# 经济代写|微观经济学代写Microeconomics代考|ECON2516

## 经济代写|微观经济学代写Microeconomics代考|Proft Maximization and WAPM

The standard assumption about the behaviour of firms is that they aim to maximize profit. ${ }^{21}$ The meaning of ‘profit’ here and in the entire chapter is the marginalist/neoclassical one, i.e. in this chapter ‘profit’ stands for what is left to the ‘entrepreneur’ (the owner or owners of the firm) after paying all costs including interest on capital advances 22 . Even when the apparent aim of the firm is another one, e.g. sales maximization or maximization of growth rate, a case can usually be made that this does not entail significantly different choices from the ones aimed at maximizing long-run profit. Giving up profits today means to have less money to reinvest, and so to have one’s wealth increase slower: so sales maximization entailing less profits today than with fewer sales seems reasonable only if this entails other advantages, e.g. an increase in market power which will allow raising price and hence profits in subsequent periods, but then it is part of a strategy of long-period profit maximization (in an imperfectly competitive market). Growth maximization entailing persistently lower profits than with a lower growth rate is analogously difficult to justify unless it brings other advantages, e.g. a greater chance of survival in a world with increasing returns to scale, but then again the ultimate reason is to avoid a loss of profits in subsequent periods. A different argument is that perfectly efficient profit maximization cannot be assumed, owing to limited managerial capacities and to the costs of maximization. But as long as management strives for profit maximization the fact that the goal is only imperfectly realized does not alter the broad pattern of industry behaviour. For example, the tendency to invest more in the industries that offer better profitability prospects will exist even if on average management is not very good at minimizing costs. And the occasional episodes of managers pursuing strategies of personal enrichment at the expense of the profitability of their firm usually end up rather quickly in the disappearance or takeover of the firm. I accept profit maximization as broadly valid as a survival condition in competitive industries. ${ }^{23}$ A monopolist entrepreneur not threatened by takeovers might indulge in other aims, e.g. to have political influence, or play golf, or be generous towards employees; firms in competitive environments and under threat of takeovers end up being taken over or going bankrupt if they do not struggle to obtain good results, and this requires profit maximization or at least attempting it.

## 经济代写|微观经济学代写Microeconomics代考|Optimal Employment of a Factor

Consider a competitive, i.e. price taking, firm that produces a single output and has a differentiable production function $q=f(\mathbf{x})$. Profit is $\pi=p q-\mathbf{x}=p f(\mathbf{x})-\mathbf{v x}$ where $q$ is output (a scalar), $p$ its price, $\mathbf{x}$ the vector of inputs (measured as positive quantities), $\mathbf{v}$ the vector of input rentals. How does the firm decide how much to employ of a single input? It must maximize $\pi$ with respect to the employment of each factor $i$. Assuming an interior solution, $x_i>0$, all $i$, the first-order condition for each factor is
$\left(p \cdot \partial f / \partial x_i\right)-v_i=0$, or in the usual symbols : $p \cdot \mathrm{MP}_i=v_i$.
i.e. the equality between marginal revenue product of the factor and ‘price’ (i.e. rental) of the factor. The marginal revenue product of a factor is, intuitively speaking, the increase in revenue due to the increase in output caused by one more small unit of the factor; under price taking it is $p$. $f f \partial x_i$. This first-order condition can also be expressed as $\mathrm{MP}_{\mathrm{i}}=v / p$ where $v / p$ is the real rental of the factor measured in terms of the product.

A third way of writing this equality, $p=v / \mathrm{MP}{\mathrm{i}}$, has the following interesting intuitive interpretation: the reciprocal of $\mathrm{MP}{\mathrm{i}}$ is the increase in input $i$ needed for output to increase by one (small) unit; therefore $v / \mathrm{MP}{\mathrm{i}}$ is the increase in cost if the increase in output by 1 unit is obtained by increasing only factor $i$. Note that if profit is maximized the condition must hold for all inputs, which means $$p=v_i / \mathrm{MP}{\mathrm{i},}=v_j / \mathrm{MP}{\mathrm{j},}=\alpha\left(v_i / \mathrm{MP}{\mathrm{i}}\right)+(1-\alpha)\left(v_j / \mathrm{MP}_{\mathrm{j}}\right), \quad \text { for } 0<\alpha<1 .$$
This means that the increase in cost to obtain a very small output increase is the same whether obtained by increasing only one input, or two (or, generalizing, even all inputs). This increase in cost, the derivative of total cost relative to output, is called marginal cost, MC, and it must be equal to $p$ for profit maximization. All this is of course rigorously true only for infinitesimal output variations, but it is sufficiently exact as long as output is measured in very small units. (The name ‘marginal cost’ reflects the fact that $\mathrm{MC}$ is the derivative of the cost function with respect to $q$, as explained in the next section.)

The second-order condition for the optimal employment of a factor, since $p$ and $v_i$ are given for the firm, is $\partial^2 f / \partial x_i^2<0$, that is, $\mathrm{MP}_{\mathrm{i}}$ must be decreasing in $x_i$ at the optimal employment of the factor.

It can happen that no positive value of $x_i$, however small, avoids a marginal revenue product inferior to the given $v_i$ In this case profit maximization implies a ‘corner solution’ with $x_i=0$ and $p \cdot \mathrm{MP}_{\mathrm{i}} \leq v_i$.

## 经济代写|微观经济学代写Microeconomics代考|Optimal Employment of a Factor

$\left(p \cdot \partial f / \partial x_i\right)-v_i=0$ ，或通常的符号: $p \cdot \mathrm{MP}i=v_i$. 即要素的边际收益产品与要素的“价格”（即租金) 之间的相等性。一个要素的边际收益乘积，直观地说， 就是多一个小单位的要素所带来的产量增加所带来的收益增加；在价格之下是 $p . f f \partial x_i$. 这个一阶条件也 可以表示为 $\mathrm{MP}{\mathrm{i}}=v / p$ 在哪里 $v / p$ 是根据产品衡量的要龶的实际租金。

（小）单位的产量；所以 $v / \mathrm{MPi}$ 是如果仅通过增加因子获得产量增加 1 单位的成本增加 $i$. 请注意，如果利 润最大化，则条件必须对所有输入都成立，这意味着
$$p=v_i / \mathrm{MPi},=v_j / \mathrm{MPj},=\alpha\left(v_i / \mathrm{MPi}\right)+(1-\alpha)\left(v_j / \mathrm{MP}{\mathrm{j}}\right), \quad \text { for } 0<\alpha<1 .$$ 这意味着，无论是通过增加一个输入还是两个 (或者概括地说，甚至是所有输入)，获得非常小的输出增 加所增加的成本都是相同的。这种成本的增加，即总成本相对于产出的导数，称为边际成本，MC，它必 须等于 $p$ 为了利润最大化。当然，所有这一切都严格地适用于无限小的输出变化，但只要输出以非常小的 单位测量，它就足够精确了。（”边际成本”这个名称反映了这样一个事实： $\mathrm{MC}$ 是成本函数关于 $q$ ，如下一 节所述。) 因子最优使用的二阶条件，因为 $p$ 和 $v_i$ 是给公司的，是 $\partial^2 f / \partial x_i^2<0$ ，那是， MP $\mathrm{M}{\mathrm{i}}$ 必须在减少 $x_i$ 因素的 最佳使用。

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