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Simple Analytics of Public Sector Pricing in a Capacity Constrained Mixed Economy

This exercise on public sector pricing focuses on a mixed economy which faces explicit capacity constraints in one or more sectors. Of particular interest among the results is that the decisions on either pricing the public sector product or the mode of financing losses do not have simple consequences for the price of the private sector product or the measure of profitability in either sector.

Simple Analytics of Public Sector Pricing in a Capacity Constrained Mixed Economy

This exercise on public sector pricing focuses on a mixed economy which faces explicit capacity constraints in one or more sectors. Of particular interest among the results is that the decisions on either pricing the public sector product or the mode of financing losses do not have simple consequences for the price of the private sector product or the measure of profitability in either sector.

VISHWANATH PANDIT, BADAL MUKHERJI

IIIII
IntroductionIntroductionIntroductionIntroductionIntroduction

T
he last many years have witnessed a wave of new economic policies, aimed at liberalisation and structural adjustment, being undertaken in many developing economies. The centrepiece of the guiding philosophy of these policies has been to allot a wider space and role to market forces. Consequently, the so-called “commanding heights of the public sector” and more widely the role of the state as an economic agent in these economies are seen to be de-emphasised. These developments have given rise to a debate on the role of the state (versus that of the market) in shaping the destinies of economies and societies [Datta-Chaudhuri 1990]. Whatever may be the outcome of this debate, it is premature to assume away the continued role of the government and more particularly of the public sector in most developing economies over the near future. For, the public sector in countries like India, China, Russia and many developing economies is still very large and in command of some basic goods crucial for both production and consumption. India’s recent experience with regard to pricing of petroleum products, foodgrains under the public distribution system, power tariffs and coal prices all bear witness to this fact. Importantly, the decisions in this regard are not simple insofar as they have on the one hand important welfare effects and on the other complex fiscal implications. Some of these issues which remain relevant even under the new economic policy regime were discussed by us earlier [Mukherji, Pandit and Sundaram 1992].

As a sequel to that paper the present exercise focuses on a mixed economy which faces explicit capacity constraints in one or more sectors. The economy is mixed in the sense that one commodity producing sector (of two) is owned by the state and its profits are a net revenue of the state. But if one assumes as we do that the economy is indecomposable then a capacity constraint even in one sector constrains output in all sectors. For a less developed economy it is quite natural to assume that the constraints on capacity are imposed by inadequate capital stock, i e, plant and machinery.

In the earlier paper we had outlined an analytical framework within which one can answer questions relating to the determination of output and prices when the government of a mixed economy tries to mobilise resources by acquisition of commodities (or services) in specific physical magnitudes. The proposed framework highlighted an interaction between the production system, income generation, levels of demand and price adjustments, on the one hand, and government’s expenditure decisions subject to its budget constraint and implied variations in money supply on the other. It was assumed all along that capacity in all sectors was unconstrained so that the system was primarily driven by demand. The paper also analysed how factor shares of income varied under alternative regimes. This paper will subsequently be referred to as MPS.

The questions considered in this and the earlier paper have been the subject of debate in India since the early 1980s. The main contributions in this debate, summarised earlier (in MPS) were either based on partial analysis or not focused on broader issues raised here [Chetty and Ratha 1987; Dasgupta 1992; Jha and Mundle 1987; Panda and Sarkar 1990; Pandit and Bhattacharya 1987 and Sundaram and Tendulkar 1987]. We established the important proposition that in a regime of excess capacity it is preferable to finance increased government expenditure by a budgetary deficit than by raising administered prices of public sector goods and services not only because the former leaves prices unchanged but also because the latter depresses output.

IIIIIIIIII
The ModelThe ModelThe ModelThe ModelThe Model

For reasons of continuity and analytical simplicity we shall consider a miniature closed economy with two commodities and one service. Commodity 1 is produced in the public sector. We stick to the earlier notation which was as follows: xi (i=1,2) Gross levels of output. ci (i=1,2,3) Total consumption or demand. cip(i=1,2) Private consumption or household demand. gi (i=1,2) Real government purchases. pi (i=1,2,3) Unit prices. w Wage rate. θi (i=1,2) Unit mark-up rates. Wo Government expenditure on wages and salaries for administration, etc. ms, md Stock of money supplied, demanded. Y Total Household (private) income.

It should also be useful to recall the assumptions underlying this as well as the earlier exercise. They are: (i) The production structure is of a Leontief type and indecomposable specified by the intermediate input coefficients aij (i, j=1,2) and the labour coefficients aoi (i = 1, 2); (ii) The utility function of households is of a Cobb-Douglas type yielding constant budget share demand functions. These shares are denoted by α, β, γ and δ for the two commodities, one service and money; (iii) All savings are held as money because no interest-bearing financial assets exist;

  • (iv) Unit mark-ups are additive over prime production costs;
  • (v) Nominal wage rate w and government expenditure on wages and salaries Wo are exogenously fixed; and (vi) Government imposes no direct or indirect taxes.
  • At this stage it would be legitimate to claim that the assumption that there are no external transactions is unrealistic in today’s world. Yet, we have deliberately made this assumption so that the problem in its other dimensions is clearly set out. Needless to add that in further deliberations the model must assume an open economy.

    For the present analysis we impose the restriction that productive capacity in the two commodity producing sectors is subject to an effective ceiling. Thus, x1 ≤ x1* and x2 ≤ x2* which implies that net outputs c1 and c2 are subject to upper limits c1* and c2* respectively. The problem for the state then is to choose the markup rate θ1 and money supply m consistently with overall equilibrium. It should be obvious that the introduction of effective capacity constraints makes the problem far more difficult to handle but equally more relevant and meaningful to a developing mixed economy.

