Estudio comparativo entre productos y servicios

Three static properties are observed in a general equilibrium solution, reached with a free competitive market mechanism:

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(a) Efficient allocation of resources among firms (equilibrium of production) (b) Efficient distribution of commodities produced between the two consumers

(equilibrium of consumption)

(c) Efficient combination of products (simultaneous equilibrium of production and consumption).

These conditions are called marginal conditions of Pareto optimality.

(a) Equilibrium of production (Efficiency in Factor Substitution): We saw in Fig.

3.2 that the equilibrium of the firm requires that 𝑀𝑅𝑇𝑆_𝐿,𝐾 =𝑤

𝑟 … … … (3.3)

where w and r are the factor prices prevailing in the market and MRTS is the marginal rate of technical substitution between the two factors.

The joint equilibrium of production of the two firms in our simple model can be derived using the Edgeworth box of production.

Figure 3.2: Edgeworth Box of production

The locus of points of tangency of the x and y isoquants is called the Edgeworth contract curve of production and is given by ox-oy in Figure 3.2. The curve contains the efficient allocations of K and L between the firms. Each point shows a specific allocation of K and L in the production of commodities x and Y that is efficient. Since the Edgeworth contract curve is a locus of tangencies of X and Y isoquants, at each one of its points the slopes of the isoquants are equal:

That is: slope of x isoquant = slope of y isoquant or 𝑀𝑅𝑇𝑆_𝐿𝐾^𝑥 = 𝑀𝑅𝑇𝑆_𝐿,𝐾^𝑦 … … … (3.4)

Thus, in our simple general equilibrium model, the firms will be in equilibrium only if they produce somewhere on the Edgeworth contract curve. This follows from the fact that the factor prices facing the producers are the same and the profit maximization requires that each firm equates it’s 𝑀𝑅𝑇𝑆_𝐿,𝐾 with the ratio of factor prices ^𝑤

𝑟

𝑀𝑅𝑇𝑆_𝐿𝐾^𝑥 = 𝑀𝑅𝑇𝑆_𝐿,𝐾^𝑦 =^𝑤

𝑟 … … … ....(3.5) This will be the same for all firms.

It thus appears that the production equilibrium is not unique since it may occur at any point along the Edgeworth contract curve. However, with perfect competition (in which all producers face the same factor prices) one of these equilibria will be realized: that is

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the one at which the (equalized between firms) MRTL,K is equal to the ratio of factor prices ^𝑤

𝑟. That is, with perfect competition general equilibrium of production occurs where condition (3.5) is satisfied.

If factor prices are given, from the Edgeworth box of production we can determine the amounts of x and y which maximize the profits of firms. However, in a general equilibrium analysis, these quantities of x and y must be equal to what the consumers want to buy in order to maximize utility or satisfaction. In order to bring together the production side of the system with the demand side, we must define the equilibrium of the firms in the product space, using the production possibility curve or frontier of the economy as shown in Figure 3.3.

Figure 3.3: Production possibility curve.

This is derived from the Edgeworth contract curve of production, by mapping its points on a graph on whose axes are measured the quantities of the final commodities x and y.

From each point of the Edgeworth contract curve of production, we can read off the maximum obtainable quantity of one commodity, given the quantity of the other, with the given factors 𝐾̅ 𝑎𝑛𝑑 𝐿.̅

The production possibility curve of an economy is the locus of all Pareto-efficient outputs, given the resource endowment ( 𝐾̅ 𝑎𝑛𝑑 𝐿̅) and the state of technology. It is also called the product transformation curve because it shows how a commodity is transformed into another, by transferring some factors from the production of one commodity to the other. Its negative slope is called the marginal rate of product transformation, MRPTx,y, and, it shows the amount of y that must be sacrificed in order to obtain an additional unit of x. by definition

𝑀𝑅𝑃𝑇_𝑥,𝑦 =𝑑𝑦

𝑑𝑥… … … . . (3.6)

It can be shown that MRPTx,y is equal to the ratio of the marginal costs of the two products.

𝑀𝑅𝑃𝑇_𝑥,𝑦 =^𝑑𝑦

𝑑𝑥 =^𝑀𝐶𝑥

𝑀𝐶𝑦………..(3.7)

In perfect competition, the profit maximizing firm equates the price of the commodity produced to the long-run marginal cost of production:

𝑀𝐶_𝑥 = 𝑃_𝑥 𝑎𝑛𝑑 𝑀𝐶_𝑦 = 𝑃_𝑦… … … (3.8) Thus given the commodity price, the equilibrium is reached at the point on the production transformation curve that has a slope equal to the ratio of these prices, as shown in Figure 3.4.

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Figure 3.4: General Equilibrium of production in perfect competition.

At this point, the product mix is given by x* and y*

Proof of Equation. 3.7:

By definition 𝑀𝐶_𝑥 = 𝑑^𝑇𝐶

𝑑𝑥𝑎𝑛𝑑 𝑀𝐶_𝑦 = 𝑑^𝑇𝐶

𝑑𝑦 … … … . . (1) Thus ^𝑀𝐶𝑥

𝑀𝐶𝑦 = 𝑑^{(𝑇𝐶)𝑥}

𝑑𝑥 × ^𝑑𝑦

𝑑(𝑇𝐶)𝑦 =^{𝑑(𝑇𝐶)𝑥}

𝑑(𝑇𝐶)𝑦 𝑑𝑦

𝑑𝑥……….(2)

𝐵𝑢𝑡 𝑑(𝑇𝐶)𝑥 = 𝑤(𝑑𝐿_𝑥) + 𝑟. (𝑑𝐾_𝑥) 𝐴𝑛𝑑 𝑑(𝑇𝐶)𝑦 = 𝑤 (𝑑𝐿_𝑦) + 𝑟. (𝑑𝐾_𝑦)

