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מאמריםמאמריםמאמריםמאמרים סדרתסדרתסדרתסדרת DISCUSSION PAPER SERIES
31905 חיפה ,הכרמל הר ,לכלכלה החוג ,חיפה אוניברסיטת University of Haifa, Department of Economics, Mt. Carmel, Haifa 31905 Israel
http://hevra.haifa.ac.il/econ/index.html
Sticky Prices and Sequential Trade
Benjamin Eden
Discussion Paper No. 01-05 July 2001
STICKY PRICES AND SEQUENTIAL TRADE
Benjamin Eden*
The University of Haifa
July 2001
I consider a cash-in-advance model in which agents arrive at the
market-place sequentially and choose goods according to McFadden's
random utility maximization model. I allow for price dispersion and a
two stage production process to get a positive relationship between
money and output. It is argued that a single-sticky-price model does not
deliver a monotonic relationship between money and output. The paper
thus makes a connection between the sticky-price literature and the
uncertain and sequential trade model.
JEL Classification numbers: E000, E300, E400.
Mailing address: Department of Economics, The University of Haifa, Haifa
31905, Israel.
E-mail: b.eden@econ.haifa.ac.il
2
1. INTRODUCTION
Sticky price models attribute the short run real effects of money
to the presence of price rigidities. It is typically assumed that (a)
prices do not adjust immediately to changes in the money supply and (b)
there is a commitment on the part of the firm to supply any quantity
demanded at the not fully adjusted prices.
The failure to fully adjust all prices to changes in the money
supply is often rationalized by the existence of fixed menu type cost
for changing nominal prices. Akerlof and Yellen (1985) and Blanchard and
Kiyotaki (1987) show that even when these fixed menu costs are small
they can cause large aggregate effects in a monopolistically competitive
environment.
The second assumption about the commitment of firms to supply any
quantity demanded has received less attention. Here I show that once
this assumption is relaxed, we may get a negative rather than a positive
relationship between money and output. This is shown in a
monopolistically competitive, cash-in-advance environment in which
buyers arrive at the market-place sequentially and choose a single brand
according to McFadden's random utility maximization model. The reason
for this somewhat surprising result is in the assumption that money
earned today is spent in the next period. Therefore, when a seller
observes a high money supply in the current period he expects a high
next period price level and a low real wage.
To overcome this difficulty I assume a two stage production
process and allow for many prices. In equilibrium sellers know exactly
for which realizations of the money supply they will satisfy demand. We
3
may therefore think of sellers as making a contingent demand satisfying
commitments. This type of demand satisfying behavior is rather
realistic. Stores often hit a capacity constraint and do not satisfy the
entire demand. They may also require buyers who arrive "late" to pay a
higher price. For example, a store may declare an item on "sale" and
commits to satisfy demand when the money supply is low. If the money
supply is high some buyers will not be able to find the item at the
sale-price and will have to buy it at the higher list-price. It may be
argued that the proposed model is more realistic than the single-sticky-
price model. More importantly, it yields an unambiguous positive
relationship between money and output.
The proposed model is a version of the uncertain and sequential
trading (UST) model in Eden (1994), Lucas and Woodford (1994), Bental
and Eden (1996), Woodford (1996) and Williamson (1996). In previous UST
models buyers who arrive at the market-place can see all supply offers
and buy at the cheapest available offer. In equilibrium sellers face a
tradeoff between the price and the probability of making a sale: a
seller who quotes a high price will make sale with low probability. The
fraction of output sold at a given price is therefore zero or unity.
This aspect has been criticized as unrealistic and more importantly it
poses a difficulty in applying the model to explain micro data (see,
Eden [forthcoming]). Here buyers do not always buy at the cheapest
available price and therefore in equilibrium some quantity of the more
expensive goods will always be sold. The tradeoff that arises in
equilibrium is between the average fraction of output sold and the price
rather than between the probability of making a sale and the price.
4
2. A STICKY PRICE MODEL
I consider a cash-in-advance model in which the typical
household is a worker/shopper pair. To simplify, I assume that the
single period utility function of the representative household is given
by c - v(L) where c denotes consumption and L denotes the labor input
supplied by the worker. The cost function v( ) has the standard
properties (v' > 0 and v'' > 0 everywhere). The household's discount
factor is given by 0 < β < 1. I use the beginning of the period money
supply (per household) as the unit of account and call it a normalized
dollar.1
At the beginning of the period the household starts with m
normalized dollars (in equilibrium m = 1) and gets a transfer of x
normalized dollars, where -1 ≤ x ≤ ∞ is the random rate of change in
the money supply. It is assumed that x is i.i.d with a density function
φ(x).