    After some manipulations meant to weed out the unnecessary relationships the formal model simplifies to:

    *

    c=(I-A)x x ≤ x (2.1) -1 o

    p =(I-A’) {wa + θ} (2.2) p= λ1p + λ2p + λ3w (2.3)

    s ' '

    m =Wo + p .g -B1x1 + mo (2.4) ’

    Y =Wo + wao x + p3c3 + B2 x2 (2.5)

    ⎛ a /p ⎞

    ⎜⎟

    c= 1(Y+m)+g

    ⎜⎟ o (2.6)

    f /p

    ⎝ 2⎠

    c3 = y(Y + mo)/p3 (2.7) d

    m= &(Y + mo) (2.8)

    g, c, x, θ, p, ao, x (in italics) which appear in (2.1), (2.2) and

    (2.6) are 2×1 vectors whereas A is a 2×2 matrix. It is easy to see that (2.1) gives the production system; (2.2) and (2.3) price formation; (2.4) follows from the government budget constraint;

    (2.5) determines private income; (2.6) and (2.7) demand for the two commodities and the service and (2.8) is a disguised saving function.

    Before we proceed any further it will be useful to take note of the following implication of the budget constraint. (2.4) gives us: m -mo + B1x1=Wo+pg +pg

    This, together with, B1=(1 -a11)p1 -a21p2 -wao1, from (2.2) gives us Δm+[1-a11)x1-g1]p1 = Wo + wao1x1 + (a21x1+g2)p2

    where Δm ≡ m – mo is the required increase in money supply.

    Notice that (1 – a11)x1 – g1 ≡ S1 is the quantity of product 1 sold by the public sector to the private sector. Similarly, a21x1 +g2 ≡ S2 is the quantity of product 2 sold by the private sector to the public sector. Now denoting public sector’s total wage bill on both production and administration by W= Wo + wa01x1 and assuming without loss of generality that government deficit directly adds to money supply, we have, Δm– p2S2 = W – p1S1 which can be expressed as:

    Proposition 1

    The excess (or shortfall) of increased money supply over the public sector’s purchase (of product 2) from the private sector must be exactly equal to the excess (or shortfall) of public sector’s total wage bill over its sales (of commodity 1) to the private sector.

    We shall see an important consequence of this in proposition 2 below. Note that if W > p1 S1 then Δm > p2 S2 and vice versa. Two distinct adjustments are possible in such a capacity constrained set up. These are:

  • (a) Mark-up rates θ1 and θ2 are endogenously determined so as to clear markets. This necessarily implies that suitable adjustments in m, satisfying the budget constraint (2.4), must accompany changes in θ1 and θ2. This implies that monetary policy is in this case passive.
  • (b) A second possibility is that the government supplies the available output of sector 1 at a fixed price p1*. This is indeed the case closest to what occurs for several public sector products. In the present discussion we assume p1* to be distinctly below its market clearing level so that we are dealing with a case in which output of the public sector is rationed to consumers.1
  • It is useful now to recall proposition 1 from our earlier paper (MPS) which says that no matter what regime is considered and which variables are fixed and which ones are left free to adjust and how, money demand will always equal its supply. In other words, households will always be induced to hold exactly as much money as the government decides to supply. Hence ms = md = m always holds. Bearing this in mind, let us consider the two cases as follows.

    IIIIIIIIIIIIIII
    Market Clearing Prices and Passive Money SupplyMarket Clearing Prices and Passive Money SupplyMarket Clearing Prices and Passive Money SupplyMarket Clearing Prices and Passive Money SupplyMarket Clearing Prices and Passive Money Supply

    In dealing with this set-up we need to note that the mark-up rates θ1 and θ2 and price levels p1 and p2 are jointly determined. While (2.2) gives us the supply prices, these have to match with the demand side given by (2.6) to ensure market clearing for given government demands g1 and g2. However, money supply is not a free variable as it depends on θ1, the mark-up rate on public sector production. Since θ2 depends only on p1, p2 and θ1, it is convenient to let θ2 to adjust ex post once p1, p2 and θ1 are adjusted. It is interesting to note that the asymmetry between θ1 and θ2 is due to the presence of θ1 in the government budget constraint. After some rearrangements we end up with the three equilibrium conditions as:

    (1− a11) p1 − a21p2 − B1 = wao1 ⎫

    ⎪⎪
    (& 1p − ag1) p1 − ag2 p2 + ax1B1 = aWo ⎬

    (3.1)

    and, − fg1p1 + (& 2p − fg2) p2 + fx1B1 = fWo⎪⎪

    where c1p ≡ c1 − g1 andc2p ≡ c2 − g2 are quantities of private demand for the two goods. The three equations yield the solution:

    ΔΔ Δ

    12 3

    p1 = , p2 = and B1 =

    ΔΔ Δ where Δ, Δ1 Δ2 and Δ3 are appropriate determinants.

    In particular

    Δ= &(f 1pS2 + a 2 pS1 − & 1pc2 p ) (3.2)

    where S1 and S2 denote, as before, quantities of product 1 sold by the public sector to the private sector and of product 2 sold by the private sector to the public sector, respectively. It turns out that if W denotes total wage income in the economy, then Δ1 =−a& 2 pW andΔ2 =−f& 1pW are both negative. Thus, the two price levels p1 and p2 are positive only if Δ < 0. This requires one of the following three conditions. First, using (3.2)

    S2 S1

    Δ< 0 iff & > f − a 2 p 1p

    cc

    which sets a lower bound to the propensity to save. The critical parameters here are the ratios of intersectoral transactions of the two goods to their private sector consumption. Second, if we let

    am S fm

    c =− a 1 and c =

    1p 2 p . 11p 2

    &p c &p

    Δ < 0 implies that m >p2 S2 − p1S1 which means that the private sector must hold as much money as the government is constrained to supply. This is obviously possible only if the propensity to save is adequately large. Finally, an equivalent condition is that

    aS1c2 + (& 2 − fa21x1)c1p

    g <

    2

    aS1 + (f + &)c1p

    which sets an upper limit to government demand for goods supplied by the private sector, given various technological and behavioural parameters.

