Producer Theory & Existence of equilibrium - Assignment Example

Producer Theory ANd Existence of equilibrium By: Macro & Micro Economics Lecturer: 13 June Question Producer Theory) -State, explain and justify the propositions which constitute Arrow-Debreu (or neoclassical) producer theory (both the ‘cost minimizing’ and ‘profit maximizing’ versions). You may use prose, and/or diagrams and/or formal mathematics. Comment briefly on the complications which can arise when producers are operating under conditions of increasing returns to scale. From Arrow-Debreu model a producer chooses an action plan for operating in future markets.

There are J producers in an economy. For a producer j, yj represents combination of the quantities of all inputs and outputs. Yj is the set of input and outputs that are technically feasible for the producer j. Without any production in the economy the feasibility condition would imply that the aggregate consumption (input-output) is not larger than the aggregate endowment. If there is production, then the total resources allocated for consumption increase depending on net outputs of corresponding goods.

In Arrow-Debreu model the producer’s want ‘profit maximization’ meaning that the input-output combination selected by a firm on a given a price should be such that no other input-output combination gives more profit. We have assumed here that the prices are fixed and firms choose inputs-outputs so as to maximize profit limiting itself to technically feasible combination i.e. yj ? Yj. There are other assumptions that need to be considered for example the one that is often called the ‘free disposal’ assumption which is- the firm can add any amount of input without reducing output or if we look at it from the cost minimizing concept then it means to add input which can be disposed of at zero cost.

If we “let p be an original price called by the Walrasian Auctioneer and let yj(p) the associated profit maximizing response. Let p? and yj(p?) be a new price-supply quantity configuration… … ? (p? - p)T( yj(p?) - yj(p)) ? 0 So the Law of Supply holds for A-D producers by virtue of the profit maximization hypothesis (assuming of course a profit maximizing supply response exits in the two price situations).” (Lecture 5, p.10-11) Increasing returns to scale raises productivity, but it has a problem of organizing returns efficiently.

A firm having increasing returns to scale cannot be organized without producing a loss. “Increasing returns to scale creates complications for the productivity output. Small firms tend to have increasing returns to scale because there are a smaller number of employees that have specialized interest in one or two areas of production.” (Peterson, N 2009, “Returns to scale”” “The market system cannot take advantage of the potentiality for gain through division of labor and increasing returns to scale.

” ( McCain, “Dark side of the force”) Question 2 (Existence of equilibrium) -Outline the institutional features of the Arrow-Debreu model. Sketch an argument which leads to an existence result for Walrasian equilibrium in an Arrow-Debreu economy, (be careful to explain any assumptions you might make). Briefly discuss an application of the Arrow-Debreu model with which you are familiar. Give an example of something which might prevent the existence of equilibrium. “The outcome of competitive markets in a private ownership economy is modeled as a Walrasian equilibrium, which is an allocation together with a price system.

” (Border, “Brief notes on the Arrow-Debreu-McKenzie model”) It is formulated to reach at the level of input-output that creates equilibrium by making demand and supply equal so as to eradicate the problems of excess demand and excess supply. The assumptions we take are that consumers want to maximize utility, producers struggle to maximize profits and that prices are fixed as perfect competition exists. The two approaches we use for the existence question are the primitives approach and the excess demand approach; and the two basic assumptions we take with these approaches are the ‘function’ and the ‘homogeneity of degree zero’.

Another major assumption states that “Walras Law holds for the economy so for any p? Sl–1, pTZ(p) = 0…This means that for any price vector (not just the equilibrium price vector), the price weighted sum of excess demands is equal to zero.” (Lecture 6, pp.5) Theorems are a product of the conditions imposed on the mappings of which one is : “Felix Brouwer: Let f : X > X. If f is a continuous function and if X is a non-empty, compact and convex then ? x*? X such that f(x*) = x*.” (Lecture 6, pp.9) All such mathematical theorems, the adjustment rule, together with numerous assumptions like the ones mentioned above and also the boundary condition on excess demands, the interior endowment assumption etc.

lead to existence of equilibrium. The major application of Arrow-Debreu theory of general equilibrium is to help governments efficiently allocate risk. “This theory teaches that under certain conditions—in particular, if contingent-claims markets are complete—the allocation of risk will be Pareto efficient. In other words, with complete markets, society can let the invisible hand allocate risk.”(Ball, pp 523) Government make policies to allocate this risk specially using the social security system.

Reason being that the invisible hand in the free markets allocates risk but not efficiently, so the government intervention is important. It requires regulating the system of supply and demand so that equilibrium prevails. This also applies to the risk sharing and investment. Considering the proposition four we can conclude that “In essence, the government here centrally plans the economy. It holds all assets and enforces the complete-markets allocation of consumption.” (Ball, pp 537) Equilibrium in markets can be prevented by many external factors for example in case of natural calamities when supply and demand both are affected.

It depends whether the calamity struck the industrial zones or the residential areas to decide whether it would create excess demand or excess supply. Moreover, the government policies that regulate risk allocation efficiently can be affected due to change in government itself or even foreign influence, especially if the markets belong to developing countries. References Lecture 5, p.10-11 Peterson, N 2009, “Returns to scale”, Ezine Articles, viewed 12 Jun 2011, McCain, R 2007, “Dark side of the force”, Drexel, Drexel University, Department of Economics, viewed 12 Jun 2011, < http://faculty.lebow.drexel.

edu/McCainR//top/Prin/txt/MPCh/RtoS4.html> Border, KC 2000, “Brief notes on the Arrow-Debreu-McKenzie model”, California Institute of Technology, Division of the Humanities and Social Sciences, vol. 2010.10.19::10.11, pp.4 Lecture 6, pp.5 &9 Ball, L 2007, “Intergenerational risk sharing in the spirit of Arrow, Debreu, and Rawls, with applications to social security design”, The Journal of Political Economy,  The University of Chicago Press, vol. 115 no. 4, pp 523 & 537.