1
2 Model Setup
Representative Firm We assume a competitive economy with identical firms, then we use representatives to build the model. Production function of representative firms is Cobb-Douglas form with Harrod-neutral,
Neoclassical Growth Model
Weijie Chen Department of Political and Economic Studies
University of Helsinki 15 June, 2011
In this article, we use dynamic programming to solve neoclassical growth model which is based on microfoundation. Objectives of agents are formed explicitly, households choose saving in order to maximize utility, firms maximize their profits by decisions of employment and investment. Sometimes Solow model is also called neoclassical growth model, however Solow model is very different from the what we are presenting here, the most prominent feature is that savings rate is exogenous in Solow model.
In order to coordinate decision in decentralised economy, we need to introduce markets, such as labour market, capital market, assets market and etc. Neoclassical growth model is a decentralised economy model, it is the baseline model for DSGE modeling. Households and firms will optimise their preferences based on dynamic budget constraints. A general equilibrium will be reached between representative household, representative firm and governmeLt,It
Dt + 1 + r V (Kt+1)
1
= max
Lt,It
Yt
−
wtLt
−
It
+
1
+
V r
(Kt+1)
= max
Lt,It
Ktα(AtLt)1−α
−
wtLt
−
It
+
1
1 +
V r
(Kt+1)
(5)
2
subject to,
Yt = Ktα(AtLt)1−α Kt+1 = (1 − δ)Kt + It
F.O.C. w.r.t. Lt,
(1 − α)KtαAt1−αL−t α − wt = 0
(6)
F.O.C. w.r.t. It, and due to the second constraint,
1 The Decentralised Economy
What we have seen previously, single individual economy or Robinson Crusoe model is called centralised economy, representative agent make decisions on consumption, saving, leisure and works, which means it can be replicated by a central planner making decision for all individuals. There is no need for market structure, all decisions automatically coordinate.
Yt = Ktα(AtLt)1−α with 0 < α < 1
(1)
Note that we use capital letter emphasize they are all aggregate variables. Law of motion of the capital,
Kt+1 = (1 − δ)Kt + It with 0 < δ < 1
establish the value function of representative firm,
∞
s
1
Vt =
s=t
s =t+1 1 + rs Ds
(4)
1
1
1
= Dt + 1 + rt+1 Dt+1 + 1 + rr+1 1 + rr+2 Dt+2 · · ·
Set up Bellman equation, with Lt, It being controls,
(2)
Since it is competitive economy, Yt is firm’s real revenue in period t, thus dividends for shareholders in t is,
Dt = Yt − wtLt − It
(3)
where wt is real wage. The reason of setting up a dividends equation is to