    The foregoing results may effectively be summarised as follows:

    Proposition 2a

    Market determined prices for the two products will be positive only if the propensity to save is above a critical level so that the private sector is inclined to hold as much money as the government is constrained to supply to maintain its solvency.

    The above condition may be called the “solvency condition” which requires the value of total sales of the public sector (to the private sector) plus budgetary deficit (≥ 0) to strictly exceed the value of total purchases from the private sector. A moment’s reflection reveals why. In reality, in any mixed economy, the public sector not only has a very large presence in major infrastructure enterprises and elsewhere but also the sheer volume of intra-public sector trade is enormous. Taken singly, a lot of them can show a profit but only at each other’s or of the private sector’s expense. Analytically, the solvency condition requires us to set out the value of all these intra-public sector trades.2

    Considering the difficulty of ensuring positive profits in the regulated sectors, despite positive prices (proposition 2b) our concerns would seem to be justified. Thus, if the solvency condition holds, there is no way for the government to pay for its own pre-commitment to employees, i e (Wo+ wao1 x1). This is indirectly meant to ensure that the intersectoral terms of trade do not turn too much against the public sector. What should be obvious is that given other things, the government cannot induce the households to hold more than a certain amount of m. Thus, the saving propensity assumes importance. Hence non negative solutions in equilibrium will not be assured if the solvency condition breaks down.

    Turning now to the profit margins θ1 and θ2 let us first reiterate that conditions stated earlier only ensure the positivity of the two prices. This, by itself does not ensure that margins in the two sectors (θ1 and θ2) will be positive. Consider θ1 first. Since θ = Δ3/ Δ ,θ > 0 holds if and only if Δ3 < 0 , given that p1 and p2 are positive, i e, Δ < 0. To this end, note that

    11

    Δ3= &[wao1(& c1pc2p -f c1pg2 -a c2pg1) + Wo{a21 f c1p -(1-a11)a c2 p}]

    Δ = &[(a c2 pg1 + f c1pg2 -& c1pc2p) + x1{f a12x1p -a(1-a11)c2p}] (3.3)

    which implies that

    B1= Δ3 / Δ = -wao1 + & K W / Δ

    where K ≡ f a12c1p -a(1 -a11)c2p

    and, W ≡ W + wa x

    o o11

    Now, since with p1 > 0, p2 > 0 we have Δ < 0 and also W > 0, the sign of K is of critical importance. In particular K > 0 implies θ1 < 0. For positive θ1, K must be negative and large. Going back to the expression for Δ3 a necessary condition for θ1 > 0is

    c1p wa01(& c2p -f g2) + a21 f Wo a > (3.4)

    c2p (1 -a11)Wo + wa01g For θ2 > 0 we obtain

    f > c2p wa02(& c1p -a g1) + a a12Wo + a wx1[a02(1-a11) + a01a12]

    (3.5)

    c1p (1-a22)Wo+wa02g2+wx1[a01(1-a22)+a02a21]

    which looks quite analogous to the condition needed for θ1>0 except for the two additional terms appearing, the last in the numerator and the denominator. The intuition behind the two restrictions is that if the demand for either product is weak (α “too low” or β “too low”) then the profitability for either sector cannot be assured. We can summarise these results as follows.

    Proposition 2b

    Positive prices do not by themselves ensure positive profit margins. For either sector to earn positive profits it is necessary that the share of income allocated to the expenditure on the product of either sector remains above a critical minimum. In each case the relevant critical minimum depends on the technological parameters, policy variables, capacity constraints and other behavioural parameters.

    We now turn to comparative statics of the quantity constrained case. To this end, take total differential of the fundamental system (3.1), rearrange terms and end with

    ⎡ 1---1⎤⎡ dp ⎤⎡ 0 ⎤

    a11 a12 1

    ⎢ ⎥⎢⎥⎢ ⎥ ⎢&c1p -α g1 a g2 -a x1⎥⎢dp2⎥ = ⎢ -(a + &)p1dg1-a p2dg2⎥

    ⎢ ⎥⎢⎥⎢ ⎥(3.6)⎢ ⎥⎢⎥⎢ ⎥

    f g f g-& c-f xdB1-f p dg -(f + &)p dg

    ⎢⎣ 1 22p 1⎥⎦⎣ ⎦⎣ 11 22⎦

    Focusing on the comparative statics of the system, with respect to g1 and g2 we observe that

    ∂ p & p2p

    1 12

    = [(a + f + &)c2 p − f(1− a22 )x2 ]

    ∂ g afmW

    1

    ∂ p & p2p (1-)x

    1 21 a11 1

    = >0

    ∂ g2 f m(Wo + wao1x1) (3.7)

    ∂ p2 & p12p2 a12x1

    =-<0

    ∂ g a m(Wo + wao1x1)

    1 2

    ∂ p & pp

    2 12

    = [(a + f + &)c1p + a a12 x2] > 0 ∂ g2 af m(Wo + wao1x1)

    Observe that while price changes associated with changes in g2 are expected and unambiguous those associated with changes in g1 are indefinite or counter intuitive. Let us consider ∂ p2/∂ g1 first by examining how m adjusts to variations in g1. We have

    m=W +pg +pg -B

    o 11 22 1x1

    For simplification of algebra we set m0 = 0 in the above expression so that in substituting for p1, p2 and θ 1 their solution values, we obtain

    & c2 pc1p W

    m= (& c1pc2p + aS1c2p -f S2c1p) From this expression we can readily establish that: ∂ m-a& W a12x2c2

    2p

    =

    2 (3.8)

    ∂ g1 D

    where D = & c1pc2p + aS1c2p -f S2c1p

    Clearly ∂ m/∂ g1 < 0. It is this that explains why ∂p2/∂g1 < 0 holds under mild conditions. Clearly, the upward pressure on p2 due to increased demand is dominated by the downward adjustment in m, other things remaining unchanged.

