\documentclass[10pt]{article}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}
\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
\usepackage[round]{natbib}
\let\cite\citep
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\begin{document}
\title{Ch. 6~ Nominal Rigidity: Part B (macro)}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\sloppy
\section*{}
{\label{408107}}\par\null\par\null
\section*{}
{\label{408107}}
\section*{Part B~ Microeconomic Foundations of Incomplete Nominal
Adjustment}
{\label{408107}}
Some type of incomplete nominal adjustment appears essential to
understanding why monetary changes have real effects. Although
individuals really care about real prices and quantities, and it doesn't
cost much to update prices, these little nominal frictions have a large
effect in the aggregate. This makes it worth analyzing, seeing when the
microeconomic conditions causing changing price costs lead to
significant nominal stickiness in response to onetime monetary shocks.
In particular, we will focus on a static model where firms face
a~\emph{menu cost} of price adjustment -- a small fixed cost for
changing a nominal price.~
\section*{6.5~ A Model of Imperfect Competition and
Price-Setting}
{\label{439332}}
Before turning to stickiness, we'll examine an economy of imperfectly
competitive price-setters with complete price flexibility. We are
interested in causes and effects of barriers to price adjustment, but it
is essential to see the foundation first.~
\subsection*{Assumptions}
{\label{574286}}
There is a continuum of differentiated goods indexed
by~\(i \epsilon [0,1]\). Each good is produced by a single firm with
monopoly rights. Firm i's production function is just
\begin{quote}
\(Y_i = L_i\)
\end{quote}
Firms hire labor in a perfectly competitive labor market and sell output
in an imperfectly competitive goods market. Thus firms can set their
prices freely and any profits they earn are accrued to the households.~
The utility of the representative household depends positively on its
consumption and negatively on the amount of labor it supplies. It takes
the form
\begin{quote}
\(U = C - \frac{1}{\gamma} L^{Y}, \gamma > 1\)
\end{quote}
Crucially, C is not the household's total consumption of all goods
(since all goods would be perfect substitutes for one another and firms
would have no market power). It is rather an index of the household's
consumption of the individual goods, taking the
constant-elasticity-of-substitution form:
\begin{quote}
\(C = [\int _{i = 0} ^1 C_i ^{(\eta - 1)/ \eta}]^{\eta / (\eta - 1)}\)
\end{quote}
This formulation parallels the production function in the Romer model of
endogenous technological change. Government purchases and international
trade are absent, so output matches consumption.
Households choose their labor supply and purchases of consumption goods
to maximize utility. Firms choose prices to maximize profits, taking
demand curves and wages as given. The aggregate demand side of the model
will be:
\begin{quote}
\(Y = \frac{M}{P}\)
\end{quote}
which implies an inverse relationship between price level and output.~
\subsection*{Household Behavior}
{\label{663250}}
It is easiest to start by considering how households allocate their
consumption spending among the different goods. Consider a household
that spends S. The Lagrangian for its utility-max problem is
\begin{quote}
\(\mathcal{L} [\int_{i = 0} ^1 C_i ^{(\eta - 1)/ \eta} di] ^{\frac{\eta}{\eta - 1}} + \lambda [ S - \int _{i = 0} ^{1} P_i C_i di]\)
\end{quote}
The first-order condition for Ci is
\begin{quote}
\(\frac{\eta}{\eta - 1} [\int_{j = 0} ^1 C_j ^{(\eta - 1)/ \eta} dj] ^{\frac{1}{\eta - 1}} \frac{\eta -1 }{\eta} C_i ^{-1/ \eta} = \lambda P_i\)
\end{quote}
The only terms that depend on i are consumption and price. Thus it must
take the form:
\begin{quote}
\(C_i = AP_i ^{- \eta}\)
\end{quote}
Finding A in terms of the given variables, we can substitute it into the
budget constraint and solve, yielding:~
\begin{quote}
\(A = \frac{S}{\int_{j = 0} ^1 P_j ^{1 - \eta} dj}\)
\end{quote}
Finally, substituting this to solve for the definition of C we find:
\begin{quote}
\(C = \frac{S}{(\int_{i = 0} ^1 P_i ^{1 - \eta} di) ^{1/(1 - \eta)}}\)
\end{quote}
This means that when households allocate their spending across goods
optimally, the cost of obtaining one unit of C is the denominator (which
is the price index corresponding to the utility function). These
expressions can simplify to
\begin{quote}
\(C_i = (\frac{P_i}{P}) ^{- \eta} \frac{S}{P} \Longrightarrow (\frac{P_i}{P}) ^{- \eta} C\)~
\end{quote}
Thus the elasticity of demand for each individual good
is~\(\eta\).
