem: (Q/K)K + (Q/L)L = Q = (AK -1 L .K) + (AK L-1.L) =  (AK L ) + (AK L) = + (AK L) Q = + (Q) Value of output = value of inputs or factor prices Prabha Panth 11
12. 12. If the above function exhibits: 1) Constant returns to scale: + = 1, so RHS = LHS, (TR = TC), and value of output = sum of factor incomes. 2) Increasing returns to scale:  + > 1, then sum of factor incomes will be greater than value of output. TC > TR, Q <  + (Q). The TR earned will not be enough to pay factor incomes, and there will be deficit. 3) Decreasing returns to scale:  + < 1, then sum of factor incomes will be less than value of output, Q >  + (Q) or TR. Thus there will be undistributed surplus after paying the factor incomes. Only if there are constant returns to scale, will the entire product be exhausted, when output is distributed to factors as per their marginal productivity. Prabha Panth 12
13. 13.  According to Wicksteed, LRAC is linear, i.e. constant returns to scale.  Otherwise TR TC, and there will be surplus or deficits.  But a linear LRAC cannot be used to achieve equilibrium. So there is a conflict here.  Walras and Wicksell tried to solve this in the following way: “The condition required for marginally determined rewards to exhaust the total product, i.e. CRTS, is fulfilled only at one point, i.e. the minimum point of the LRAC of a perfectly competitive firm in long run equilibrium”  That is at point C in figure 4. Prabha Panth 13
14. 14.  If product exhaustion has to take place, the production function should exhibit constant returns to scale throughout.  The LRAC should be a horizontal straight line.  Samuelson and Hicks stated that if there is perfect competition in the product market, LRAC is U-shaped curve, to fulfill conditions of equilibrium. (i.e. MC = MR, and MC).  The U-shaped LRAC exhibits all three types of returns – increasing, constant and decreasing.  At C, in the figure, TR = TC wL + rK) = 0PCQx 14 LRMC LRAC C, R 0 P Q AR=MR C Qx Figure 5 Samuelson Hicks solution
15. 15. Assuming 2 factors of production, K and L: TR = TC = P x Q = wL + rK According to MP theory, w = VMPL= P. MPPL r = VMPK = P.MPPK Therefore, P x Q = L(P.MPPL) + K(P.MPPK) Dividing both sides with P Q = L(MPPL) + K(MPPK) Thus, for a given price, if factors are paid according to their MPPs, the total payments to K and L would be equal to TR. The entire product would be exhausted. This is possible only at the lowest point of LRAC, where the firm experiences CRTS. At this point, product exhaustion is satisfied, and it is not necessary to assume a linear, homogeneous production function. Prabha Panth 15
16. 16.  If there the product market is a monopoly or other imperfect markets, there will be no product exhaustion.  Factors and consumers will be exploited.  If the firm is not operating at constant costs, but has increasing or decreasing returns, again there is no product exhaustion. Prabha Panth 16