There is a large number (n) of households. Each worker produces a
different brand of the consumption good using a constant returns to
scale technology: One unit of output per unit of labor. The typical
worker first chooses his price. He then observes the money supply shock
x and then chooses output.
Buyers arrive sequentially and in equilibrium each spends 1 + x
normalized dollars on a single brand. I follow McFadden (2000) random
1 Thus, I divide all nominal magnitudes by the pre-transfer money
supply.
5
utility maximization model and assume that when n brands are available
the probability of choosing brand i is:
(1) Prob(i) = w i/Σnk=1 w
k,
where w i = exp(bZ
i), Z
i is a vector of attributes of brand i and b is a
vector of parameters.
I start from the case in which the only relevant attribute of a
brand is its relative price:
(2) w i = exp[α(p/p i)],
where p i is the price of brand i and p is the average price of all other
brands. It will be shown that under (2) the demand for brand i is
similar to what one gets when employing the Dixit-Stiglitz (1977)
utility function, which is more common in the macro literature. Here I
use the random utility maximization model because of the possibility to
introduce non-price competition. This will be attempted later.
Assuming that all other sellers quote the price p and that all the
n alternatives are available, the probability that the buyer will buy
brand i at the price p i is (approximately)
2:
(3) exp[α(p/p i)]/n[exp(α)].
2 The exact expression is: exp[α(p/p i)]/{(n-1)[exp(α)] + exp[α(p/p i)]}.
This is equal to (3) when n is large.
6
Note that when p i = p, the probability that the buyer will choose brand
i is 1/n. When p i > p the probability of choosing brand i is less than
1/n but is greater than zero. This is the monopolistic competition
aspect of the environment.
Seller i expects that other sellers will satisfy demand when
x ≤ ζ, where ζ is a cut-off parameter that will be determined in
equilibrium. Therefore when he observes x ≤ ζ, he assigns the
probability (3) to the event that a buyer will choose his product. He
can then choose whether to satisfy demand or not.
If he satisfies demand after observing x ≤ ζ, he will get:
(4) R(p i, p, x) = (1 + x)n{exp[α(p/p i)]}/n{exp[α]}
= (1 + x){exp[α(p/p i)]}/{exp[α]},
normalized dollars. The computation of (4) uses the assumption that n is
large and therefore revenues are equal to total spending, (1 + x)n,
times the probability that the good will be chosen. If, after observing
x ≤ ζ, the seller does not choose to satisfy demand he will get p iL
normalized dollars. Revenues when x ≤ ζ are therefore given by:
min[R(p i, p, x), p
iL].
When x > ζ, seller i expects that other sellers will not satisfy
demand and that he will be able to sell any amount at the price p i. His
revenues in this case are simply p iL.
Since sellers are not required to satisfy demand, buyers who
arrive late may not make a buy. The probability of making a buy is
denoted by Π(x).
7
Because of symmetry all sellers face the same revenue functions
and the same maximization problem. Labor is supplied to create money
that will be spent in the next period. To set the labor choice problem,
I use V(m) to denote the maximum expected utility that a household can
obtain when it starts the period with m normalized dollars. This value
function will soon be defined by a Bellman equation.
A seller who quoted the price p i and observes that other sellers
have quoted the price p and x ≤ ζ will choose labor by solving:
(5) G(p i, p, x) = max
L≥0 - v(L)
+ Π(x)βV{min[R(p i, p, x), p iL]/(1 + x)}
+ [1 - Π(x)]βV{[m + x + min[R(p i, p, x), p iL]]/(1 + x)}.
The first row in (5) is the disutility from supplying labor. The sum of
the second and third rows is the expected future utility which depends
on the beginning of next period normalized balances. These are:
m' = min[R(p i, p, x), p
iL]/(1 + x) if the buyer makes a buy and
m' = [m + x + min[R(p i, p, x), p
iL]]/(1 + x) if he does not make a buy.
Note that we divide current normalized dollars by 1 + x to convert them
to next period's normalized dollars.
A seller who observes x > ζ will choose labor by solving:
(6) g(p i, x) = max
L≥0 - v(L) + Π(x)βV[p iL/(1 + x)]
+ [1 - Π(x)]βV[(m + x + p iL)/(1 + x)].
