NUMERICAL EXAMPLE SHOWING THE APPLICATION OF THE LITTLE - MIRRLEES AND UNIDO APPROACHES TO PROJECT APPRAISAL

(Source:Thirlwall A.P. Planning, Resource Allocation and Choice of Techniques, page no. 212/213)

Let us assume that world prices are measured in dollars ($) and domestic prices in rupees  (R), that the shadow exchange rate is 1.25 rupees per $, and that the standard conversion factor for converting non-traded goods prices into world prices is 0.8, i.e. 1/1.25. We further assume that;

1.         All output is exported with an annual value of $3000.

2.         The cost of the investment has a foreign component of $1000 and a local (non-traded) component of 1000 rupees.

3.         There are traded inputs of $1000 and non - traded of 1000 rupees.

4.         The accounting rate of interest is equal to the consumption rate of interest.

We can now apply out net present value formula in equation (1) using the two approaches. Using the Little-Mirrlees approach we must convert all the prices of non-traded goods into world prices using the standard conversion factor of 0.8, and using the UNIDO approach we must convert all values at world prices ($) into domestic prices using the shadow exchange rate of 1.25. We will do the analysis for three periods only (as shown in table). The results of using Little-Mirrlees approach and the UNIDO approach will differ only to the extent that the shadow exchange rate is different from the actual exchange rate. To obtain net present value, the net benefit streams in years 1 and 2 must be discounted by the appropriate discount factors which we here to be the same using both approaches. Assuming a discount rate of 10 percent we have

Using Little-Mirrlees

NPV  = $1200  +  $1200  -  $1800 = $282.6

              (1.1)         (1.1)²

Using UNIDO

NPV  = R1500  +  R1500  -  R2250 = R353.2

              (1.1)         (1.1)²

A comparision of the Little-mirrlees and UNIDO approaches to project appraisal

Little - Mirrlees				UNIDO
Cost of investment (K)	Year 0	Year 1	Year 2	Cost of investment (K)	Year 0	Year 1	Year 2
1. Foreign component	$1000			1. Foreign cost converted into Rupees at shadow exchange rate of  1.25	R1250
2. Local component =1000 Rupees x conversion factor  0.8	$800			2. Local Component	R1000
Input Costs (C)				Input Costs (C)
1. Traded inputs		$1000	$1000	1. Traded inputs converted into Rupees at shadow exchange rate of 1.25		R1250	R1250
2. Non-traded inputs = 1000 rupees X conversion factor 0. 8		$800	$800	2. Non-traded inputs		R1000	R1000
Benefit flow (V)		$3000	$3000	Benefit flow in Rupees		R3750	R3750
Net benefit	$1800	$1200	$1200	Net benefit	R2250	R1500	R1500

The Little-Mirrlees result will yield the same rupee value if the actual exchange rate of dollars into rupees is equal to the shadow exchange rate of 1.25.

Little and Mirrlees conclude their own evaluation of the two approaches by saying 'there is no doubt that the two works adopt basically the same approach to project evaluation. Both treatments single out the values of foreign exchange, savings and unskilled labor, as crucial sources of a distorted price mechanism. Both go on to calculate accounting prices which will correct these distortions and both carry out these corrections in an essentially similar manner. Both advocate DCF (Discounted Cash Flow) analysis and the use of PSVs (Present Social Values). Both method works making explicit allowance for inequality and distributional considerations in project choice through manipulation of the shadow wage. In the UNIDO approach this is done through giving greater weight to the increased consumption of the poor than of the rich, which reduces the present value of lost consumption. In the Little -Mirrlees approach distributional considerations are taken account of by working out the value of   S₀ taking account of the standard of living of the particular extr4a workers employed., Extra consumption of the rich (or out of  profits) may be given no future value  and treated as a pure cost. This would reduce S₀  and reduce the shadow wage, favoring more labor intensive projects and present consumption.

THE SHADOW EXCHANGE RATE

The shadow exchange rate or the shadow price of foreign exchange is required in the conventional approach to social cost-benefit analysis where goods are measured in domestic prices so that traded goods in foreign prices need to be converted into domestic prices, but not necessarily at the official exchange rate if there are distortions in the foreign exchange market and the official rate does not reflect the true opportunity cost of foreign exchange. The shadow price of foreign exchange may be written as:

P_F = ∑f_i [P^D / P^E]  ...         ...         ...         (14)

Where P^D is the domestic price of the ith import and P^E is the official exchange rate price of the  ith import and the fis are the weights reflecting the proportional importance of the commodity in the total import bill.

WORTH VALUING OF ALL GOODS AT WORLD PRICES

 The justification in the Little-Mirrlees approach for valuing all goods at world prices is that it avoids the use of the exchange rate in order to value in a single currency some goods measured at world prices (traded goods) and others measured at domestic prices (non - traded goods). To avoid the trouble worth using a foreign exchange rate some economists feel it is a lot of trouble for accuracy because of need to disaggregated non - traded goods into their traded and non - traded inputs which requires input - out put data  which do not exist in many cases. It may be just as accurate, it is argued, to convert the prices of non-traded goods into a single currency by the exchange rate approximately for under or over valuation.

