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{\LARGE  A Simple Model of Keynesian Fiscal Policy}

\bigskip

\bigskip

July 1,  2009



  Eric Rasmusen \\[0pt]

\bigskip

\textit{Abstract}
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\vspace*{-12pt}A simple Keynesian model based on fixed wages.

\bigskip


{\small \noindent \hspace*{20pt} Rasmusen: Dan R. and Catherine M.
Dalton
Professor, Department of Business Economics and Public Policy, Kelley
School
of Business, Indiana University. BU 438, 1309 E. 10th Street,
Bloomington,
Indiana, 47405-1701. (812) 855-9219. Fax: 812-855-3354.
\href{mailto:erasmuse@indiana.edu}
{ erasmuse@indiana.edu}, \url{http://www.rasmusen.org}. }

{\small \noindent This paper:
\url{http://www.rasmusen.org/papers/keynes-rasmusen.pdf}. }

{\small \noindent Keywords: Keynes, fiscal policy. }


{\small \noindent I  thank xxx  for helpful comments.  }



\newpage

\noindent
{\bf 1. Introduction}


It's useful to have as simple  a model as possible of Keynesian
stimulus. That's what I try to provide here.  It is even easier to
understand than the aggregate supply/aggregate-demand model of the
textbooks, I think, but more rigorous too. It achieves this simplicity
by
omitting money, time, and interest rates from the model.



 \bigskip

\noindent
{\bf 2. The Model without Taxes}

 The economist consists of a government and a  large   number of
workers and identical firms, which are owned by the workers.

 The number of firms is fixed, and production has diminishing returns
so in equilibrium each firm produces the same output.   Firms  hire
workers at a fixed wage $w$, contractually fixed at the start of the
model's one period,  to
produce one   good.  Each unit of output requires one unit of
labor and an amount of depreciation of the existing stock of capital
that depends on how much labor is used, so the cost to the economy of
producing $Q$ units of output is
\begin{equation} \label{e1}
 c(Q) = wQ + r d(Q),
\end{equation}
 where we assume that $d'>0, d''>0$.


 The aggregate  supply curve is the industry
marginal cost curve, which is
\begin{equation} \label{e2}
Aggregate Supply (AS): P(Q) =  c'(Q) = w  + r d'(Q).
\end{equation}

Private income is the sum of labor income and capital income, which is
\begin{equation} \label{e2}
Income  =  wQ    + (PQ- (wQ+ r d (Q))  =  wQ    +  [ w  + r d'(Q)]Q-
wQ- r d (Q))  =  wQ       + r d'(Q) Q - r d (Q))  = [w     + r d'(Q)]Q
- r d (Q).
\end{equation}

 Private quantity demanded, $y$,   falls in the price directly and
rises in national income. Since national income rises with price, it
is possible that $y(P)$ rises with price. We will rule that out by
assumption.


Demand for output is the sum of private quantity demanded y(P)  plus
government deficit  quantity demanded  G:
\begin{equation} \label{e2}
Aggregate Demand (AD): Q(P) =  y(P) + G.
\end{equation}

  I  call $G$ the government deficit because balanced-budget
government spending, if financed by lump-sum taxes, will not affect
aggregate demand or supply.


xxx How about saying that in the second period, the worker retires and
consumes his entire wealth  at a particular  longterm equilibrium
price-wage. YES.  Zero time preference, concave utility will do that.
I still need the slope assumption.


  


G can be any exogenous demand source. Foreign demand for our exports,
for example. Or investment demand. 



\bigskip

\noindent
 {\bf  3. Using the Model}



Now we can put the two curves together.

 For each possible wage level $w$  there is a different AS and AD
curve
combination. AS and AD both  shift  up with the wage.

For each government deficit level $G$ there is a different AD curve,
though G does not affect the AS curve. Bigger $G$ shifts up AD.



There is
one wage level  that would equilibrate the system if wages were
flexible and $G=0$. FALSE--  THE OUTPUT IS PINNED DOWN, BUT NOT THE
NOMINAL WAGE We will call that the long-run wage and its  AS-AD
equilibrium output  the long-run output.


 We finally come to the payoff of this model. What happens if the
government deficit increases?

What happens after $G$ increases is that AD shifts up. There is a
new intersection of AS and AD at a bigger output and higher prices.

 The story is that government demand tends to raise the price, which
makes production more profitable, so employment goes up. As people
become richer, they demand more too, raising prices still further,
which increases employment further.


Note that this reasoning applies whether the economy starts below
long-run equilibrium output or above.

Could the economy permanently be above long-run equilibrium output?
Yes, in this myopic model. But that wouldn't be optimal-- people are
working too hard.


Can private consumption rise with G (a multiplier of more than 1?)


 Also discuss a supply shock-- banking crisis. 

Also discuss a coordination expectations shock. 





 I don't have anything that pins down prices in this model in the long
run.


 \bigskip

\noindent
{\bf 4. The Model with  Taxes}

 Now let us add  two kinds of taxes to the model.   The first is a tax
on workers of   $t$ per dollar of labor income. The second is a sales
(or value-added)
tax on
firms   of $s$ per  unit of output.

Private aftertax income, the sum of labor and capital income,
is now
\begin{equation} \label{e2}
 Income  =  wQ (1-t) + ([P-s]Q -  [wQ+  r d(Q)])
\end{equation}



Now the aggregate supply curve becomes
\begin{equation} \label{e2}
Aggregate Supply (AS): P(Q)   = w  + s+  r d'(Q)
\end{equation}





\newpage

\noindent
{\bf References}

Barro, Robert J. and Herschel Grossman. 1971. "A General Disequilibrium Model of Income and Employment." American Economic Review. March, 61:1, pp. 82-93.


Hicks, John R. 1937. "Mr. Keynes and the 'Classics.'” Econometrica. April, 5:2, pp. 147-59.


The Macroeconomist as Scientist and Engineer
N. Gregory Mankiw
Harvard University
May 2006

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