THE SHADOW WAGE RATE

In imperfect or distorted market or in dual economy, such as typical developing country, where the marginal product of labor differs between sectors and in which saving is suboptimal there are two aspects to the measurement of the social cost of the use of more labor in projects:

1.         The first is the opportunity cost of the labor in alternative uses, which could be the marginal product in agriculture, or perhaps the earnings to be had on the fringe of the industrial sector in the informal service sector (P_A).

2.         The second is the present value of the sacrificed saving that results if an attempt is made to maximize present output by equating the marginal products in the different sectors.

In practice, the shadow wage will lie somewhere between  P_A and W , depending on the value of S₀. To measure the relative valuation of future versus present consumption an approach suggested by A.K. Sen and supported by Little and Mirrlees is taken to calculate the present value of the future consumption gain arising from investment now, relative to the current consumption sacrifice.

Thus,

S₀  =  [C₁ / (1+ i)] +  [C₂ / (1+ i)²] + ... +  [C_t / (1+ i)^t]   

                                            C₀

= ∑ [C_t / (1+ i)^t]     ...         ...         ...         (4)

                      C₀

The value of S₀ will depend on the marginal product of capital, the length of the time horizon (T) taken, and the discount rate (i) chosen. The longer the time horizon, and the lower the discount rate, the higher S₀ and the higher shadow wage rate. If  S₀ = 3, for example, then assuming P_A and m to be very small, the shadow wage would be approximately two -thirds of the industrial wage. In the Little-Mirrlees approach to project appraisal, the shadow wage is the principal means by which the scarcity of foreign exchange is allowed for . The lower the shadow wage, the greater the use of domestic resources.

RELATION BETWEEN CONSUMPTION IN INDUSTRY AND AGRICULTURE

Marginal propensity to consume out of wages is assumed as unity and that all 'profits' are saved. It has also been assumed that consumption falls in agriculture to the extent of the migrant's consumption. Practically the marginal propensity to consume out of wages may be less than unity; the marginal propensity to save out of 'profits' may be less than unity; and consumption in agriculture may not fall by the extent of the migrant's consumption. A more general formulation of the shadow wage is called for which allows for these possibilities.

The change in consumption in industry as more labor is employed  is given by,

C = Wc + c* (P_I - W)                                      ...         ...         ...         (5)

Where W is the industrial wage; (P_I - W) is 'profit' per worker; c is the marginal propensity to consume out of wages, and c* is the marginal propensity to consume out of profits.

The change in consumption in agriculture as labor is drawn away may be written as:

m = d ( 1 - c')                                                  ...         ...         ...         (6)

Where d is the consumption of the migrants from agriculture, and c' is the propensity to consume of those remaining in agriculture. Clearly if those remaining increase their consumption by as much consumption as migrants 'release', so that c' = 1, agricultural consumption will not fall as labor migrates.

We know that the shadow wage is equal to the loss of agricultural output, plus the increase in consumption, less that part of the increase in consumption which is treated as benefit and is given by

P_I = P_A + (C - m) (1-1/S₀)                                ...         ...         ...         (7)

Putting the values of equations (4) and (5) in equation (6) we get,

P_I = P_A + [Wc + C*(P_I - W) - d (1 - c')](1-1/S₀) ...      ...         ...         (8)

This is the optimal shadow wage (W*).