So that ^{𝑑(𝑇𝐶)𝑥}

𝑑(𝑇𝐶)𝑦 = ^{(𝑤(𝑑𝐿𝑥)+ 𝑟.(𝑑𝐾𝑥))}

(𝑤 (𝑑𝐿_𝑦)+ 𝑟.(𝑑𝐾_𝑦))… … … (3)

In order to remain on the production possibility curve (PPC) the factors released from the decrease in commodity Y must be equal to the factors absorbed by the increase in the production of x, that is

𝑑𝑙_𝑥 = − 𝑑𝑙_𝑦 𝑎𝑛𝑑 𝑑𝑘_𝑥 = −𝑑𝑘_𝑦… … … . (4) Substituting (4) in (3) we get:

𝑑(𝑇𝐶)𝑥

𝑑(𝑇𝐶)𝑦=(𝑤(−𝑑𝐿_𝑦) + 𝑟. (−𝑑𝐾_𝑦))

(𝑤 (𝑑𝐿_𝑦) + 𝑟. (𝑑𝐾_𝑦)) = −1 and substituting (5) in (2) we obtain

𝑀𝐶_𝑥

𝑀𝐶_𝑦 = −1 (𝑑𝑥

𝑑𝑦) = 𝑑𝑦

𝑑𝑥 = 𝑠𝑙𝑜𝑝 𝑜𝑓 𝑃𝑃𝐶

= 𝑀𝑅𝑃𝑇_𝑥,𝑦… … … . 𝑄𝐸𝐷 … … … . (6)

F¹ x*

y*

0 F¹ y

F

𝑃_𝑥

𝑃_𝑦 = 𝑀𝐶_𝑥 𝑀𝐶_𝑦

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Equilibrium of Consumption (Efficiency in Distribution of Commodities) From Figure 3.1 we saw that the consumer is in equilibrium when 𝑀𝑅𝑆_𝑥,𝑦 =^𝑃𝑥

𝑃𝑦. Since both consumers in a perfectly competitive market are faced with the same prices, the condition for joint or general equilibrium of the consumer is:

𝑀𝑅𝑆_𝑥𝑦^𝐴 = 𝑀𝑅𝑆_𝑥𝑦^𝐵 =𝑃_𝑥

𝑃_𝑦 … … … (3.9)

This general equilibrium of consumption for the product mix is shown in Fig 3.6.

Figure 3.5: Equilibrium of Consumption (in a perfect market)

In the figure above, the Edgeworth contract curve of consumption, OT, represents efficient distributions. At each point on this curve, the following equilibrium condition is satisfied

𝑀𝑅𝑆_𝑥,𝑦^𝐴 = 𝑀𝑅𝑆_𝑥,𝑦^𝐵 … … … (3.10)

Thus, for a given product mix such as T, there is an infinite number of possible efficient or optimal equilibrium distributions (indicating that equilibrium of consumption is not unique). However, with perfect competition, only one of these points is consistent with the general equilibrium of the system. This is the point of the contract curve where the (‘equalized’) MRSx,y of the consumers is equal to the price ratio of the commodities, that is, the condition (3.9) is satisfied.

In the figure, the equilibrium of the consumers is defined by point, T. Consumer A reaches the utility level implied by the indifference curve A3, buying OM of x and ON of y. Consumer B reaches the utility level implied by the indifference curve B3. Buying MXB of x and Ny0’^s of Y.

(b) Simultaneous Equilibrium of Production and Consumption (Efficiency in Product – Mix)

In addition to the conditions specified in (a) and (b) above, a third condition is required to be satisfied if the general equilibrium of the system as a whole is to be achieved. This

F¹ x*

y*

0 F¹ y

T A

B M

N B5

B4

B3

B2

B1

A1

A2

A3

A4

A5

B’s indifference curves

A’s indifference curves

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condition is that the marginal rate of product transformation (slope of the PPC) be equal to the marginal rate of substitution of the two commodities between the consumers.

𝑀𝑅𝑃𝑇_𝑥𝑦 = 𝑀𝑅𝑆_𝑥,𝑦^𝐴 = 𝑀𝑅𝑆_𝑥,𝑦^𝐵 … … … .3.11) In perfect competition, this condition is satisfied since, from equation (3.8)

𝑀𝑅𝑃𝑇_𝑥,𝑦 =𝑃_𝑥

𝑃_𝑦… … … … . . … … … . . (3.12) and from equation (3.9) 𝑀𝑅𝑆_𝑥,𝑦^𝐴 = 𝑀𝑅𝑆^𝐵_𝑥,𝑦…….…(3.13)

so that:

𝑀𝑅𝑃𝑇_𝑥,𝑦 = 𝑀𝑅𝑆_𝑥,𝑦^𝐴 = 𝑀𝑅𝑆^𝐵_𝑥,𝑦………(3.14)

(as demonstrated by the equality of slopes at T and T in Fig. 3.6). This is the third marginal condition of Pareto efficiency. It refers to the efficiency of product substitution or optimal composition of output.

In summary, with perfect competition, the simple two factors, two-commodities, two consumer systems have a general equilibrium solution in which three Pareto efficiency conditions are satisfied.

1. The MRS between the two goods is equal for both consumers (optimal allocation of the goods among consumers)

2. The MRTS between the two factors is equal for all firms (efficiency in factor substitution-implying optimal allocation of factors among the two firms)

3. The MRS and the MRPT are equal for the two goods (optimal composition of output and thus optimal allocation of resources).

SELF ASSESSMENT EXERCISES

1. What do you understand by the term general equilibrium?

2. In an economy of two individuals (A and B) and two commodities (X and Y), state the condition(s) at which general equilibrium of exchange is reached.