    With regard to ∂p2/∂g1 we note that

    S2 a21x1 + g2 a + &

    ≡=

    (3.9)

    c -g f

    2 pc2 2

    Since x1, x2, c1 and c2 are held fixed, the ratio depends only on g2. Thus, a higher g2 tends to give a higher ratio so that ∂p1/∂g1 < 0. This is because while the primary impact of increased g1 on p1 is positive there is a secondary impact through

    m. As g1 rises, p1 rises along with θ 1. The latter causes m to fall. But since the effect of increased S1 on p2 is negative, p2g2 will decline – the magnitude of decline depending on how large g2 is. Thus, a large g2 will imply that the secondary negative impact on m and thus on p1 will outweigh the primary positive impact. The asymmetry between the two pairs of derivatives is thus to be traced to the budget constraint (with flexible m). While θ 1 affects it θ 2 does not.

    We show (below) that as g1 rises, if p1 also rises (under the very mild condition stated above) then θ 1 rises even more sharply (i e, ∂θ1/∂g1 > 0 if ∂p1/∂g1 positive). Therefore m falls. This reduces nominal income and therefore reduces p2 (with g2 held constant). As observed earlier, the root of this asymmetry lies in the fact that θ 1 enters the government budget constraint whereas θ 2 does not.

    Turning now toθ1 andθ2 as stated above. It is easy to verify that

    ∂ B p2 p

    1 12

    = [(a + f + &){1 -a11) c2p -f | I-A | x2} (3.10)

    ∂ g afmW

    1 It follows that ∂θ 1/∂ g1 > 0 if m

    a12a21

    (1 -a22)p x2<(a + f + &) + px2

    2 2 (3.11)

    & 1

    a11 Clearly, therefore, ∂p1/∂g1 > 0 implies ∂θ1/∂g1 > 0 Similarly,

    (a + f + &) (a21p12)

    ∂B / ∂ g = -< 0 unambiguously. (3.12)

    21

    a W

    The consequence on sectoral profits of a ceteris paribus increase in g2 are more complicated as they depend on the distribution of outputs and consumptions. Thus,

    2

    ∂ & pp

    B1 12

    =[a |I -A|x2 -(a + f + &)a21c1p]

    ∂ g af mW

    2

    ∂ B1

    So that > 0 iff (3.13)

    ∂ g

    2

    b21c1p a | I-A| c1p a | I-A |

    < or <

    x2 a + f + & x2 (1-y)a21

    where bij is the (i, j)th element of B = (I-A)–1. This means that the cumulative total need of the private sector product to support a unit increase of consumption of the public sector product is bounded above by the constant α/(α + β + δ) determined by the budget shares of the commodities. Similarly,

    ∂ & p2

    B2

    = 1 [(1-a22)(a + f + &)c1p

    ∂ g af mW

    2

    +a(a12(1-a22)x2-a21(1-a11)x1)] (3.14)

    Thus, a sufficient condition for ∂θ2/∂g2 >0 is x2 a21(1 -a11)

    > (3.15)

    x1 a12(1 -a22)

    i e, the private sector’s requirement of the public sector product exceeds the public sector’s requirement of the private sector product. This depends clearly on the technological coefficients particularly a12 and a21 as well as the levels of production in the two sectors. Let us now summarise the foregoing discussion as follow:

    Proposition 3

    In a regime of constrained capacity and flexible prices in both sectors, an increase in the purchase of the private sector product by the government (say, for investment) will raise prices in both sectors. However, an increase in the use of its own product by the government has a depressing effect on the price of the private sector product; the effect on the price of its own (public sector product) is positive if the prevailing purchase of the private sector product is below a certain magnitude and negative if it is above that magnitude – all other parameters remaining unchanged.

    Proposition 4

    If capacity is constrained and prices for both sectors are flexible, then an increase in the use of its own (public sector) product by the government depresses the unit mark-up rate for the private sector; and raises its own unit mark-up rate if it increases the price of its own product. However, the corresponding effects associated with an increase in the governments’ purchase of the private sector product are indefinite. The unit mark-up rate for the public sector product is raised (lowered) if the magnitude of this sector’s product for final use by the government is above (below) a certain level. The effect on the private sector’s unit mark-up rate too is positive if the quantum of intermediate requirement of the public sector product by the private sector exceeds the intermediate requirement of the private sector product by the public sector.

    IVIVIVIVIV
    Fixed Public Sector Price and RationingFixed Public Sector Price and RationingFixed Public Sector Price and RationingFixed Public Sector Price and RationingFixed Public Sector Price and Rationing

    We have so far considered the case in which outputs in the two commodity producing sectors are capacity constrained, but prices are assumed to be market clearing for both the commodities. This meant that the two mark-up rates θ1 and θ2 were endogenously determined so as to equate supply and demand in all the markets, including those for money and services. The consequences of the government decision to increase investment by raising either g1 or g2 or both were then traced.