8
Here next period's nominal balances are given by m' = p iL/(1 + x) if the
buyer makes a buy and by m' = (m + x + p iL)/(1 + x) if he does not make
a buy.
The representative household takes the parameter ζ, the price
quoted by others, p, and the functions Π(x) and R(p i, p, x) as given
and solves the following Bellman equation:
(7) V(m) = ∫∞-1 Π(x)[(m + x)/p]φ(x)dx
+ max
p i ∫ζ-1 G(p
i, p, x)φ(x)dx + ∫∞ζ g(p
i, x)φ(x)dx
s.t. (5) and (6).
We now solve for the labor supply decisions, which takes
(p i, p, x) as given. When x ≤ ζ the individual seller solves (5). The
first order condition for this problem are:
(8) R(p i, p, x) ≥ p iL ;
(9) v'(L) ≤ βV'p i/(1 + x), with equality if R(p i, p, x) > p iL.
Condition (8) says that it is not optimal to produce more than the
quantity demanded. Condition (9) says that the marginal cost, v'(L),
must be lower than the marginal benefit. The marginal cost must equal to
the marginal benefit if there is excess demand for the firm's output.
9
When x > ζ the individual seller solves (6). Since in this region
the seller can sell as much as he wants at the price p i, the first order
condition for this problem requires that the marginal cost is equal to
the marginal benefit:
(10) v'(L) = βV'p i/(1 + x).
We can also solve for the constant marginal utility of money:
(11) V' = π/p + (1 - π)βV' = π/p(1 - β + πβ),
where π = ∫∞-1 Π(x)φ(x)dx denote the probability that the shopper will
buy. To derive (11) note that an additional unit of money will buy in
the current period with probability π and if it buys it yields 1/p
utils. With probability 1 - π the additional unit of money will not buy
and will be carried to the next period yielding βV' utils. The marginal
utility of money is a constant because we assume risk neutrality.
Equilibrium is a pair of scalars (p, ζ) and the functions
[Π(x), R(p i, p, x), L(x)] such that (4) is satisfied and
(a) Given (p, ζ) and the functions [Π(x), R(p i, p, x)], the price
p i = p and the output L(x) solve (7);
(b) Π(x) = min{1, pL(x)/(1 + x)};
(c) 1 + x = pL(x) for all x ≤ ζ and 1 + x > pL(x) for x > ζ.
Equilibrium condition (a) requires that [p, L(x)] will solve the
household's problem. Equilibrium conditions (b) and (c) require rational
10
expectations. The probability of making a buy is given by (b): In the
case of excess demand it is the ratio of nominal supply to nominal
demand. Condition (c) says that excess supply occurs when x ≤ ζ and
excess demand occurs when x > ζ. As in the disequilibrium literature the
quantity transacted is the minimum between supply and demand. (See Barro
and Grossman [1971], for example).
In an Appendix which is available upon request I show that there
exists an equilibrium if the demand elasticity α is sufficiently large.
Here I analyze the properties of the equilibrium labor supply function,
L(x).
In equilibrium we can write the first order conditions (9) - (11),
as:
(12) v'[(1 + x)/p] ≤ A/(1 + x) for x ≤ ζ,
(13) v'[L(x)] = A/(1 + x) for x > ζ,
where A = βπ/(1 - β + πβ) = pβV'. Note that the marginal benefit,
A/(1 + x), does not depend on the price p. This is because the marginal
utility of money, V', is inversely related to p.
The equilibrium marginal cost is described as a function of x by
the solid line in Figure 1. When x ≤ ζ the demand constraint is binding
and the marginal cost, v'[(1+x)/p], is less than the marginal benefits,
A/(1+x). When x > ζ, there is excess demand and the marginal cost is
equal to the marginal benefit: A/(1+x).
11
Figure 1
Since labor supply is a monotonic function of the marginal cost,
it follows that
Claim 1: The equilibrium labor supply function, L(x), is increasing when
x ≤ ζ and then decreasing when x > ζ.
This is not surprising. Since money earned today is used in the
next period, the relevant real wage is the price p deflated by next
period price level, p(1 + x) which yields 1/(1 + x). A higher
realization of x therefore implies a lower real wage which in the excess
demand region leads to less supply of labor.
Problems with the equilibrium concept: It was mentioned before that the
existence of equilibrium proof is only for the case in which the demand
elasticity α is high. The reason for the difficulty in proving
existence may be in the requirement that in equilibrium all sellers must
12
post the same price. This requirement does not make sense once we remove
the demand satisfying assumption. An individual seller may want to
choose a higher price and sell only when x > ζ and there is excess
demand.
