- Email: [email protected]
Price variability, changes in demand and the rate of interest
Economics Letters 7 (1981) 7- 10 North-Holland Publishing Company
PRICE VARIABILITY, CHANGES IN DEMAND AND THE BATE OF INTEREST P.J. PHLIPS * Massachusetts
Institute of Technology, Cambridge,
MA 02139, USA
L. PHLIPS * C.O.R.E.,
1348 Louvain-la-Neuve,
Received
IO March
Belgium
198 1
It is shown that, over time, prices are almost inversely proportional to quantity demanded in the case of a monopolist who produces for inventory. For a monopolistically competitive firm, the dependence of prices on variations of demand is weak. In both market structures, the rate of interest introduces an element of exponential inflation.
It seems of interest to have a closer look at Smithies’ (1939) intertemporal price discrimination rule and to work out its implications with respect to the extent at which prices react to changes in demand and to the level of the rate of interest. Indeed, this rule provides a better understanding of the behavior of prices over the business cycle (or over the seasons) and may help explain the otherwise puzzling variability of prices in periods of exceedingly high interest rates such as the current one. Consider a firm that is on the threshold of a period during which it expects both prices and costs to be rising for a finite period, and thinks it can predict with certainty the behavior of its demand and cost schedules over the period. This firm will produce for inventory with a view to selling at some subsequent date. Its price, rate of production and rate of sales are determined by the rule that the discounted marginal revenue
* We are grateful earlier draft.
to Jacques
H. Drtze
01651765/81/0000-0000/$02.50
and Fiorella
Padoa
Schioppa
for comments
0 1981 North-Holland
on an
8
P.J. Phlips, L. Phlips / Price variability,
changes in demand, rate of interest
from sales and the discounted marginal cost of goods produced at every point of time should be both equal and constant [for a further discussion, see Phlips (1980)]. In particular, this rule implies P +
(1)
dap/aq) = A e”,
where p = p( q, t). q = r is the constant rate only costs of holding solution to (1) of the
is the rate of sales, X is a positive constant and of interest. (For simplicity, it is assumed that the inventories are interest costs.) We shall try to find a form q(t)
(2) using the method of separation differential equations theory. Substitute (2) in (l), so that
of variables,
well-known
in partial
f(4kw + q(af/addd =A err,or af h err f+qG=g(t)=constant=A. The left-hand side is a function of q only and the right-hand side is a function of t only. As a result, they can be equal for any q and l only if they are constants, and
g( t ) = (A/A)
(4
e”.
On the other hand, we have (5) an ordinary
differential
equation,
with solution,
after using (4),
p(f)=h( 1 +t--&)err, where B is another constant. To determine =p(O) at t = 0, to obtain
p(l)
Zzq(o)(q-
l),
B/A,
use the initial
condition
P.J. Phlips, L. Phlips / Price variability, changes in demand, rate of interest
9
so that finally
PO>= ~+(P(w)q(t) [
4(O) ,f 1
e.
Notice that, in perfect competition, A ‘p(O) according to eq. (1). In perfect competition, prices will thus fully reflect storage costs in the sense that p(t) =p(O) e”, or p/p = r, as required by social optimality: see Pyatt (1978) and Phlips and Thisse (1981). In the same way, competitive prices reflect transport costs in a spatial economy. Monopolistic competition implies that h N p(O) or more correctly A/p(O) = 1 -e,
O
1,
remembering that h
p(t)=[i -c(“E;,l)]err. Under monopolistic competition, prices continue to reflect storage costs, although not as fully as in perfect competition. Indeed P = rP c e”@“, so that we could say there is ‘absorption’ of storage costs when Q > 0, by analogy with ‘freight absorption’ as defined in spatial economics. The recent rise of r to historically record levels should thus lead to a concomittant increased variability of prices of storable goods sold under conditions of monopolistic competition. Quite interestingly, eq. (8) also indicates that the dependence of p on variations of q is weak (because E< 1). One might thus expect changes in demand to be less fully transmitted into prices in competitive industries than in concentrated industries, a conclusion reached by other means in Phlips (1980), or to show up (in a price equation) with a regression coefficient that is not statistically different from zero. When monopoly power gets really big, the picture changes entirely. Suppose, for convenience, that VP(O)
= 6,
0<6<<1,
10
P.J. Phlips, L. Phlips / Price variabiliry, changes in demand, rate of interest
because h gets small compared to p(O). Then eq. (7) becomes
