Partial market model

The simple partial market model is one of the most basic models in economics . It describes a market for a single good .

General

A partial market is a market for a single good that is sufficiently homogeneous that there is a single price. It is also assumed that there are no spatial or temporal differences in the transactions between providers and buyers. Furthermore, there is complete market transparency. This means that the market price is known to all economic actors.

Contrary to the consideration of the interactions of all markets and products, which is necessary for total analyzes , only one market is considered in the partial analysis. The effects of a decision on other parameters are neglected. This strong simplification of reality means that decisions in a market can be viewed in isolation. In reality, meaningful results can be generated if the elasticities of other markets to changes in the market under consideration are low.

Description of the partial market

supply and demand

In the partial market model, it is assumed, for the sake of simplicity, that the market can be clearly represented by a demand and a supply function. The demand function expresses how many units are demanded by consumers at a fixed price, and the supply function expresses how many units are offered by producers at a fixed price.

${\displaystyle D\colon \,P\to \mathbb {R} _{+},\;p\mapsto D(p)\quad \quad }$ (Demand function)

${\displaystyle S\colon \,P\to \mathbb {R} _{+},\;p\mapsto S(p)\quad \quad }$ (Offer function)

Excess demand function

The difference between demand and supply is known as excess demand.

${\displaystyle E\colon \,P\to \mathbb {R} _{+},\;p\mapsto D(p)-S(p)\quad \quad }$ (Offer function)

Importance of the partial market model

The partial market model is a hypothetical ceteris paribus view. So the question is - assuming the rest of the economy doesn't change - how price relates to demand and supply. In principle, no particularly strong assumptions are made. The only assumptions are that supply and demand are functions. This means that there is a clearly determined demanded and a clearly determined offered quantity of the good for a price.

Market equilibrium

The equilibrium price

An equilibrium price is a price where supply equals demand.

${\displaystyle p^{*}:E(p^{*})=0}$

existence

For the existence of a market equilibrium , according to the intermediate value theorem, the following is sufficient if the excess demand is continuous

${\displaystyle \exists p_{1},p_{2}\in P:E(p_{1})<0,E(p_{2})>0}$

Uniqueness

Strict pseudo-monotony is sufficient for the unambiguousness of a market equilibrium:

${\displaystyle \exists p_{1}\neq p_{2}:(p_{1}-p_{2})E(p_{2})\leq 0\Rightarrow (p_{1}-p_{2})E(p_{1})<0}$

Stability of market equilibria

One problem with market equilibrium as a theoretical concept is that it does not represent what happens when in the partial market the price is not the equilibrium price. To do this, a dynamic price change process must be added to the model.

Dynamic system

A movement , where is the set of all possible prices and the set of all points in time, is called a dynamic system if ${\displaystyle \varphi :T\times P\to P}$ ${\displaystyle P\subseteq \mathbb {R} _{+}}$ ${\displaystyle T\subseteq \mathbb {R} }$

${\displaystyle (i)\quad \varphi (0,p_{0})=p_{0}}$

${\displaystyle (ii)\quad \varphi (t_{1},\varphi (t_{2},p_{0}))=\varphi (t_{1}+t_{2},p_{0})}$

A frequently used dynamic system is the têtonnement process, in which

${\displaystyle {\frac {dp(t)}{dt}}=E(p(t))\quad \quad ,p(0)=p_{0}}$

The interpretation behind this is that the price tends to rise when there is positive excess demand and decrease when there is excess supply.

stability

A dynamical system converges locally asymptotically stable to if ${\displaystyle p^{*}}$

${\displaystyle \exists \delta >0\;\forall p_{0}\in P|p_{0}-p^{*}|<\delta \Rightarrow \lim _{t\to \infty }|p_{t}-p^{*}|=0}$

A dynamic system converges globally asymptotically stable to if ${\displaystyle p^{*}}$

${\displaystyle \forall \delta >0\;\forall p_{0}\in P|p_{0}-p^{*}|<\delta \Rightarrow \lim _{t\to \infty }|p_{t}-p^{*}|=0}$

Rationalizability

Demand and supply in a partial market means that they can be rationalized if one can assign an associated utility function to the demand function and an associated cost function to a supply function .

demand

If an invertible and integrable demand function is given, where p is a price and x is a demanded quantity on a partial market, then the following applies to the utility function of the representative agent ${\displaystyle D\colon \,P\to \mathbb {R} ,\;p\mapsto x}$

${\displaystyle D^{-1}(x)=u'(x)}$

if a quasi-linear utility function is assumed. For the utility function of the representative agent, there results , therefore ${\displaystyle U(x)=u(x)-px}$ ${\displaystyle p=D^{-1}(x)}$

${\displaystyle U(x)=\int _{0}^{x}D^{-1}(x')dx'-D^{-1}(x)x}$

The associated preference relation is then obtained with

${\displaystyle U(x_{1})>U(x_{2})\Leftrightarrow x_{1}\succ x_{2}}$

${\displaystyle \int _{0}^{x_{1}}D^{-1}(x)dx-D^{-1}(x_{1})x_{1}>\int _{0}^{x_{2}}D^{-1}(x)dx-D^{-1}(x_{2})x_{2}\Leftrightarrow x_{1}\succ x_{2}}$

offer

If an invertible and integrable supply function is given, where a price and an offered quantity are on a partial market, then the following applies for the cost function of a representative company ${\displaystyle S\colon \,P\to \mathbb {R} ,\;p\mapsto x}$ ${\displaystyle p}$ ${\displaystyle x}$

${\displaystyle S^{-1}(x)=c'(x)}$

if a profit function is assumed. There applies ${\displaystyle \pi (x)=px-c(x)}$ ${\displaystyle p=S^{-1}(x)}$

${\displaystyle \pi (x)=S^{-1}(x)x-\int _{0}^{x}S^{-1}(x')dx'}$

Other equilibrium models

• Generalized partial market model
• General equilibrium model
• Edgeworth box model
• Robinson Crusoe economy
• Walrasian equilibrium model
• Arrow-Debreu equilibrium model

literature

• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .