Price-Demand And Price-Supply

Price-Demand And Price-Supply
Equations From Basic Assumptions.

V.I.G.Menon 1

Chief Consultant, Yahovan Centre for Excellence,12/403,W Block, Anna Nagar West Extension, Chennai,India-600101,Email:v.i.g.menon@gmail.com, Phone: : :+91984 0600251, +919962952559.Author is grateful to Giri Menon for his Valuable Suggestions.
Abstract

The dependence of demand or supply on price is established in the form equations from simple assumptions. Also this dependence is explored in the Matrix form ,as well as using the mean of the parameters Price,Wage ,Supply and Demand Quantities for a community as a market. The paper thus explores the conditions under which the standard demand/supply relations hold with respect to price.Analysis of the wage,price,demand graph shows expected trends.And the improvised approach to understanding the elasticities of demand based on the basic equations show that for goods with -ve price elasticities of demand ,income elasticities of demand is positive,and vice versa
Introduction
The dependence of price on demand or supply is the most basic relation that supports the micro economic thought everywhere. In the following we discuss why this “Natural Law” holds.
Price-Demand Dependency
We assume a buyer with a fixed wage Wb for any given period of the wage cycle, buying Qi amounts of a good at price Pi. Figue 1 Gives the Matrix where the rows 1 to B are the buyers and Columns 1 to S are the sellers .If all the wage/income of this buyer for this period is spent on N goods and services, we can write

Wb = ? Pi *Qi …………………………………………… (1)
,where summation is done from i =1 to i = N.
If we assume that as long as Wb , is constant for the individual “b”, then we get
? Pi *Qi = Constant. …………………………………………… (2)
Now if we assume that the consumption profile ( Pi *Qi /? Pi *Qi ) is a constant Ki for each i, then we have
Pi *Qi = ( Pi *Qi /? Pi *Qi ) = Ki*Wb …………………………………………… (3)

Where Ki = Pi *Qi /? Pi *Qi in equation (3) is the percentage or fraction of “b”s income Wb spent on item “i”.This equation tells us that if individuals spend generally fixed ratios of their wages on various goods, then demand Qi depends inversely with price.
That is , the Quantity Qi demanded at wage Wb is indirectly proportional to the price Pi,which is what we wanted to derive
Price-Supply Dependency
Similarly, now consider a supplier who is supplying “M” number of goods to the market, including the goods that are bought by the buyer above. His total wage or income Ws from the trade can be represented as
Ws = ? PJ QJ …………………………………………… (4)
where J is summed from 1 to M.

If we assume that the supply profile ( PJ *QJ/? PJ*QJ ) is a constant KJ for J
then as above we can write
PJ QJ = ( PJ QJ/? PJQJ ) = KJ Ws …………………………………………… (5)
Where , QJ is the total quantity of “J” sold in the market by him at price PJ , KJ is the percentage or fraction of suppplier’s income Ws received on item “J”
and so we have,

.Since PiQi represents the total paid by the buyer for the Quantity Qi of “ith” product at price Pi ,this amount will go to the seller as a fraction of his income.
For the same item where buyer’s item i = J th item for the seller ,we write
(PiQi / PJQJ ) = KiWb/KJWs ,,we can also note KiWb/KJWs = C ,where C is the fraction of the income generated due to product J for the supplier “s” by the buyer b”.
Or, QJ = [(1/ C ) * (Qi / PJ ) ]*Pi …………………………………………… (6)
QJ = C* Pi …………………………………………………… (7)
Where “constant ” C = [(1/ C ) * (Qi / PJ ) ] showing that the Quantity Supplied, QJ is proportional to the purchase price Pi which is the supply-price relation we wanted to establish,under the assumption of constant consumption and supply price profiles for the product i=J (not necessarily Pi =Pj the equillibrium price)
As the terms involving buyer or supplier are not entering here, these equations are independent of any buyer “b” or supplier “s”.
Generalised Results
Figure shows a community of buyers purchasing varying quantities of products and services ,each according to their wages Wb.

For any buyer “b” and similarly for any supplier “s”, we can rewrite the above equations as in Matrix representations,
Wb = ? Pi *Qi = P.Q …………………………………………… (8)

which represents the “Matrix product” of P with Q,treating P as 1X N
and Q as NX1 Matrices.
Let us,for clarity,choose to write P.Q for buyer “b” as Pb.Qb and for supplier “s” as Ps.Qs noting that
Wb = Pb.Qb …………………………………………… (9)
for the buyer and
Ws = Ps.Qs …………………………………………… (10)
for the supplier.
Summing for all buyers “b” from b = 1 to B and for all suppliers “s” from s = 1 to S, we have (see Figures)

KB*WB = ? KbWb = ? Pb.Qb = (? Pb).(?Qb) = PB*QB, ……………… (11)
for all the buyers and
KS*WS = ?KSWs = ? Ps.Qs = (? Ps).(?Qs) = PS*QS ………………… (12)
for all the suppliers,where (? PB) = PB ,etc.,

PB*QB, and PS*QS represent Matrices whose elements Pij.Qkl are as shown in the figures 2 & 3 above. KB*WB and KS*WS are single column matrices

So that we can write,

Generalised Matrix Laws
We again start with
KBWB = PBQB …………………………………………… (13)
KSWS = PSQS …………………………………………… (14)
Both these look exactly like our single buyer case except that these are Matrix Equations.
Assuming as in the previous case that Consumption Profiles will be same when the wage rates are fixed, we have
(P-1B)* KSWS = QB …………………………………………… (15)
Which tells us that the demand matrix QB is inversely dependent on the price matrix PB.
This is our Generalised Price-Demand Law in Matrix form involving many goods ,buyers and suppliers.
Now Multiplying KSWS = PSQS in equation (14) by the inverse of KBWB we have
(KBWB) -1* KSWS =( PBQB) -1* PSQS ………………………………… (16)
ie.,
C =( PBQB) -1* PSQS ………………………………… (17)
Where C is a constant Matrix, Or,
( PB)*( QB) * P-1S * C* = QS ………………………………… (18)
Which again shows that the Demand Matrix QS is proportional to
Price matrix PB, under the assumption that ( QB) * P-1S is a constant.
This is our Generalised Supply-Price Law in Matrix form involving many goods , buyers and suppliers.
The Matrix relations will enable us to study the system using linear equations and linear programming
Derivation Based On Mean of the Parameters
For a community of average wage rate is W is consuming an average Quanity Qi of the ith Commodity at an average price Pi ,
Then, as in the individual case equations (1) to (7) ,we have
Ki W = Pi*Qi, ………………………………… (18)
where Ki is the average fraction of the wage spent on the ith product .

This again leads us to the conclusion that if mean wage W is constant and the mean fraction of the wage spent on the ith product is fixed,then demand Qi for this product in this community depends inversely on its price Pi.
Similarly it is easy to see that
QJ = C* Pi ………………………………… (19)

that is, the Mean Quantity QJ of item “J ” supplied is proportional to the Mean Purchase Price Pi for the same commodity (where generally, ith commodity bought by buyer is same as the Jth commodity supplied- pl note that for enumeration, the i th product bought by the buyer need not be the same as the ith product supplied by the supplier)

Wage and Price
Relation between the individual,(or mean), wage, price and demand is seen by plotting Ref P = K.W/Q,with K =1,and Q on the X,W on the Y and P on the Z axes, using the online 3D plotter .
The results are very encouraging .Apart from showing the expected price-demand dependence,on its red edge,the green edge of the 3D “sheet” shows that as wage level increases ,price level also increases Ref.
http://www.livephysics.com/ptools/online-3d-function-grapher.php?ymin=1&xmin=0&zmin=0&ymax=5&xmax=5&zmax=15&f=y*x^-1
Calculation Of Price Elasticities

a) Price Elasticity Of demand ED is given by
(?Q/Q) dlnQ ……………… (20)
ED = (?P/P) = dlnP

for infinitesimal changes in Q & P .

From equation (3) dlnQ/dlnP can be worked out as follows

PdQ + QdP = KdW, dividing both sides by PQ,

dQ/Q + dP/P = K(dW/PQ) = K(dW/KW) ,since PQ = KW,

ie, dlnQ + dlnP = dlnW, dividing by dlnQ gives,

( dlnQ/dlnP) + 1 = (dlnW/dlP)
( dlnQ/dlnP) = (dlnW/dlnP) – 1 ……………………………… (21)
Multiplying both sides of Equation (21) by ( dlnP/dlnQ) ,gets us
1 = (dlnW/dlnQ) + ( dlnP/dlnQ), that is,
1 = (dlnW/dlnQ) + [1/(dlnQ/dlnP)] ……………………………… (22)
Representing (dlnW/dlnQ) as a and (dlnQ/dlnP) as ß, we write Equation (22) as
a + 1/ß = 1

( dlnW/dlnQ) = 1 – [1/(dlnQ/dlnP)] ……………………………… (23)

ie,

a = 1 – 1/ß ……………………………… (23)
Which shows how wage ( income) elasticity of demand ( dlnW/dlnQ) is related to

Price elasticity of demand.Thus we see that increase of ß increases a.The relation

between price elasticity of demand and income elasticity of demand are explored in

the graph ref,showing that for goods with -ve price elasticities of demand ,income

elasticities of demand is positive, such that as wage increases, the demand increases for

goods with -ve elasticity .