Another problem is in the behavior of sellers in the range of
excess supply. We expect the seller to promote sales whenever he wants
to raise his price but cannot. This argument is not new. Stigler (1968)
starts his article on price and non-price competition by saying that:
"When a uniform price is imposed upon, or agreed to by, an industry,
some or all of the other terms of sale are left unregulated. The setting
of taxi-meter rates still allows competition in the quality of the
automobile. The fixing of commission rates by the New York Stock
Exchange still allows brokerage houses to compete in services such as
providing investment information". In his article he uses advertising as
a prototype of non-price variables. For other examples see, Spence
(1977) and Dixit (1979).
In the context of this paper we may note that McFadden (1978) has
found that non-price variables, like the time it takes to get on a bus,
are important in determining the probability of choosing the mode of
transportation. We may therefore expect that a bus company which want to
but cannot increase its price, will increase the frequency at which
buses arrive at the station.
In the Appendix I substitute (2) by:
(14) w i = exp[α(p/p i) - γ(Y/Y i)],
13
where Y i is the amount of labor employed by firm i, Y is the average per
firm amount of labor and γ > 0 is a parameter. The specification (14)
assumes that the firm can increase the probability that its brand will
be chosen by increasing the amount it produces relative to others. Under
one possible interpretation, we may think of the good as being sold in
many locations. The more McDonnalds are in town and the less time you
have to wait for getting serviced, the higher is the probability that
the buyer will choose McDonnalds over the alternatives.3 Advertisement
is another possible interpretation.
It is shown in the Appendix that under (14), the equilibrium labor
supply function L(x) may be decreasing in the entire range.
To overcome these difficulties I now propose a model that allows
for equilibrium price dispersion and a two stage production process.
3. A SEQUENTIAL TRADE MODEL
I assume a two stage production process. Fish restaurants may be a
good example. Fresh fish are bought and cleaned before lunch time. Then
the fish is cooked when customers actually sit down and order. We may
assume that cleaning the fish (creating capacity) occurs before the
realization of x is observed. Serving the meal occurs after the
realization of x, when demand is actually realized. Since the variable
3 Here we define goods by its physical characteristics only. In the
standard model goods are characterized by location as well. We may
therefore think of the variable Y/L as measuring the relative number
of spatially indexed goods produced by household i.
14
costs (of serving the meal) are relatively small, the restaurant will
choose to satisfy demand up to its capacity limit.
I allow for price dispersion. When there are many prices and
capacity limits, a commitment to satisfy demand may take two forms. A
store may plan to sell at the announced price up to the capacity limit.
A store may also plan to sell a certain quantity at a lower "sale" price
and then, if demand is high, to sell at a higher "list price" to late
arrivals. Our model allows for both types of demand satisfying
behavior. In general sellers can make contingent demand satisfying
commitments of the type: "I will satisfy the demand at the price qs if
the money injected to the economy is less than x s".
I keep the cash-in-advance structure of the previous section. It
is assumed that the transfer payment is done by helicopter (as in
Friedman [1969]) which drops money on the agents in the economy. This is
a process rather than a single shot. Everyone observe the money rain
("helicopter money") falling but no one knows when it will stop.
Capacity is chosen before the beginning of the money transfer process.
Capacity may then be converted to output at zero cost if there is demand
for it.
It is assumed that the total amount of transfer x may take S
possible realizations: x 1 < x 2 < ... < x S. The realization x
s occurs
with probability Π s. Buyers spend the money immediately after getting
it. They will first spend 1 + x 1 normalized dollars per seller (buyer).
Then, with probability ψ 2 = ΣSs=2 Π s a second transfer will arrive and
everyone will spend an additional amount of ∆ 2 = x 2 - x
1 normalized
dollars per seller and so on until the transfer process ends.
There are S markets. The price in market s is denoted by q s, where
15
q 1 ≤ q 2 ≤ ...≤ q S. A good supplied to market s will be bought by
fractions of the transfers: 1,...,s. Thus, if the first s transfers are
realized then all the goods supplied to market s are sold. If only
j < s transfers are realized then only a fraction of the goods supplied
to market s are sold. This is different from previous sequential trade
models in which buyers always buy at the cheapest available price and
therefore all good supplied to market s are sold if the first s
transfers are realized but none of the goods supplied to market s are
sold if only j < s transfers are realized.
