First Fundamental Theorem of Welfare Economics - Essay Example

The third condition for competitive equilibrium is that the allocation maximizes the profit of each firm at the given price system. A simple proof of the theorem is shown in the following notation. Proof of the first fundamental theorem of welfare economics Let [(x0i), (y0j), (Ф)] be a competitive equilibrium, and under the condition of non-satiation, for each: i, ui(x) = ui (x0i)……… eqn. 1 implies Ф (x) ? Ф (x0i). Instead, if we denote this as: ui(x) = ui (x0i), and Ф (x)< Ф (x0i).

by the local non-satiation condition, it is possible to let {Xn} be a sequence in Xi converging to x such that ui(xn)> ui(x)= ui (x0i), 1, 2, …… Since Ф is continuous, this condition implies that, for a big n, Ф (xn)< Ф (x0i). However, by the second condition in the definition of competitive equilibrium, ui(xn) > ui (x0i), implies that Ф (xn)>Ф (x0i). Therefore, the contradiction implies that eqn. 1 is true. Using this contradiction, we can suppose that the initial allocation [(x0i), (y0j), (Ф)] is not Pareto optimal, which implies that there is another allocation of resources [(x’i), (y’j)] such that ui(x’i) > ui (x0i).

this condition holds for all i with strict inequality for some i. Employing the second condition in the definition of competitive equilibrium, gives that for some instances of i, ui(x’i) > ui (x0i) gives the implication that Ф (x’i)> Ф (x0i). From eqn. 1 and the linearity of Ф, it can be seen that k?i, where ui(x’k) > uk (x0k), ?k Ф(x’k) ??k Ф(x0i). For l?k, where ul(x’l)> ul(x0l), ?l Ф(x)> ?l Ф(x0i). Finding the sum of the equations across all i; , which contradicts the third condition of competitive equilibrium. 2. The theorem proved above is mathematically true; however, some drawbacks are associated with it, for example, when public goods and externalities are introduced.

This is because the theorem assumes that in the economy, there are no public goods or externalities (Jehle and Reny, 2001). This means that the theorem will not hold in an exchange economy where an individual’s utility depends on another individual’s consumption as well as the original individual’s consumption. Also, the theorem does not hold if the production possibility set of one firm in an exchange economy depends on the production set of another firm in the same economy. The presence of externalities and public good sin the market will cause market failure iof they are not corrected, since there are no markets for these goods. 3. The above proposition can be proved by the following example, where externalities and public goods are introduced into an economy.

In this case, an externality is used to mean the situation where the actions of an individual or firm affects the actions of another individual or firm other than through the effect on prices (Jehle and Reny, 2001). For example, one production firm could be increasing the costs of production for another firm by the production of smoke, which forces the other firm to increase costs. One factory could be producing electronic gadgets, a process which requires the emission of smoke. The factory could be located upwind, meaning that the smoke emitted harm another