    We now consider the more likely case in which the first commodity, produced in the public sector, is subject to rationing. To be more specific it is postulated that the price for this (first) commodity is fixed by the government and the available fixed quantity is sold to households at that price. Four points need to be underlined before we proceed further. First, as in the preceding case outputs of both the commodities are capacity constrained. Second, for the rationing to be meaningful the price fixed by the government must be lower than the price that would clear the market. Third, price for the second commodity is left free to find its market clearing level. Fourth, as stated above we do not consider any specific rationing scheme for the scarce commodity.

    Let us now turn to the nature of adjustments envisaged. As mentioned earlier we must set p1 = p1* as a policy parameter. Since p2 is market clearing variations in both θ1 and θ2 are required so as to ensure that p1 remains fixed at p1*. With the money wage w being exogenous, changes in θ1 and θ2 must be in opposite directions. Further, since the government budget must balance, money supply cannot be held fixed. For, it must adjust passively to any changes in θ1 and θ2 brought about by changes in g1 and/or g2. The question of an exclusive choice between changes in money supply and those in administered prices does not arise. Failure to articulate these issues has frequently led to much confusion in discussions relating to price policy and resource mobilisation.

    To formalise the problem let us start with household demand. Following the earlier framework we have

    afy

    Max u =c1pc2pc3p(m/p1)δ d

    Subject to: Y + mo = p1c1p + p2c2p + p3c3 + m (4.1)

    Also, p1 = p1* which is below the market clearing level so that the entire supply of commodity 1 is purchased. Thus, we have

    c1p = c1* – g1 = c*1p (say)

    Assuming without loss of generality that mo = 0 the demand functions are:

    f′

    *

    *

    c2p =c2 − g = (Y-pc1p)

    21

    p

    2

    *

    c3 = y (Y -p1*c1p), and,

    p (4.2)

    3

    d **

    m= &′(Y -p c1p)

    It must be that f′ + y′ + &′ = 1.

    This, in turn implies, as shown in proposition 1, that md = ms = m (say). Thus, we must have from (4.2)

    m = δ'(Y – p*1c*1p) which implies that

    ** m

    − p1c1p =

    (4.3)

    &'

    From the preceding section we note that under market clearing condition we have

    o a c2p(Wo + wao1x1) 1

    p= f′ c1p(a21x1+ g2)-a′ c2 p(x1-a11x1-g1) -&′ c1pc2 p

    (4.4)=-a′ c2 pW/[ &′ c1pc2p + a′ c2 pS1-f′ c1 pS2]

    where W ≡ Wo + wao1x1denotes as before the government’s total wage bill. Now, if rationing is meaningful p1 must be set at a level below p1o ie, p1* < p1o where p1* is the administered and p1o the market clearing price for the public sector product. This implies that the following condition must hold.

    *

    a′ c2pW > p1[&′ c1pc2p + a′ c2pS1 -f′ c1pS2]

    **

    ⇒ a′ c2p(W -p1S1)>p1c1p(&′ c2p -f′S2) (4.5)

    Noting that c2 p = f′m/(&′ p ) and by proposition 1 that W – p1S1=

    2 m–p2S2 > 0 (5.5) can be written in many alternative forms, e g, *

    a′ p2c2p -f′ p1c1p>0 or, p2> f′ p1c1p/ a′ c2p (4.6) Thus, we have:

    Proposition 5

    If in a regime of rationing and capacity constraints, the price of the public sector product is set below its market clearing level, then the market clearing level of the price of the private sector product is not only positive but also subject to a lower bound – depending directly on the level of the public sector product price.

    Under this regime we have x1, x2, c1 and c2 fixed as in the preceding case. g1 and g2 are policy variables. But once these are fixed c1p and c2p are also fixed. The endogenous variables are: θ1, θ2, p2 and m. Note that both θ1 and θ2 must adjust so as to ensure that p1 is rigidly fixed and p2 is market clearing. Thus, with p2 and θ1 both endogenous, m also must adjust as required. Thus, we have a basic system of four equations as:

    (i) B + ap = (1− a ) p − wa

    1 212 111 o1

    (ii) B − (1− a ) p =−ap − wa

    2 222 121 o1

    (iii) f′x1B1 + (&′c2p − f' g2 ) p2 = f′(Wo + p1g1) (4.7)

    (iv) m = Wop g1 + pg − B x (4.7)

    1 22 11

    (Note that from now on we drop the asterisk (*) on p1.)

    Of these four equations (i) and (iii) can be solved for θ1 and p2. These can then be substituted in (ii) to obtain θ2 and in (iv) to get m. Equations (i)-(iii) can be solved to yield

    f(W -pS) ⎞

    p= 11 ⎟

    2(& c2p -f p2) ⎟

    B = (1 -a11)p1 -wa01 -f a21(W -p1S1) ⎟

    1(& c2p -f p) ⎟

    2 ⎟

    (a12p1 + wa02) + f(1 -a22)(W -p1S1)

    B =

    2

    (& c2p -f p) ⎟

    2

    ⎟ Further, ⎟ (4.8)

    ⎟ ⎟ & c2p(W -p1S1) ⎟

    m=

    ⎟⎟

    (& c2p -f p)

    2 ⎠

    Since p2 > 0 it is obvious that m > 0. Of particular concern

    B

    2

    /[f(1-a )c1p+( + )a c ]/

    Δ

    (4.15)

    f &

    p=

    1

    22 12 2p

    in the present context are the signs and magnitudes of θ1 and

    θ2. Let us turn to these. From (4.7) and (4.8) we have

    δ

    S2/ c / we have /p 0

    1

    Thus, if unambiguously.