The seller chooses capacity before the beginning of trade and
allocates it among the S potential markets. We may think of this
allocation as a choice of price tags: A seller who chooses the price tag
q s for a given unit supplies it to market s. We may also say that a
seller who supplies to market s is committed to satisfy the demand until
transactions in market s are completed. But this commitment is not
binding in equilibrium: Sellers make a plan which is time consistent and
have no incentive to change their plans during the trading process. The
sequential trade process is illustrated by Figure 2.
16
Figure 2
After the completion of trade in market j-1 the available price
offers are (q j, ..., q
S) and the average price offer is:
(15) p j = ΣSi=j q
i/(S+1-j).
I assume that after the completion of trade in market j-1 the
probability of choosing a good with the price tag q s (s ≥ j) is given
by: w s/ΣSi=j w
i, where w
i = exp[α(p j/q
i)]. Thus here only the relative
price play a role in the buyer's choice. Unlike the case of the single-
sticky price model in the previous section, here the main results are
robust to allowing for non-price competition.
A seller who quotes the price q s will get a fraction:
(16) F j(p
j, q
s) = exp[α(p j/q
s)]/ΣSi=j exp[α(p j/q
i)],
17
of transfer j ≤ s if it is realized.4 The expected revenue for a seller
who quotes the price q s and satisfy demand until transactions in market
s are completed is therefore:
(17) Σsj=1 ψ j∆ jF j(p
j, q
s),
normalized dollars, where ψ j = ΣSs=j Π s is the probability that transfer
j will be realized. In terms of next period's normalized dollars this
is:
(18) Σsj=1 ω jψ j∆ jF j(p
j, q
s),
where ω j = ΣSi=j Π i/ψ j(1 + x i) is the expected value of a current
normalized dollar in terms of next period's normalized dollar given that
transfer j was realized.
The quantity required for satisfying the demand at the price q s
until transactions in market s are completed is:
(19) k s = Σsj=1 ∆ jF
j(p
j, q
s)/q
s.
4 Note that when α is large,
F j(p
j, q
s) = exp[α(p j/q
s)]/ΣSi=j exp[α(p j/q
i)] approaches unity for
s = j and zero otherwise. In this case, buyers always buy at the
cheapest available price as in previous uncertain and sequential
trading (UST) models of the type studied by Eden (1994) and Lucas and
Woodford (1994).
18
The expected revenue per unit supplied at the price q s is denoted by Γ s
and is obtained by dividing (18) by (19). Thus,
(20) Γ s = q sΣsj=1 ω jψ j∆ jF
j(p
j, q
s)/Σsj=1 ∆ jF
j(p
j, q
s).
The representative household takes the prices (q 1,...,q
S), the
expected per unit revenues (Γ 1,...,Γ S) and the functions F s(p
s, q
i) as
given and chooses the quantities (k 1, ..., k
S) to solve the following
Bellman's equation:
(21) V(m) = (m + x 1)ΣSi=1 F
1(p
1, q
i)/q
i + ΣSs=2 ψ s∆ sΣSi=s F
s(p
s, q
i)/q
i
+ max
{k s} - v(ΣSs=1 k
s) + βV(ΣSs=1 Γ sk
s),
where risk neutrality is used to substitute EV(m') by V(Em').
The first order conditions for this problem are:
(22) βΓ sz = v'(ΣSs=1 k s) for all s,
where z = V' = ΣSi=1 F 1(p
1,q
i)/q
i is the expected purchasing power of a
normalized dollar held at the beginning of the period. Condition (22)
says that the benefits of supplying a unit to market s must equal to the
marginal cost.
Equilibrium is a vector (q 1,...,q
S; Γ 1,...,Γ S; k
1,...,k
S) which
satisfies (19), (20) and (22).
19
Note that the equilibrium concept is competitive in the sense that
sellers take the real price (the expected revenue per unit, Γ s) in each
market as given. Sellers equate marginal cost to the real price. We
restrict the choice of prices (markets) to the set (q 1,...,q
S). This
restriction is a problem if x s - x
s-1 are large. In what follows we will
assume that x s - x
s-1 are small so the probability distribution of x is
close to a continuous distribution.