The household's other choice variable is its labor supply. Its spending
equals~\(WL + R\) where W is wage, R is profit income and so
the problem for choosing L is therefore
\begin{quote}
max~\(\frac{WL + R}{P} - \frac{1}{\gamma} L^\gamma\)
\end{quote}
The first-order condition for L is
\begin{quote}
\(\frac{W}{P} - L^{\gamma - 1} = 0\)
\end{quote}
Implying that~
\begin{quote}
\(L = (\frac{W}{P}) ^{\frac{1}{\gamma - 1}}\)
\end{quote}
Thus labor supply is san increasing function of the real wage, with an
elasticity equal to the exponent. This represents the aggregate value of
L.~
\subsection*{Firm Behavior}
{\label{292314}}
The real profits of the monopolistic producer of good i are its real
revenues minus its real costs:
\begin{quote}
\(\frac{R_i}{P} = \frac{P_i}{P} Y_i - \frac{W}{P} L_i\)
\end{quote}
Recall that the amount of the good produced must equal the amount
consumed. Substituting in the values of Yi and Li from the previous
equations yields:
\begin{quote}
\(\frac{R_i}{P} = (\frac{P_i}{P}) ^{1 - \eta} Y - \frac{W}{P} (\frac{P_i}{P}) ^{- \eta} Y\)
\end{quote}
Solving the first-order condition for Pi/P gives us:
\begin{quote}
~\(\frac{P_i}{P} = \frac{\eta}{\eta - 1} \frac{W}{P}\)
\end{quote}
That is, a producer with market power sets price as a markup over
marginal cost, with the size of the markup determined by the elasticity
of demand. This is our intuitive assumption.~
\subsection*{Equilibrium}
{\label{146755}}
Because the model is symmetric, its equilibrium is also symmetric. This
means that all households supply the same amount of labor and have the
same demand curves. Similarly, the fact that all firms face the same
demand curve and the same real wage implies that they all charge the
same amount and produce the same amount. That is, in
equilibrium,~~\(Y = C = L\). Starting with the expression of real
wage as a function of output:
\begin{quote}
\(\frac{W}{P} = Y^{\gamma - 1}\)
\end{quote}
Substituting this expression into the price equation yields an
expression for each producer's desired relative price as a function of
aggregate output:
\begin{quote}
\(\frac{P_i ^*}{P} = \frac{\eta }{\eta - 1} Y^{\gamma - 1}\)
\end{quote}
We know each producer charges the same price, and that the price index
equals this common price. Thus we can write:
\begin{quote}
\(Y = (\frac{\eta - 1}{\eta}) ^{\frac{1}{\gamma - 1}}\)
\end{quote}
This is the equilibrium level of output. Using the aggregate demand
equation,~\(Y = \frac{M}{P}\), we can find the equilibrium price level:
\begin{quote}
\(P = \frac{M}{Y}\)
\(P = \frac{M}{ (\frac{\eta - 1}{\eta}) ^{\frac{1}{\gamma - 1}}}\)
\(P = \frac{M}{Y}\)
\end{quote}
\subsection*{Implications}
{\label{774944}}
When producers have market power, they produce less than the socially
optimal amount. The fact that producers face downward-sloping demand
curves means that the marginal revenue product of labor is less than its
marginal product. As a result, real wage is less than the marginal
product of labor. Thus the gap between equilibrium and optimal levels of
output is greater when producers have more market power.~
The fact that equilibrium output is inefficiently low under imperfect
competition has important implications for fluctuations. The booms and
busts will still be away from the optimum. The gap between the actual
and ideal also implies that pricing decisions have externalities.
Suppose there is a marginal reduction in all prices. Marginal
productivity rises and so aggregate output does too. The real wage
increases and the demand for each god shifts out. Thus under imperfect
competition, there is an \emph{aggregate demand externality}~where
pricing externalities operate through the overall demand for goods.~
Finally, an example of this can be seen in the graph below.~
\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null\par\null
Remember, small frictions in economy are enough to cause dramatic
swings.~~
\selectlanguage{english}
\FloatBarrier
\end{document}