THE EQUIVALENCE OF LITTLE-MIRRLEES FORMULATION OF SHADOW WAGE AND UNIDO APPROACH

In the UNIDO approach, the optimal shadow wage is given by,

W* = P_A + s* (P^INV - 1)W       ...         ...         ...         (9)       

Where  s* is the capitalists' propensity to save; W is the wage and  P^INV measures the price of investment in terms of consumption. Assuming that the propensity to save of capitalists is unity (s* =1) and all wages are consumed, equation (8) may be written as:

 W* = P_A +  (P^INV - 1) C          ...         ...         ...         (10)     

Where (P^INV - 1) C represents the present value of aggregate consumption lost              assuming (P^INV >1) because of  increased consumption of workers which reduces saving for future consumption.

Now from above for the two approaches to give equivalent cash flows, whichever numeraire is taken, P^INV must be equal S₀ , so that the equation (9) may be written as:

W* = P_A + (S₀ - 1)C                ...         ...         ...         (11)

Now Little - Mirrlees formulation of the shadow wage from equation (6) is:

W* = P_A +  (1 - 1/S₀)(C-m)     ...            ...      ...         (12)     

Which, on the assumption that  P_A = m , may be written as:

W* = P_A / S₀ + (1 - 1/S₀)C      ...         ...         ...         (13)     

Comparing equation (12) with the UNIDO equation (11) , it can be seen that the UNIDO formula is  S₀ times the Little-Mirrlees formula because the UNIDO numeraire is 1 / S₀ times as valuable as the Little-Mirrlees numeraire.

The calculation of the shadow wage will be different according to the two approaches assuming (P^INV = S₀ ) only to the extent that they classify goods into traded and non-traded in a different way and that the standard conversion factor (SCF) to convert    non - traded goods into world prices is not the reciprocal of the shadow exchange rate (SER) to convert traded goods prices into domestic prices.

VALUATION OF PRODUCTION FORGONE AND INCREASED CONSUMPTION

Using the Little - Mirrlees approach, the valuation if agricultural production forgone (P_A), and the value of increased consumption (C - m ), must be measured at world prices. To value agricultural production forgone, a bundle of goods must be taken to represent the marginal physical product which then must be priced according to whether they are traded or non - traded, as described earlier. To value the increased consumption also requires taking a bundle of goods, distinguishing between traded and non - traded goods. To recap, standard conversion factors are recommended to re price non - traded goods, which are calculated as the average of ratios of the domestic costs of the non - traded goods in question to the world price of these inputs costs

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*).

CHOICE OF NUMERAIRE

All benefits and costs have to be converted into a common unit of account or numeraire. The traditional approach is to take consumption measured at domestic market prices as numeraire. By contrast, Little and Mirrlees take the savings contribution of the project as the objective and accordingly take as numeraire what they call ' uncommitted social income , measured at border prices (or free foreign exchange in the hands of government). Clearly the numeraire taken will have a bearing on the estimation of the Net Present Value formula.

To evaluate NPV we first find the value of benefit flow V_t as shown from the  equation (1). Conventional approach is considered (as UNIDO guidelines). Annual flow of social benefit is measured in consumption stream in domestic currency. If Little Mirrlees approach is considered we take the value of benefit as the projects cash flow valued at world price with future consumption revalued in terms of saving. Both the approaches give the same result as to the price of investment in terms of consumption.

Since income and consumption are expected to rise over the time we discount future benefits and costs with their aggregates as;

B_t = B₀ + V₁ B₁ + ... +V_T B_T   ...         ...         ...         (2)

Where the weights attached to future consumption streams, V₁ , ...   ...  ..., V_T  , decline over time. The weights V₁ , ...   ...  ..., V_T   are discount factors.

If weights decline over time at a constant percentage rate (r), then it can be shown as,

B* = ∑ [B_t / (1+ r)^t]            ...         ...         ...         (3)

Where r = social discount rate.

If the Little-Mirrlees approach is taken future benefits and costs should not be discounted by the so-called consumption rate of interest (CRI) but by the rate at which the marginal utility of public saving falls, since public saving expressed in foreign exchange is the numeraire and this rate is called accounting rate of interest (ARI) which is equal to rate of return on public money. Hence project with positive net present value at this rate should be chosen. In Little-Mirrlees ARI is determined by a process of trial and error such that its value does not pass more projects as profitable than the investment budget allows.

If investment is totally done on consumption the social investment cost is measured by the current sacrifice of consumption, if it is numeraire.

If investment of one project is partly of another then a part of cost of the sacrifice of consumption is deferred until the time displaced investment would itself have yielded consumption.