Solving for equilibrium:
I say that market s opens when transfer s arrives. (In our
framework some goods allocated to market s are sold before transfer s
arrives and therefore market s leaks before it opens). I use
π s = Π s/ψ s-1 to denote the probability that market s will open given
that market s-1 open. I now solve for equilibrium under the assumption
that δ s = π sω s/ω s-1 is close to unity.5
We choose q S arbitrarily. We now imagine that transfer S-1 does
occur and market S-1 opens. In equilibrium all units with a price tag
q S-1 can be sold when market S-1 opens and a seller has no incentive to
change price tags. We therefore look for a price q S-1 that will make the
seller indifferent between selling the unit in market S-1, at the price
q S-1, to allocating it to market S. For this purpose we first compute
5 This is the case when the probabilility distribution of x can be
approximated by a continuous distribution with a mass point at x = xS.
20
the expected revenue (in terms of next period's normalized dollars) when
allocating the unit to market S. This is:
(23) ∆ S-1ω S-1F S-1(p
S-1, q
S) + π S∆ Sω S,
where p S-1 = (
1/2)(q S + q
S-1) is the average price when market S-1
opens; π S = Π S/ψ S-1 is the probability that market S will open given
that market S-1 open; ω S-1 = (1/2)[1/(1 + x
S-1) + 1/(1 + x
S)] and
ω S = 1/(1 + x S) are the values of a normalized dollar in terms of next
period normalized dollars.
The quantity required to satisfy demand at the price q S is:
(24) [∆ S-1F S-1(p
S-1, q
S) + ∆ S]/q
S.
The revenue per unit allocated to market S, conditional on market S-1
being opened, is obtained by dividing (23) by (24). This yields:
(25) γ S = q Sω S-1[∆ S-1F
S-1(p
S-1, q
S) + δ S∆ S]/[∆ S-1F
S-1(p
S-1, q
S) + ∆ S],
where δ S = π Sω S/ω S-1 < 1.
When market S-1 opens, the seller can sell the entire unit in
market S-1 and get for it q S-1ω S-1 next period's normalized dollars. We
require that the expected per unit revenue will be the same for both
price tags. Thus:
(26) q S-1ω S-1 = γ S.
21
Since, δ S < 1, (25) implies that q Sω S-1 > γ S and therefore (26)
implies: q S > q S-1. Note also that γS is an increasing function of q
S-1.
When q S-1 = 0, F
S-1 = 0 and γ S = π Sq
Sω S. Since we assume that
δ S = π Sω S/ω S-1 is close to unity, the derivative of γS with respect to
q S-1 is small. Under this condition there is a solution, q
S-1(q
S), which
is illustrated by Figure 3.
Figure 3
We can now use q S-1(q
S) to find the price q
S-2 that will make the
seller indifferent to changing the price tag on a given unit from q S-2
to q S-1(q
S), when market S-2 opens. We then use
q S-2(q
S) = q
S-2[q
S-1(q
S)], to find q
S-3(q
S) and so on until we find all
prices: q 1(q
S), ...,q
S-1(q
S). It can be shown that
q 1(q
S) < q 2(q
S) < ...< q S.
22
We now use (19) to compute the total amount required to satisfy
the demand in market s:
(27) k s(q
S) = Σsj=1 ∆ jF
j[p
j(q
S), q
s(q
S)]/q
s(q
S).
Since q 1(q
S) < q 2(q
S) < ...< q S, total demand,
d(q S) = ΣSs=1k
s(q
S), is large when q
S is small. Since
q s(q
S) > δ s+1q
s+1(q
S), total demand d(q
S) is small when q
S is large.
Supply is given by (22) which can be written as:
(28) v'(k) = βΓ 1(q S)z(q
S) = βq 1(q
S)ω 1z(q
S),
where z(q S) = ΣSi=1 F
1[ΣSj=1 q
j(q
S)/S, q
i(q
S)]/q
i(q
S). Since v'(0) = 0 and
v'( ) is increasing, there exists a solution, s(q S) = k(q
S), to (28).
We now equate supply and demand:
(29) s(q S) = d(q
S).
We denote the solution to (29) by q̂ S and compute all equilibrium prices
and quantities: q s(q̂
S), k
s(q̂
S).
Note that the real wage term (the right hand side of [28]) is:
βω 1ΣSi=1 [q 1(q
S)/q
i(q
S)]F
1[ΣSj=1 q
j(q
S)/S, q
i(q
S)]. Therefore, if the
elasticity of q i with respect to qS is unity, the real wage will not be
affected by the change in q S. The intuition is that when all prices
change by the same proportion, the change in the purchasing power
completely offset the change in current prices. It is therefore possible
that supply is flat as in Figure 4.