Results and Discussions

Under the assumptions of constant income for the buyer and the supplier, as well as constant consumption and supply profiles for the buyer and the supplier respectively, it is found that the Classical price-demand as well as the price-supply relations hold exactly with respect to an individual or a community (ie market).Such relations can be expressed in Matrix form as well as through Mean of Price, Quantity Wage and the associated Constants.
Graphical analysis of the equations in 2D shows the expected pattern, and the 3D graph involving, wage, price and demand equation shows that the Quantity demanded increases with wage. Analysis of price elasticities show that for goods with -ve price elasticities of demand ,income elasticities of demand is positive.

Fig.1:The Matrix Shows Suppliers 1 to S ,each supplying products 1 to S for the Market comprising Of Buyers 1 to B. The Column total for ,say buyer “i” ?PiQJ for any product “i” for all “J” ,gives his total budget to spend which may be a fraction Ki of his wage Wi,as given by KiWi. Along the column, Pi remains same,but QJ varies for all buyers for the same product at the same price Pi .The Row total Gives the income or revenue for each product to Supplier, where CJWJ is income of Supplier from all buyers for product J sold in this market =?PiQJ for any J for all “i” .
Notes On The Transaction Accounting Notation in Fig 1.

Products are assumed to be numbered from “I” =1 to N

Supplies are assumed to be numbered from “I” =1 to S

Buyers are assumed to be numbered from “I” =1 to B

Quantities are assumed to be numbered from “I” =1 to M

Value of any transaction is given by Price multiplied by Quantity,ie,P*Q

Price Matrix is [ P] which is a 1X N row matrix
Quantity Matrix is [ Q] which is a MX 1 Column matrix

Transaction Value Matrix or Transaction Matrix [T] = [PQ] is an NXM matrix

The sum of the rows of [T] is given by ?PiQJ ,where PiQJ =0
for i ? J.

bP1 Represents the price of “item” “1” bought by any buyer “b”
SQ1b Represents the Quantity of “item” “1” bought by any buyer “b” from supplier “S”.
bPS1 *SQ1b This product of bP1 with SQ1b represents the total amount transacted by any buyer “b” for “item” “1” supplied by any supplier “S” at price bP1 for quantity SQ1b
bPSn *SQnb This product of bP1 with SQ1b represents the total amount transacted by any buyer “b” for “item” “n” supplied by any supplier “S” at price bPn for quantity SQnb
bW represents the Wage of any buyer “b”

bK represents the fraction of the budget allocation by “b”

bWS represents the fraction of the revenue of any supplier S due to any buyer “b”

bKS represents the fraction of the budget allocation by “b” for supplier “S”.

bKSn represents the fraction of the budget allocation by “b” for supplier “S”

due to a product “n”.

bKSn * bWS gives budget allocation of “any” buyer “b” for “any” product “n” offered by “any” supplier “S” obtained by multiplying bKSn with bWS

? bKSn * bWS = bKS * bWS gives total budget allocation of “any” buyer “b” for all
n=1 to N
products 1 to N offered by “any” supplier “S”.
? bKS * bWS = KS * WS gives total budget allocation of all buyers “1 to B” for all
b=1 to B
products 1 to N offered by “any” supplier “S”.

? KS * WS = K * W gives total budget allocation of all buyers “1 to B” for all
S=1 to S
products 1 to N offered by suppliers 1 to “S”.

? bKS * bWS = bK * bW gives total income of all suppliers “1 to S ” for all products
S=1 to S
1 to N bought by any buyer “b” .

Transaction Account for Fig1.
Quantity
Fig 2:Sample Price-Deamand graph calculated from Equation (3) using online graph plotter (http://graph-plotter.cours-de-math.eu/) setting X axis for Price and Y-axis for Quantity , K=5,W = 1000, and X (that is ,Price) set to change from 100 to 500 units
Price

Fig3.Sample Price-supply graph calculated from Equation (7) using online graph plotter (http://graph-plotter.cours-de-math.eu/) setting X axis for Price and Y-axis for Quantity , K=5,W = 1000, and X (that is ,Price) set to change from 10 to 50 units.

Fig.4.Apart from showing the expected price-demand dependence,the 3D “sheet” plotted based on equation (3) shows that as wage level increases ,price level also increases Ref. http://www.livephysics.com/ptools/online-3d-function-grapher.php?ymin=1&xmin=0&zmin=0&ymax=5&xmax=5&zmax=15&f=y*x^-1
a
ß
Fig.5.Plot for income elasticity of demand a as a function of price elasticity of demand ß as in equation (23). As the values of ß is varies from -5 to + 5, the values of a take a hyperbolic shift from -ve to + ve range as can be seen.(plotted using plotter at (http://graph-plotter.cours-de-math.eu/)