    ∂B

    f

    2p 2

    Next, it turns out that

    (1 -a11)p1 -wao1 -f a21p1c1p/ (a c2p)

    ⎞⎟⎟⎠

    B

    dp ={f pdg +( + f)pdg +[fg-f(1 -a11)x1]dp} / Δ

    (4.9)

    -a12p1 -wao2 + f(1 -a22)p1c1p/(a c2p)

    so that

    B

    Clearly, the signs of both θ1 and θ2 are indefinite. However,

    ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟

    ∂∂

    p

    2

    g

    1

    given the technological parameters, θ2 is more likely to be

    positive whereas θ1 is more likely to be negative. It is worth noting

    = f p /

    1

    Δ

    that the upper bound for θ1 and the lower bound for θ2 do critically depend on the level at which p1 is set and the relative magnitudes of the net availability of the two commodities, c1p and c2p. We have a proposition quite drastic for its massage to many a mixed

    ∂ ∂

    p

    2

    g

    2

    =(f + )p /

    2

    Δ

    (4.16)economy, which is:

    ∂∂

    p

    2

    p

    1

    =-fS1 /

    Δ

    Proposition 6

    If the public sector product is subject to rationed distribution

    Thus, with S2/c2p / f we must have

    at a fixed price (below the market clearing level) then the mark

    22 2

    up rate in the public sector is subject to a ceiling and that in

    p g

    1

    p g p p

    1

    0, 0 and

    0

    unambiguously.

    the private sector subject to a floor. Both, the ceiling as well

    ∂∂∂

    2

    as the floor depend critically on the relative net availability of

    Finally, turning to the endogenously determined increase in

    the two products for private consumption.

    money supply m, we note from

    Turning now to comparative statics let us totally differentiate

    c2p(W -p1S1)

    the first three equations of (4.7), which gives us (4.8) that m =

    ⎤ ⎥ ⎥ ⎥ ⎥⎦

    ( c2p -f p2)

    21 1

    ⎤ ⎤

    (1 -a11)dp1

    10

    d

    Simple differentiation shows that:

    B

    a

    ⎢ ⎢ ⎢ ⎢⎣

    ⎢ ⎢ ⎢ ⎢⎣

    ⎢ ⎢ ⎢ ⎢⎣

    0 1 -(1-a ) d dp

    1

    = -a

    12

    B

    22 2

    (4.10)

    ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

    dp pdg+( + )pdg + gdp

    11 211

    m

    f

    c-f g

    2p

    f f f

    ⎥ ⎥ ⎥ ⎥⎦

    x

    1

    ⎥ ⎥ ⎥ ⎥⎦

    2 2

    Δ

    2

    S1 / 0=- c

    2p

    p

    1

    Simple calculations show that

    ∂ ∂

    B1 g

    1

    =

    f

    a21

    p/

    1

    ∂ ∂

    B

    g

    1 2

    Δ

    =-(+ p/

    )a 2

    Δ

    ; m

    f

    21

    p /

    1

    Δ

    0= c

    2p

    g

    1

    (4.17)

    It follows that

    θ1

    /

    g10 and

    θ1

    /

    g20 if S2/c2p δ/β (4.11)

    m

    2

    (W-pS)(c +p) /

    11 2

    0

    Δ

    = f

    2p

    (see proposition 3).

    m m m S2

    Further, we have 0, 0 and 0 if B1

    a condition that has been extensively discussed in Section 3

    g

    2

    .

    Clearly,

    .

    p g gc f

    11 2p

    2

    ={(f + )(1-a11)c2p + f a21-c1p -f | I-A | x2}/

    Δ

    Thus we have:

    p

    1

    (4.12)

    =| I-A |{(f + )b22c2p + f b21C1p -f x2}/

    Δ

    Proposition 7

    where bij are elements of (I-A)-1

    If the private sector sales to the public sector relative to

    what is left for sale to the households is less than a critical

    S

    Notice that if 2 then the numerator as well as the

    f

    c2P

    minimum (depending on the propensity to save of households)

    denominator in (4.12) are positive. Hence

    then (i) increase in g1 or, g2 will raise θ1, θ2, p2 and m;

    (ii) increase in p1 raises θ1 but lowers θ2, p2 and m.

    S2

    c2 p

    δβ

    implies that

    ∂∂

    0,

    ∂ ∂∂∂

    θ1

    1

    θ1

    2

    θ1

    p

    1

    0

    (4.13)

    0 and,

    g g

    With regard to θ2 we note that

    VVVVV

    B

    /

    g

    1

    = f

    (1-a )p /

    1

    Δ

    and

    B

    2

    /

    g

    2

    =( + )(1-a )p/

    2

    Δ

    Some Illustrative SimulationsSome Illustrative SimulationsSome Illustrative SimulationsSome Illustrative SimulationsSome Illustrative Simulations

    f

    22 22

    (4.14)

    Conditions stated in (4.11) ensure that In this section we report the results of a few numerical exercises

    focusing on the nature of solutions for prices and profit rates

    ∂θ2/

    of change unambiguous. On the other, we illustrate the kind of

    /

    p1 = {f(1-a22)g1-a12( c2 p -f g2) -f | I-A|x1} /

    Δ

    Note that x1=

    [(1 -a22)c1 + a12c2]

    |I-A| so that

    parameter combinations which might lead to alternative and

    in some cases unacceptable solutions. In cases relating to

    comparative statics the primary focus is on changes in the governments’ acquisition of the two goods. In all exercises the following parameters remain fixed at values

    indicated. a11 = 0.36, a12 = 0.15, a21 = 0.08 a22 = 0.80, a01 = 0.07, a02 = 0.15 λ1 = 0.15, λ2 = 0.35, λ3 = 0.20 mo = 1000, Wo = 3000, w = 50

    Values of the behavioural parameters, α, β, γ and δ, and the capacity outputs levels x1 and x2 are fixed in different exercises to ensure that any one or more specific conditions do or do not hold as the case may be. Policy variables g1 and g2 and, in some case p1, are naturally varied across different exercises.