Now prices are approximatively inversely proportional to 4, as 6 < 1. It is not surprising, then, that monopolists are seen to increase prices dramatically when demand goes down, and to stabilize prices when demand goes up and rates of interest are low. But when r is high, even monopolists will have to rise prices, whether demand goes down or not (when selling storable goods). In fact, they appear to tend to make sure that total revenue P(t)Q(t) grows along the path traced by e”. Eq. (9) implies j/P N r - o/Q, so that prices are rigid when r = Q/Q. When r is large and Q/Q < 0, stagflation occurs, while Q/Q > r would lead to decreasing prices. The relationship between price dynamics and the rate of interest under different market structures, as discussed above, should not be too difficult to implement empirically. [Our findings actually do not seem to be contradicted by the available empirical evidence. See, e.g., Eckstein and Wyss (1972).] And two things are clear: (a) a negative correlation between administered prices and demand over time should not be discarded off-hand as implausible, and (b) stagflation coupled with high interest rates fits perfectly well into our model. The policy implications are interesting. In a world such as the one described above, it is hard to see how a reduction of demand could reduce inflation: in monopolistically competitive markets, demand has no perceptible influence on prices, while under monopoly a reduction in demand can only increase prices. It is even harder to see how high interest rates could help control inflation-except through a suicidal policy of bankruptcy or putting an end to the production of storable goods. References Eckstein, Otto and David Wyss, 1972, Industry price equations, in: The econometrics of price determination (Board of Governors of the Federal Reserve System, Washington, DC). Phlips, Louis, 1980, Intertemporal price discrimination and sticky prices, Quarterly Journal of Economics, 525- 542. Phlips, Louis and Jacques-F. Thisse, 1981, Pricing, distribution and the supply of storage, European Economic Review 15, 225-243. Pyatt, Graham, 1978, Marginal costs, prices and storage, The Economic Journal, 749-762. Smithies, Arthur, 1939, The maximization of profits over time with changing cost and demand functions, Econometrica, 3 12- 3 18.
PRICE VARIABILITY, CHANGES IN DEMAND AND THE BATE OF INTEREST P.J. PHLIPS * Massachusetts
Institute of Technology, Cambridge,
MA 02139, USA
L. PHLIPS * C.O.R.E.,
1348 Louvain-la-Neuve,
Received
IO March
Belgium
198 1
It is shown that, over time, prices are almost inversely proportional to quantity demanded in the case of a monopolist who produces for inventory. For a monopolistically competitive firm, the dependence of prices on variations of demand is weak. In both market structures, the rate of interest introduces an element of exponential inflation.
It seems of interest to have a closer look at Smithies’ (1939) intertemporal price discrimination rule and to work out its implications with respect to the extent at which prices react to changes in demand and to the level of the rate of interest. Indeed, this rule provides a better understanding of the behavior of prices over the business cycle (or over the seasons) and may help explain the otherwise puzzling variability of prices in periods of exceedingly high interest rates such as the current one. Consider a firm that is on the threshold of a period during which it expects both prices and costs to be rising for a finite period, and thinks it can predict with certainty the behavior of its demand and cost schedules over the period. This firm will produce for inventory with a view to selling at some subsequent date. Its price, rate of production and rate of sales are determined by the rule that the discounted marginal revenue
* We are grateful earlier draft.
to Jacques
H. Drtze
01651765/81/0000-0000/$02.50
and Fiorella
Padoa
Schioppa
for comments
0 1981 North-Holland
on an
8
P.J. Phlips, L. Phlips / Price variability,
changes in demand, rate of interest
from sales and the discounted marginal cost of goods produced at every point of time should be both equal and constant [for a further discussion, see Phlips (1980)]. In particular, this rule implies P +
(1)
dap/aq) = A e”,
where p = p( q, t). q = r is the constant rate only costs of holding solution to (1) of the
is the rate of sales, X is a positive constant and of interest. (For simplicity, it is assumed that the inventories are interest costs.) We shall try to find a form q(t)
(2) using the method of separation differential equations theory. Substitute (2) in (l), so that
of variables,
well-known
in partial
f(4kw + q(af/addd =A err,or af h err f+qG=g(t)=constant=A. The left-hand side is a function of q only and the right-hand side is a function of t only. As a result, they can be equal for any q and l only if they are constants, and
g( t ) = (A/A)
(4
e”.
