经济代写|ECON121 Microeconomics

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ECON121 课程简介

Course Description: A presentation and study of national income aggregates and accounting; equilibrium analysis of output, employment and the price level; general equilibrium analysis; and an introduction to economic dynamics.

Course Prerequesites: ECON 1010 and 1020 or their equivalents; QA and MATH 2200/2350
Primary Text: Macroeconomics by N. Gregory Mankiw ( $10^{\text {th }}$ edition)
Classroom Polls: We’ll be using REEF Polling to take classroom polls. It is free to download the app to your phone, tablet or laptop.

Prerequisites 

Course Description: Microeconomic theory is the study of models economists use to describe how agents (consumers, firms, governments, etc.) make decisions and how these decisions affect market outcomes and welfare. We begin by analyzing how consumers and firms make optimal decisions given the budgetary and physical/technological constraints they face. We then study how these decisions by individuals translate into competitive market equilibria, and look at the Conditions under which the “invisible hand” of the market optimizes welfare.

The second half of the class discusses deviations from the competitive ideal. We look at important sources of market failures, including including externalities, market power (monopolies), and asymmetric information. An important tool for economists in analyzing such situations where strategic interactions between people influences outcomes is game theory. The last section of the course will be a basic introduction to game theory and its applications in economics, including imperfectly competitive markets (e.g., oligopolies), public goods, and prinicpal-agent problems.

ECON121 Microeconomics HELP（EXAM HELP， ONLINE TUTOR）

The production function given is $f(x_1,x_2) = x_1^{\alpha}x_2^{1-\alpha}$, where $\alpha$ is a parameter with $0 < \alpha < 1$.

To find the input demands, we need to maximize the profit function subject to the production function and the cost constraint. We can write the profit function as:
$$
\pi=p f\left(x_1, x_2\right)-w_1 x_1-w_2 x_2
$$
where $\$ p \$$ is the price of the output and $\$ w_{-} 1 \$$ and $\$ w_{-} 2 \$$ are the prices of the inputs.

Using the production function, we can express the profit function as:
$$
\pi=p x_1^\alpha x_2^{1-\alpha}-w_1 x_1-w_2 x_2
$$
Taking partial derivatives with respect to $\$ x_{-} 1 \$$ and $\$ x_{-} 2 \$$ and setting them equal to zero, we get:
$$
\begin{aligned}
& \frac{\partial \pi}{\partial x_1}=p \alpha x_1^{\alpha-1} x_2^{1-\alpha}-w_1=0 \
& \frac{\partial \pi}{\partial x_2}=p(1-\alpha) x_1^\alpha x_2^{-\alpha}-w_2=0
\end{aligned}
$$
Solving for $\$ x _1 \$$ and $\$ x _2 \$$, we get:
$$
\begin{aligned}
& x_1=\left(\frac{w_1}{p \alpha}\right)^{\frac{1}{\alpha}} x_2^{\frac{1-\alpha}{\alpha}} \
& x_2=\left(\frac{w_2}{p(1-\alpha)}\right)^{\frac{1}{1-\alpha}} x_1^{\frac{\alpha}{1 \alpha}}
\end{aligned}
$$

Substituting these input demands back into the production function, we get:
$$
f\left(x_1, x_2\right)=\left(\frac{w_1}{p \alpha}\right)^{\frac{\alpha}{\alpha I}}\left(\frac{w_2}{p(1-\alpha)}\right)^{\frac{1}{\alpha-1}}
$$
Finally, we can verify that the production function satisfies the property of homogeneity of degree 1, i.e., $\$ f\left(t_{-} 1, t x_{-} 2\right)=$ $\mathrm{tf}\left(x_{-} 1, x_{-}\right) \$:$
$f\left(t x_1, t x_2\right)=\left(t x_1\right)^\alpha\left(t x_2\right)^{1-\alpha}=t^{\alpha+(1-\alpha)} x_1^\alpha x_2^{1-\alpha}=t f\left(x_1, x_2\right)$
Hence, we have found the input demands and verified that the production function satisfies the property of homogeneity of degree 1.

The analysis provided is correct. The profit function is given by $\pi(y) = py – C(y)$, where $p$ is the price of the output and $C(y)$ is the cost function. The cost function is expressed in terms of the input prices $p_1$ and $p_2$ and the production function parameters $\alpha$ and $(1-\alpha)$.

Under the CRS assumption, the cost function is homogeneous of degree one in the input prices, i.e., $C(ty) = tC(y)$ for all $t>0$. This implies that the cost per unit of output is constant and independent of the level of output. Therefore, the profit function is also homogeneous of degree one, i.e., $\pi(ty) = t\pi(y)$ for all $t>0$.

For the profit to be positive and finite, we need the coefficient of $y$ in the profit function to be positive. This requires the condition $p > C'(y)$, where $C'(y)$ is the marginal cost of production, which is given by $C'(y) = p_1\left(\frac{(1-\alpha) p_1}{\alpha p_2}\right)^{\alpha-1}+p_2\left(\frac{(1-\alpha) p_1}{\alpha p_2}\right)^\alpha$.

Since the cost function is linear in $y$ due to the CRS assumption, the profit function is also linear in $y$ and has the form $\pi(y) = ay + b$, where $a$ and $b$ are constants. Therefore, the profit is maximized at the level of output where marginal revenue equals marginal cost. In this case, we have $MR = p$, and $MC = C'(y)$, which gives the condition $p = p_1\left(\frac{(1-\alpha) p_1}{\alpha p_2}\right)^{\alpha-1}+p_2\left(\frac{(1-\alpha) p_1}{\alpha p_2}\right)^\alpha$.

Hence, the condition $p = p_1\left(\frac{(1-\alpha) p_1}{\alpha p_2}\right)^{\alpha-1}+p_2\left(\frac{(1-\alpha) p_1}{\alpha p_2}\right)^\alpha$ ensures that the profit is positive and finite, and therefore, there is a positive and finite level of production.

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