    Before we proceed further let us restate the important conditions and label them as follows. Equation numbers at the end refer to those in the text for quick identification, in each case.

    S2 S1

    A: f -a

    (3.2)

    c2p c1p c1p wao1( c2p -f g2) + a21 f WoB1: a

    (3.4)

    c2p (1 -a11)Wo + wao1g1 c2p wao2(δ c1p-αg1)+αa12Wo+αwx1[ao2(1-a11)-ao1a12]B2: β (3.5) c1p (1-a22)Wo + wao2g2 + wx1 [ao1 (1-a22) + a02a22]

    a + : 2p (3.9)

    S2

    c f x2 a21(1 -a11)

    D:

    (3.15)

    x1 a12(1 -a22) c1p a |I-A|

    E: (3.13)

    x2 (1-y)a21 S2

    F: (4.13)

    c2p f

    Recall from Section III that A which can be expressed in alternative forms and was christened as the “solvency condition” ensures positive prices p1 and p2. Assuming A holds, B1 ensures positive θ1 and B2 ensures positive θ2. To be more precise, A implies p10, p20, A, B1 implies θ10 A, and B2 implies θ20. A is also necessary and sufficient for

    p p p

    122

    0, 0, 0

    g g g

    212 p 0

    No additional condition is required. However, for 1

    g

    1

    we need condition C in addition to A. Now consider cases I(a) and I(b) for which we have: x1 = 200, x2 = 400, c1 = 68, c2 = 304 α = 0.15, β = 0.50, γ = 0.10, δ = 0.25 However for I(a) we set g1 = 18, g2 = 54 and for I(b) g1 = 28,

    g2 = 134. The results are as follows

    Table 1Table 1Table 1Table 1Table 1

    Variable p1 p2 θ1 θ2 γ m
    Case (Ia)
    Base solution 32.05 21.37 15.30 4.79 9683 2670
    Deviation (Δg1 = 10) 4.29 -1.99 2.90 -2.32 -992 -248
    Deviation (Δg2 = 10) 2.07 2.33 1.14 1.55 689 172
    Case (Ib)
    Base solution 95.72 75.08 51.76 38.20 24533 6389
    Deviation (Δg1 = 10)Deviation (Δg2 = 10) -5.01 204.16 -21.71 198.48 -1.472 -16.62 115.34 122.57 -7387 54471 -1849 136

    Note: For both I(a) and I(b) A, B1 and B2 all hold giving us positive solutions for prices as well as profit rates. However while C holds for I(a) it is violated in I(b). Hence ∂p1/∂g1 is negative, in the latter case and positive in the former. Note also that ∂p2/∂g1 is negative in both the cases. We also have ∂p1/∂g1 and ∂p2/∂g2 positive in both cases, as expected.

    Comparative statics for θ1 and θ2 is rather more complex. Recall from Section IV that so long as prices are positive we always have ∂θ2/∂g1 0. For ∂θ2/∂g2 0, D is sufficient but not necessary. For ∂θ1/∂g1 0, C is necessary and sufficient if p1 0 and for ∂θ1/∂g2 0, E is necessary and sufficient. In cases I(a), D as well as E hold. Hence the signs of the deviations given in Table 1 above are, as expected, viz, ∂θ1/∂g1 is positive in I(a) and positive in I(b); ∂θ1/∂g2 0 in both cases; ∂θ2/∂g1 0 in both cases and ∂θ2/∂g2 positive in both cases.

    Next we take up case II in which A, B1, B2 and E hold, but C and D are violated as we set the capacity output etc as follows. x1 = 450, x2 = 100, c1 = 273, c2 = 44 g1 = 173, g2 = 0, c10 = 100, c2p = 44 The results are as given in Table 2.

    Since A, B1 and B2 hold both prices as well as profit rates are positive. But as C is violated ∂p1/∂g1 0. ∂θ2/∂g1 0 holds unconditionally. However, note that ∂θ1/∂g1 and ∂θ2/∂g2 are both positive even though C and D are violated. This is because both these are sufficient but not necessary conditions.

    We now turn to cases III(a) and III(b). For both of these A holds. But in III(a) B1 is violated and in III(b) B2 is violated. In both cases we set:

    x1 = 200, x2 = 400, c1 = 68, c2 = 304

    g1 = 18, g2 54, c1p = 50, c2p = 250 For III(a) we also have:

    α = 0.01, β = 0.54, γ = 0.225, δ = 0.225 and for III(b) we have:

    α = 0.35, β = 0.10, γ = 0.25, δ = 0.30.

    The results are as given in Table 3.