23
Figure 4
Discussion:
There is an ongoing debate about modeling prices. Some believe
that in the real world prices are sticky. Other believe that prices are
completely flexible. Here I propose a sequential trading model that may
bridge the gap between these two approaches. In the proposed UST model
sellers announce prices in advance but do not have an incentive to
change them during the trading process: Prices look as if they are
sticky but they are not.
Some researchers argue that in the real world sellers are eager to
sell and therefore we should assume demand satisfying behavior in our
24
models. In a UST equilibrium sellers are eager to sell but not
necessarily at the cheapest price. Thus the UST framework allows for a
"new Keynsian" interpretation which assumes sticky prices and some
demand satisfying commitment. It also allows for a competitive
interpretation that assumes flexible prices. Under both interpretations
there is a positive relationship between output (capacity utilization)
and money.
The UST model in this paper is more realistic than previous UST
models. In previous UST models buyers buy at the cheapest available
price so that cheaper goods are sold first (as in Prescott [1975]). Here
buyers may buy at prices which are higher than the cheapest alternative
and therefore in equilibrium the trade-off is between average capacity
utilization and the price. This added realism is important for the
analysis of micro data.
The UST model assumes a two stage production process and allows
for price dispersion. I argue against the alternative of a single-price
single-production-stage environment. These assumptions are not
realistic. Price dispersion is observed and sellers do not always
satisfy demand at the cheapest price. More importantly, the single-
price model does not yield a monotonic relationship between money and
output. This is because in a cash-in-advance model, money earned today
is spent in the next period and therefore when sellers observe a high
money injection they expect a high next period price level and a low
real wage. The real wage effect operates in the direction of a negative
relationship between money and output. This effect becomes even more
important when sale promotion is allowed.
25
What is the importance of the real wage effect in other sticky
price models? In a Taylor (1980) or a Calvo (1983) type model it is
possible that only a small fraction of the sellers change their nominal
prices each period. In this case a seller who observes a high money
supply will not adjust his expectations about the next period price
level by much. The real wage effect may therefore be unimportant if we
keep the assumption that sellers must satisfy demand. But if we relax
this assumption we may find that the real wage effect is important
because a high money supply is likely to have a negative effect on the
probability of making a buy in the next period (the ratio of nominal
supply to nominal demand). This will reduce the real wage because there
is now a higher probability that money earned today will be spent in the
more distant future (say 2 periods from now).
Thus, I question the micro-foundation of the new-Keynsian
economics. In particular, relaxing the assumption that demand must be
satisfied at the sticky-price makes a difference in the model we studied
and is likely to make a difference in other sticky-price models. The UST
alternative has a Keynsian flavour. Sellers are eager to sell and prices
look sticky. The policy implications however are neo-classical. Capacity
utilization and welfare will be maximized when money surprises are
eliminated (see Eden [1994] for example).
26
APPENDIX: ALLOWING FOR NON-PRICE COMPETITION IN THE STICKY-PRICE MODEL
Here I allow for non-price competition by using
w i = exp[α(p/p i) - γ(Y/Y i)], where Y
i is the amount of labor employed
by firm i, Y is the average per firm amount of labor and γ > 0 is a
parameter. I show that in a single-sticky-price equilibrium the
output/money relationship may be negative under this specification.
I start by computing the revenue function in a way which is
similar to the derivation of (4). When x ≤ ζ and all other sellers quote
the price p and produce the output Y, a seller who quotes the price p i
and produces the amount Y i will get:
(A1) r(p i, Y
i, p, Y, x) = (1 + x)exp[α(p/p i) - γ(Y/Y i)]/exp(α - γ),
normalized dollars if he satisfies demand and p iY i otherwise. When
x > ζ, the seller can sell as much as he wants at the price p i.
The household takes p, ζ and the functions Π(x) and
r(p i, Y
i, p, Y, x) as given and solves (7) after replacing the revenue
function R(p i, p, x) with r(p
i, Y
i, p, Y, x). When x ≤ ζ and other
sellers satisfy demand the labor supply choice is (5). The first order
condition for this problem are given by (11) and:
(A2) v'(L) = βV'r 2/(1 + x), if r(p
i, L, p, Y, x) < p iL ;
(A3) v'(L) = βV'p i/(1 + x), if r(p i, L, p, Y, x) > p iL ;
(A4) βV'r 2/(1 + x) ≤ v'(L) ≤ βV'p i/(1 + x),
27
if r(p i, L, p, Y, x) = p iL ;
where r 2 = ∂r/∂L = (γ/L)r(p i, L, p, Y, x) = γr/L is the return to sale
promotion. Note that the return to sale promotion is proportional to
average revenues r/L.
