There is a continuum of retail firms indexed by / across the unit interval. Each firm hires bundles of labour at the aggregate wage rate, Wt. They hire capital from the households in a perfectly competitive factor market at the rental rate of
capital, R$. The firms make the decision of how much labour, and how much capital, k fjt, to employ; and thus how much output to produce, y fit. They sell their output in a monopolistically competitive market at the price, and are constrained by production technology.
Each firm chooses an input mix to maximise profits:
yf ,tPf ,t — k fjR t ~~ h jW t
subject to its production function:
VU = z t k j / / - a)
Note th a t the technology, zt, is common across all firms.
The optimal inputs are therefore:
( 1 - a)yu Pf ,t
1 = Wt
When firms charge different prices, the optimal level of inputs varies across firms.
However, the optimal capital to labour ratio is constant across firms:
kf,t _ a Wt
l u (1 - a ) R f
W l~ aR f M C f t
= ---^
zto a{ 1 — q:)^1
I assume Calvo-type nominal rigidities in the goods market, similar to those in the labour market. In each period a randomly chosen fraction, (1 — 7 7), of the firms are free to reset their prices. These firms set new prices taking their respective demand curves as given. The remainder of the retail firms cannot reoptimise, but adjust their price by the steady state inflation, (1 + II).
If a firm has the opportunity to reset its price then it chooses the new price, P j t. The general price level is:
I assume the artifice of a perfectly competitive goods bundler employing Dixit- Stiglitz technology. Each individual firm, therefore, faces the demand curve:
In periods when the firm gets the opportunity to choose a new price, it chooses the price which maximises its expected discounted future stream of profits across the states of nature for which th a t price will hold. In other words, it maximises:
oo .
Et E w r '-f [
v,
p! {(i+nr' p'uY~e - TCr]
s = t 1
This yields the optimal price:
°o
. Et 'El(r)P{l + n ) - ,,)’- ih.P.y.M Cu
D * ___ S — t
l f t —a i 00 1 /
9 1 £,£(*>/» U + n j ' - V A . j ? # .
S = t
In other words, the firm sets the price so that its expected value is equal to a mark up, over expected marginal cost. In the case of no price stickiness (i.e. 77 = 0), Pf t = j z j M C f ]t. This is the standard result that under monopolistic competition firms set price as a mark up over marginal cost.
4.2.3. A ggregation
For tractability, I assume th a t there are full contingent claims markets. Given the ex-ante homogeneity of the households, this ensures th at consumption and wealth are constant across all households. Effectively, risk averse households will insure against not being able to adjust their wage rate. Hence, Ch,t = Q V/i and
It also entails th a t all households which are free to optimally set their wage in a given period are in exactly the same position and will choose the same wage,
Firms which are free to set their optimal price axe also all in identical positions, and so PJt = P f \/h. Furthermore, I have already shown th a t marginal cost and the capital to labour ratio are the same across all firms.
In order to aggregate output, I begin by noting th a t an individual firm’s demand and supply must be equal, and then integrating across all firms:
Am = At V/i.
w i t = w ; VA
1 1
0 0
1 1
0 o
i i ( I ) I lf-tdf = VtpSt I p' ”df
o o
yt =
„ L.OC/(!-“) ZtKt tt
pdt
l where pdt = P? j P fJ df
0
In other words, aggregate output is a decreasing function of price dispersion, pdt.
4.2.4. A sset P ric e s
The model so far expounded is a relatively standard New-Keynesian model. In this section I depart from th a t standard and introduce heterogeneous forecasting of asset prices. The behavioural finance literature suggests th a t the assumption of rational expectations in asset markets is difficult to support. The suggestion is th at asset markets are prone to uncertainty and speculation in a way in which goods markets and labour markets are not. For this reason a more complex specification of the forecasting rules employed in asset markets is needed, rather than a simple appeal to rationality.
I will, however, begin with an account of asset prices under rational expec
tations. This will provide a benchmark against which to assess the behavioural
model which I consider thereafter. It also provides us with a measure of the fun
damental value of the asset, and this will form the basis for one of the behavioural forecasting rules.
4.2 .4 .1 . A sse t P ric e s U n d e r R a tio n a l E x p e c ta tio n s. If I were to model an equity market in a rational expectations framework, I could simply introduce equity trading into the households’ optimisation problem. This would entail an equity price which is the discounted sum of all future rental payments to capital:
S = l ~ f-1
Alternatively, I can express this as the discounted value of the sum of the next period rental payment and price:
I define Q l, the price of equity in the rational model, as the fundamental value of equity.
