# 经济代写|微观经济学代写Microeconomics代考|Firm Acting in Perfect Competition—Long-Term Strategy

## 经济代写|微观经济学代写Microeconomics代考|Static Approach

Let us use the following notation:
$p>0$ – a price of a product manufactured by a firm, $\mathbf{c}=\left(c_1, c_2\right)>(0,0)-\mathrm{a}$ vector of prices of production factors, $\mathbf{x}=\left(x_1, x_2\right) \geq(0,0)$-a vector of inputs of production factors, $y=f\left(x_1, x_2\right)$-an output level, $r(y)=p y$-revenue (turnover) from sales of a manufactured product as a function of output level, $r\left(x_1, x_2\right)=p f\left(x_1, x_2\right)$-revenue (turnover) from sales of a manufactured product as a function of inputs of production factors, $c^{t o t}\left(x_1, x_2\right)=c_1 x_1+c_2 x_2+d$-total cost of production, $c^v\left(x_1, x_2\right)=c_1 x_1+c_2 x_2$-variable cost of production, $c^f\left(x_1, x_2\right)=d$-fixed cost of production, $c(y)$-minimum cost of producing $y$ output units, derived as an objective function corresponding to an optimal solution to problem (P2c), $\pi(y)=r(y)-c(y)=p y-c(y)$-firm’s profit as a function of output level, $\pi\left(x_1, x_2\right)=r\left(x_1, x_2\right)-c^{t o t}\left(x_1, x_2\right)$-firm’s profit as a function of inputs of production factors.

Problem of profit maximization with regard to inputs of production factors (P1c)

The aim of a firm is to maximize its profit expressed as a function of inputs of production factors, which can be written as a problem to solve in the following way:
\begin{aligned} \pi\left(x_1, x_2\right)= & r\left(x_1, x_2\right)-c^{t o t}\left(x_1, x_2\right) \ = & \left{p f\left(x_1, x_2\right)-\left(c_1 x_1+c_2 x_2+d\right)\right} \rightarrow \max \ & x_1, x_2 \geq 0 . \end{aligned}
Since a production function from assumption (F2) is strictly concave while a production total cost is linear, then a profit function is strictly concave. Moreover, we are interested in an optimal solution $\overline{\mathbf{x}}=\left(\bar{x}_1, \bar{x}_2\right)>(0,0)$.

Necessary and sufficient conditions for the existence of an optimal solution to problem (P1c) are given in the following theorem.

## 经济代写|微观经济学代写Microeconomics代考|Dynamic Approach

When we use the dynamic approach to present the profit maximization and the cost minimization problems we assume that part of quantities and levels taken into account by a producer changes over time. In the time horizon considered, the production technology described by a production function is assumed to be invariant. The price of a product, prices of production factors and the fixed cost of production (independent of the output level) can change over time due to different reasons. Still a firm acting in the perfect competition has no impact on the price of a product and prices of production factors in any period or at any moment of time. Let us introduce the following notation:
$t$-time as discrete $(t=0,1,2, \ldots, T)$ or as continuous ${ }^{13}$ variable $(t \in[0 ; T])$, $T$-end of the time horizon,
$p(t)>0$-a time-variant price of a product manufactured by a firm,
$\mathbf{x}(t)=\left(x_1(t), x_2(t)\right) \geq \mathbf{0}$-a vector of inputs of production factors that a producer uses in the production process in period/at moment $t$,
$\mathbf{c}(t)=\left(c_1(t), c_2(t)\right)>\mathbf{0}$-a vector of time-variant prices of production factors, $y=f(\mathbf{x}(t))$-a production function, ${ }^{14}$
$d(t) \geq 0$-time-variant fixed cost of production, that is, the cost not depending on the output level nor on inputs of production factors.

A producer, who aims to maximize the firm’s profit, in every period/at any moment $t$ determines what the optimal inputs of production factors are. When deciding about the vector $\mathbf{x}(t)$ of production factors’ inputs he/she relies on the relation between the revenue from sales of a product and the production total cost by given time variant: prices of production factors, price of a product and fixed production cost. The profit maximization problem with regard to inputs of production factors has a form:
\begin{aligned} \pi(\mathbf{x}(t)) & =r(\mathbf{x}(t))-c^{t o t}(\mathbf{x}(t)) \ & =\left{p(t) f(\mathbf{x}(t))-\left(c_1(t) x_1(t)+c_2(t) x_2(t)+d(t)\right)\right} \mapsto \max \end{aligned}
(4.105)
$$\mathbf{x}(t) \geq \mathbf{0} .$$

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Static Approach

$p>0$ – 公司生产的产品的价格， $\mathbf{c}=\left(c_1, c_2\right)>(0,0)-\mathrm{a}$ 生产要素价格的矢量，
$\mathbf{x}=\left(x_1, x_2\right) \geq(0,0)$ – 生产要素投入向量， $y=f\left(x_1, x_2\right)$-输出电平， $r(y)=p y$ –

$\pi(y)=r(y)-c(y)=p y-c(y)-$ 公司的利润作为产出水平的函数，
$\pi\left(x_1, x_2\right)=r\left(x_1, x_2\right)-c^{t o t}\left(x_1, x_2\right)$ – 公司的利润作为生产要素投入的函数。

## 经济代写|微观经济学代写Microeconomics代考|Dynamic Approach

$t$-离散的时间 $(t=0,1,2, \ldots, T)$ 或连续 ${ }^{13}$ 多变的 $(t \in[0 ; T]), T$ – 时间范围的尽头， $p(t)>0$-公司生产的产品的时变价格，
$\mathbf{x}(t)=\left(x_1(t), x_2(t)\right) \geq \mathbf{0}$-生产者在生产过程中在期间/时刻使用的生产要素投入向 量 $t$
$\mathbf{c}(t)=\left(c_1(t), c_2(t)\right)>\mathbf{0}$ – 生产要素的时变价格向量, $y=f(\mathbf{x}(t))$-生产函数， ${ }^{14}$
$d(t) \geq 0$ – 随时间变化的固定生产成本，即不依赖于产出水平，也不依赖于生产要素投 入的成本。

$(4.105)$
$$\mathbf{x}(t) \geq \mathbf{0}$$

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