On the other hand, we have (5) an ordinary
differential
equation,
with solution,
after using (4),
p(f)=h( 1 +t--&)err, where B is another constant. To determine =p(O) at t = 0, to obtain
p(l)
Zzq(o)(q-
l),
B/A,
use the initial
condition
P.J. Phlips, L. Phlips / Price variability, changes in demand, rate of interest
9
so that finally
PO>= ~+(P(w)q(t) [
4(O) ,f 1
e.
Notice that, in perfect competition, A ‘p(O) according to eq. (1). In perfect competition, prices will thus fully reflect storage costs in the sense that p(t) =p(O) e”, or p/p = r, as required by social optimality: see Pyatt (1978) and Phlips and Thisse (1981). In the same way, competitive prices reflect transport costs in a spatial economy. Monopolistic competition implies that h N p(O) or more correctly A/p(O) = 1 -e,
O
1,
remembering that h
p(t)=[i -c(“E;,l)]err. Under monopolistic competition, prices continue to reflect storage costs, although not as fully as in perfect competition. Indeed P = rP c e”@“, so that we could say there is ‘absorption’ of storage costs when Q > 0, by analogy with ‘freight absorption’ as defined in spatial economics. The recent rise of r to historically record levels should thus lead to a concomittant increased variability of prices of storable goods sold under conditions of monopolistic competition. Quite interestingly, eq. (8) also indicates that the dependence of p on variations of q is weak (because E< 1). One might thus expect changes in demand to be less fully transmitted into prices in competitive industries than in concentrated industries, a conclusion reached by other means in Phlips (1980), or to show up (in a price equation) with a regression coefficient that is not statistically different from zero. When monopoly power gets really big, the picture changes entirely. Suppose, for convenience, that VP(O)
= 6,
0<6<<1,
10
P.J. Phlips, L. Phlips / Price variabiliry, changes in demand, rate of interest
because h gets small compared to p(O). Then eq. (7) becomes
Now prices are approximatively inversely proportional to 4, as 6 < 1. It is not surprising, then, that monopolists are seen to increase prices dramatically when demand goes down, and to stabilize prices when demand goes up and rates of interest are low. But when r is high, even monopolists will have to rise prices, whether demand goes down or not (when selling storable goods). In fact, they appear to tend to make sure that total revenue P(t)Q(t) grows along the path traced by e”. Eq. (9) implies j/P N r - o/Q, so that prices are rigid when r = Q/Q. When r is large and Q/Q < 0, stagflation occurs, while Q/Q > r would lead to decreasing prices. The relationship between price dynamics and the rate of interest under different market structures, as discussed above, should not be too difficult to implement empirically. [Our findings actually do not seem to be contradicted by the available empirical evidence. See, e.g., Eckstein and Wyss (1972).] And two things are clear: (a) a negative correlation between administered prices and demand over time should not be discarded off-hand as implausible, and (b) stagflation coupled with high interest rates fits perfectly well into our model. The policy implications are interesting. In a world such as the one described above, it is hard to see how a reduction of demand could reduce inflation: in monopolistically competitive markets, demand has no perceptible influence on prices, while under monopoly a reduction in demand can only increase prices. It is even harder to see how high interest rates could help control inflation-except through a suicidal policy of bankruptcy or putting an end to the production of storable goods. References Eckstein, Otto and David Wyss, 1972, Industry price equations, in: The econometrics of price determination (Board of Governors of the Federal Reserve System, Washington, DC). Phlips, Louis, 1980, Intertemporal price discrimination and sticky prices, Quarterly Journal of Economics, 525- 542. Phlips, Louis and Jacques-F. Thisse, 1981, Pricing, distribution and the supply of storage, European Economic Review 15, 225-243. Pyatt, Graham, 1978, Marginal costs, prices and storage, The Economic Journal, 749-762. Smithies, Arthur, 1939, The maximization of profits over time with changing cost and demand functions, Econometrica, 3 12- 3 18.