Equivalently, in real terms, I have:
In this way, I can see th a t in a rational expectations model the present equity price is a function of future expectations of the equity price, or equivalently of the
Q l = E t
0
^ - i [R k+l + Q,*+1](4.1) ql = E t0^ I [rf+1 + %*+1]
infinite series of returns. It is the ability of agents to form such expectations in highly volatile equity markets th a t behavioural finance questions.
Behavioural finance suggests th at, because of the limited rationality of indi
vidual agents and effective limits on arbitrage, the actual asset price at any time can deviate from the fundamental level th a t ql represents. This is equivalent in a macroeconomic context to arguing th a t there is a bias to the expected future return on capital. In the next section, we derive th at bias.
4.2.4.2 . A sse t P ric e s U n d e r B e h a v io u ra l A ssu m p tio n s. I follow in the tra dition of Frankel and Froot (1986 [42]) and De Grauwe and Grimaldi (2006 [31]) in specifying an alternative determinant of asset prices. They highlight the impor
tance of the frictions between chartist and fundamental forecasts in the determi
nation of asset prices.
I assume th a t agents operating in the capital markets choose between two simple heuristical rules when forecasting future asset prices. At the beginning of each period chartist and fundamentalist forecasts of the asset price this period and next are formed. The forecasts of this period’s asset price then determine the actual asset price via a bargaining process. The forecasts of next period’s asset price imply a particular expected return on capital.
The chartist forecasts of the present and next period asset price are a simple extrapolation of the historical price series:
(4-2) E Cyt (qt) = qt- i + Xc(Qt-1 — q t -2) + x l ( Q t -2 — q t -3) + •••
(4.3) E Cjt (qt+1) = E Cft (qt) + x c ( E c,t ( q t ) — q t -1) + x l ( Q t -1 — q t -2) + •••
It is arguable as to whether such heuristical rules should be specified in real or nominal terms. I choose real terms on the basis that, in this paper, I am attem pting to address the issue of asset market bubbles, and hence I want to avoid the issue of money illusion.
The fundamentalist forecast is th a t the asset price will move back towards its fundamental value, q%, during the next period, unless it is already close to the fundamental, defined by the bounds EC:
(4.4) E u (qt) = qt- i - X ffa t-1 - Q t ) where \qt_i - q*t \ > C
= Qt- 1 where \qt- i - q*t \ ^ C
(4.5) E u (qt+1) = E u (qt) - X f ( E f , t ( q t ) ~ Qt ) where \qt_i - q*| > C
= Ef,t ( q t ) where \qt- ! - q*t \ < C
The actual asset price is determined via a bargaining process between those who favour the chartist rule and those who favour the fundamentalist forecast:
(4.6) qt = wCjtE Cjt (qt) + Wf,tE u (qt)
where the weights depend on the past performance of the two forecasts. I use the functional form:
(A ^ exp(i;ftc>t)
(4.7) wct
-(4.8) w f t =
exp (vQfj) + exp(r;0C;t) ______ e x p j v Qf j ) ______
exp (vSlft) + exp (i>ftCjt)
The parameter v represents the propensity with which agents switch between fore
casting rules. and Qf jt axe the excess returns over holding bonds associated with following the chartist forecast and the fundamentalist forecast respectively.
They are calculated as follows:
(4.9) n Cft = [qt - i — q t - 2 (l + r t-i)] s ig n [Ecj- 2 ( q t - 2 ) ~ q t - 2 ]
(4.10) = [qt_ 1 - qt_2 ( l + rJ_j)] .sign [EU - 2 (qt-2) - qt-2]
where qt - 2 (l + f't-i) represents the return to investing funds in bonds and qt- \ is the return to investing the same funds in equities, sign [EC j t - 2 ( q t - 2 ) — q t -2] takes the value -1 when Ecj_ 2 ( q t - 2 ) < q t -2, in which circumstances an agent following the chartist rule would choose to invest in bonds rather than equities, and takes
the value +1 when chartists choose equities over bonds, sign [-Ey,f- 2 (Qt-2) — Qt-2] is analogous.
The average expected return on capital th at is implied by these behavioural forecasting rules can be solved for by substituting the actual asset price and the weighted average forecast of next period’s asset price for the respective fundamental values in equation 4.1. Therefore:
(4.11) E bftr^+ 1 = %-^rr [wc,tEc,t (qt+1) + wf,tEf,t (<7t+i)]
P £jtM+ 1
In this way, we can think of behavioural rules as driving an asset mis-pricing by inducing a bias in the expected future return on capital.