Conditions (A2) and (A3) say that the marginal cost must equal the
real wage. The first order condition (A2) must hold when there is excess
capacity. In this case an increase in labor by one unit will increase
revenues by r 2 = γr/L current normalized dollars which will become
r 2/(1 + x) normalized dollars in the next period. Therefore the real
wage for this case is: βV'r 2/(1 + x).
Condition (A3) must hold when there is excess demand. In this
case, the real wage is: βV'p i/(1 + x).
Condition (A4) must hold when the seller satisfies demand. In this
case, if the seller increases output by one unit he will get γr/L from
his promotion efforts. If he cuts labor by one unit he will lose p i
dollars. When γr/L < p i, the marginal cost must be between the real
wage βV'r 2/(1 + x) and the real wage βV'p i/(1 + x) because at the
optimum the seller cannot benefit from either increasing output or
reducing it.
When x > ζ, there is excess demand and the seller can sell as much
as he wants at the price p i. The first order condition in this case is
(10).
Equilibrium is a pair (p, ζ) and a vector of functions
[Π(x), Y(x), L(x), r(p i, Y i, p, Y, x)] such that (A1) is satisfied and:
28
(a) Given (p, ζ) and the functions {Π(x), r[p i, Y i, p, Y(x), x]}, the
price p i = p and the output L(x) = Y(x) solve (7) after replacing R( )
by r( );
(b) Π(x) = min{1, pL(x)/(1 + x)};
(c) 1 + x ≤ pL(x) for all x ≤ ζ and 1 + x > pL(x) for x > ζ.
In equilibrium, r(p, L, p, Y, x) = 1 + x. The first order
conditions (A2) and (A4) must hold when x ≤ ζ and
1 + x ≤ pL(x). The first order condition (10) must hold when x > ζ and
1 + x > pL(x). These conditions can be written in equilibrium as:
(A5) v'(L) = γA/pL, if 1 + x < pL(x);
(A6) γA/(1 + x) ≤ v'(L) ≤ Α/(1 + x)
if 1 + x = pL(x);
(A7) v'(L) = A/(1 + x), if 1 + x > pL(x); .
where A = βπ/(1 - β + πβ).
There is a unique solution, L*, to (A5). Thus, when there is
excess supply the equilibrium labor supply does not depend on x. The
intuition is as follows. When 1 + x < pL, the benefit from increasing
labor is in promotion of sales. In equilibrium an additional unit of
labor will increase revenues by (γ/L)(1 + x) current period normalized
dollars and by (γ/L) next period's normalized dollars. Since next
period's normalized dollars is the relevant measure, the promotion
29
benefit does not depend on x and therefore when the marginal benefit is
due to sale promotion, labor supply does not depend on x.
We now distinguish between two cases: γ > 1 and γ ≤ 1. When γ ≤ 1,
the marginal cost as a function of x is given by the solid line in
Figure A1. When x ≤ pL* - 1, there is excess capacity and the marginal
benefit (due to sale promotion) is constant and is equal to the marginal
cost: γA/pL*. Capacity is fully utilized when x = pL* - 1. When
pL* - 1 ≤ x ≤ ζ, the seller satisfy demand and the marginal cost is
given by v'[(1 + x)/p]. In this region the marginal benefit from selling
an additional unit at the price p (= A/1+x) is greater than the marginal
cost but the seller does not increase output because he is constraint by
demand. When x > ζ, there is excess demand and the seller can sell as
much as he wants at the quoted price p. He therefore chooses marginal
cost equal to the marginal revenues in this range.
Figure A1
30
Since labor is a monotonic function of the marginal cost, it has
three regions as in Figure A2.
Figure A2
We now turn to the case γ > 1. In this case, condition (A6) is
never satisfied. At x = ζ, the marginal cost drops from the constant
promotion level (γA/pL*) to the unconstraint level (A/1+x). This is
illustrated by Figure A3. The implied equilibrium labor supply function
is in Figure A4.
32
Claim 2: When sale promotion is allowed labor supply is not a strictly
increasing function of x even in the excess supply range (x ≤ ζ). When
γ > 1, the equilibrium labor supply function is weakly decreasing in x
in the entire range. When γ ≤ 1, the equilibrium labor supply is not a
monotonic function of x: It is flat at first, then increasing and then
decreasing.
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