    In the final exercise we set α =0.05, β=0.85, γ = 0.05, δ =0.05 leaving other parameters unchanged with the result that A no longer holds. Theoretically, equilibrium solutions for p1 and p2

    Table 2Table 2Table 2Table 2Table 2

    Variable p1 p2 θ1 θ2 γ m
    Case (II) Base solution 643.2 4872.4 18.3 3794.5 427678 107031
    Deviation (Δg1 = 40)Deviation (Δg2 = 10) -144.6 1153.3 -2606 3483.6 116 459.8 -2063.4 2613.3 -229283290290 -57259 72975
    Table 3Table 3Table 3Table 3Table 3
    Variable p1 p2 θ1 θ2 γ m
    Case (IIIa) Base solution 9.77 105.54 -5.69 75.47 47880 11013
    Deviation (Δg1 = 10)Deviation (Δg2 = 10)Case (IIIb) Base solution 2.07 4.17 31.57 -3.19 51.34 1.80 1.58 -1.44 16.56 -2.87 40.44 -10.79 -147920871 3511 -333 4703 1353
    Deviation (Δg1 = 10)Deviation (Δg2 = 10) 4.28 0.16 -0.16 0.09 2.75 0.10 -0.77 0.05 -58723 -124 7

    Note: In III(a) since B1 is violated θ1 0. Also, as E is violated ∂θ1/∂g2 0. However, since other conditions hold all comparative static results are as expected.

    Table 4Table 4Table 4Table 4Table 4

    Variable p2 θ1 θ2 γ m
    Case IV
    Base Solution 35.36 9.67 17.03 15284 4425
    Deviation (Δg1 = 10) 4.52 -0.36 3.61 1671 566
    Deviation (Δg2 = 10) 13.15 -1.06 10.52 4768 1406
    Deviation (Δp1 =5) -9.83 3.98 -8.60 -3932 -1238

    should exist though these are negative. However, the algorithm fails to generate a solution. It is our intuition that the problem here is related to the stability of equilibrium. A general equilibrium with negative prices exists but presumably it is unstable. It is this that prevents the solution algorithm to converge.

    The same non-convergence is encountered in dealing with the rationing case (section V). Expected solutions were obtained (as given in Table 4) with p1 = 25 satisfying condition F. No convergence was obtained with p1 = 5 which violates F. We set

    α = 0.2, β = 0.6, γ = 0.1, δ = 0.1 x1= 200, x2= 400, g1 = 18 g2=54 mo= 1000, W=50, Wo= 3000.

    which satisfy condition F.

    VIVIVIVIVI
    A Summing UpA Summing UpA Summing UpA Summing UpA Summing Up

    We hardly need to reiterate that under the changed policy regimes in the erstwhile planned economies, while some prices will be allowed to be market determined several others will remain administered. The present exercise is thus of continued relevance insofar as we do consider both cases namely, flexible and administered prices of public sector products. What is indeed striking is the wide variety of possibilities that arise even in this admittedly simple model of a mixed economy. Of particular interest is the fact that the decisions of either pricing the public sector product or the mode of financing its losses does not have simple consequences either with regard to the price of the private sector product or the measure of profitability in either sector. Technological and behavioural parameters tend to play critical roles that are not usually obvious.

    Recapitulating some of the results let us note the following. First, under flexible price regime if the propensity to save of the private sector is not adequately high then positive prices may not exist. On the other hand maintenance of adequate effective demands is necessary to ensure profitability (Propositions 2a and 2b). Under the same regime, whether government’s increased demand is for its own product or of the private sector product determines the direction of its impact on the two prices as well as on the profit margins (Propositions 3 and 4).

    Second, under administered pricing of the public sector product, subsidising it sets a lower (!) bound to the price of the private sector product. This also sets a floor to the profit margin in the private sector and a ceiling to that in the public sector (Propositions 5 and 6). Again, with the administered public sector product price raised, government demand for either product raises not only the price and the profit margin for the private sector product but also the profit margin for the public sector itself, by and large, as expected. However, while an increase in the administered price of the public sector product raises the profit margin in this sector it also lowers the price as well as the profit margin of the private sector product (Proposition 7).

    Two overall impressions of these results are worth a restatement. First, in a mixed economy, possibilities of loss making (at positive prices) are very serious; much more so than has been assigned to the familiar bureaucratic bungling. Second, once arbitrary rationing constraints are imposed, the system can be endemically unpredictable.

    Finally, one may reiterate that the foregoing analysis is based on an oversimplified model. Two obvious ways in which one would need to extend this analysis is to endogenise the interest rate and to consider the economy to be open to trade and capital flows.

    rnr

    Email: vnpandit@rediffmail.com

    NotesNotesNotesNotesNotes

    1 We need not specify any particular rationing scheme. 2 The different magnitudes involved here may not be easy to decipher from the way national accounts are written in developing economies.

    ReferencesReferencesReferencesReferencesReferences

    Chetty, V K and D K Ratha (1987): Inflation Growth and Public Sector Price Policy, Indian Statistical Institute, Delhi, mimeographed.

    Dasgupta, D (1992): ‘Administered Prices and Deficit Financing: A Macro View’ in K Basu and P Nayak (eds), Development Policy and Economic Theory, Oxford University Press, Delhi.

    Datta, Chaudhuri, M (1990): ‘Market Failure and Government Failure’, Journal of Economic Perspectives, Vol 4, No 3. Jha, S and S Mundle (1987): ‘Inflationary Implications of Resource Mobilisation through Administered Prices’, Economic and Political Weekly, August 5.

    Mukherji, B, V Pandit and K Sundaram (1992): ‘General Equilibrium in a Miniature Mixed Economy’, Indian Economic Review, Vol XXVII, Special Number in Memory of Sukhamoy Chakravarty.

    Panda, M and H Sarkar (1990): ‘Resource Mobilisation through Administered Prices’ in L Taylor (ed), Socially Relevant Policy Analysis, Cambridge, Mass.

    Pandit, V and B B Bhattacharya (1987): ‘Resource Mobilisation, Growth and Inflation: A Trade-off Analysis for India’ in M Dutta (ed), Asia Pacific Economics: Promises and Challenges, Jai Press, New York.

    Sundaram, K and S D Tendulkar (1987): ‘Policy on Administered Prices and Deficit Financing’, Economic and Political Weekly, June 21.

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