# 经济代写|微观经济学代写Microeconomics代考|BEA470

## 经济代写|微观经济学代写Microeconomics代考|WACm; Kuhn–Tucker Conditions and Cost Minimization

Cost minimization has an implication analogous to the WAPM. If for a given output $q^{\circ}$ and given input rentals $v$ the firm finds it optimal to utilize an input vector $\mathbf{x}^{\circ}=h\left(\mathbf{v}, q^{\circ}\right)$, it must mean that any other input vector capable of producing $q^{\circ}$ (or more) must cost at least $\mathbf{v x}^{\circ}$, in other words, $\mathbf{v x}^{\circ} \leq \mathbf{v x}$ for all $\mathbf{x}$ such that $f(\mathbf{x}) \geq q^{\circ}$, a result sometimes called Weak Axiom of Cost Minimization, WACm for short. If at input prices $v^{\circ}$ the firm chooses input vector $\mathbf{x}^{\circ}$ and at input prices $\mathbf{v}^$ the firm chooses input vector $\mathbf{x}^$ to produce the same output, proceeding in the same way as for the WAPM one reaches the conclusion
$$\left(\mathbf{v}^0-\mathbf{v}^\right)\left(\mathbf{x}^0-\mathbf{x}^\right) \leq 0,$$
more often expressed as
$$\Delta \mathbf{v} \cdot \Delta \mathbf{x} \leq 0 .$$
This implies, for example, that if only one input price changes, the demand for that input cannot change in the same direction (and if it changes at all it must change in the opposite direction). (Note that if inputs were measured as negative quantities the inequality sign would be reversed, still this is not the same result as was derived from the WAPM, because here output is kept fixed.)

In general, not all possible inputs will be used in positive amounts by a firm; the condition $v / v_j=\mathrm{MP}i / \mathrm{MP}{\mathrm{j}}$ must hold for inputs both used in positive quantities; the more general necessary first-order conditions for cost minimization are derivable from the Kuhn-Tucker theorem ( Sect. 4.8). In the present case, with $q^{\circ}$ the given output, the function to be maximized is $-\mathbf{v x}$ and the constraints are $f(\mathbf{x})-q^{\circ} \geq 0$, and $x_i \geq 0, \mathrm{i}=1, \ldots, \mathrm{n}$. The Lagrangian is $-\mathbf{v} \mathbf{x}+i^{\circ}\left(f(\mathbf{x})-q^{\circ}\right)+\sum_{\mathrm{i}} \lambda_i x_i$ indicating $\partial f f \partial x_i$ as $f_i$, the first-order conditions are therefore:
Kuhn-Tucker first-order necessary conditions for cost minimization:
$$-v_i+\lambda_i+\lambda_i=0, \quad i=1, \ldots, n,$$
with complementary slackness conditions $\lambda^{\circ}\left(f(\mathbf{x})-q^{\circ}\right)=0$ and $\lambda_r x_i=0$, that is, if $\left(f(x)-q^{\circ}\right)>0$, then $\lambda^{\circ}=0$, and if $\lambda_i>0$, i.e. if $v_i>\lambda f_i$, then $x_i=0$.

## 经济代写|微观经济学代写Microeconomics代考|Supply Curves: Short-Period Marshallian Analysis, Quasi-Rents

The traditional, Marshallian approach distinguishes short-period from long-period profit maximization and short-period from long-period supply curves of firms. As Marshall puts it, the long-period or normal value of a commodity “is that which economic forces tend to bring about in the long run. It is the average value which economic forces would bring about if the general conditions of life were stationary for a run of time long enough to enable them all to work out their full effect’ (1972, V, iii, 6; p. 289). At the level of the single firm, the distinction between short-period and long-period decisions is traditionally described as follows: in the short period a firm cannot alter the quantities of some of the inputs, in the long period all inputs can be varied.

This is often put in terms of whether the more durable part of the capital goods used by the firm, its fixed plant, is given or variable. However, it takes very little time for a firm to sell or rent out its fixed plant to other firms, or to buy or rent the fixed plants of other firms (a firm is a legal entity, not to be confused with its plants): for example, a firm can rent a building or flat if needed for a particular production or period. So the idea that the short period is characterized by each firm having a given fixed plant is difficult to defend. If one re-reads the inventor of the short-period/long-period distinction, Alfred Marshall, one finds that the fixity he had in mind referred not to the individual firm but to the industry: the short period was the analytical period within which there was not enough time significantly to change the amounts available to the entire industry of specialized durable inputs necessary to the industry and requiring considerable time for their production. Thus Marshall’s short-period analysis of the fishing industry takes as given the number of fishing ships (and of experienced fishermen), not the number of firms nor how the fishing ships are divided among different firms (1972, V, v, 4; p. 307). In the language of plants, one can say that in the short period the total number of fixed plants available to an industry is given, how they are divided among firms is secondary: a well-run firm owning two separate plants will behave in the same way as two firms each owning one of the two plants because it will want to maximize the profit from each plant.

However, once the point is grasped, then precisely because how the plants are divided among firms is secondary if all firms are efficient, to assume that the distribution of plants among firms is given will make no difference for the determination of the industry’s supply decisions (which are what is relevant to determine the equilibrium of the market one is studying.). Therefore we can follow the standard textbook approach after all.

## 经济代写|微观经济学代写Microeconomics代考|WACm; Kuhn–Tucker Conditions and Cost Minimization

$$\Delta \mathbf{v} \cdot \Delta \mathbf{x} \leq 0$$

$$-v_i+\lambda_i+\lambda_i=0, \quad i=1, \ldots, n,$$

## 经济代写|微观经济学代写Microeconomics代考|Supply Curves: Short-Period Marshallian Analysis, Quasi-Rents

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