# 数学经济学|MATHEMATICAL ECONOMICS ECON113代写

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## Instructions:

1. Calculus: Calculus is an important tool in mathematical economics, and students will need a strong foundation in differential and integral calculus. Topics covered may include limits, derivatives, optimization, partial derivatives, and integration.
2. Linear Algebra: Linear algebra is used extensively in economics, particularly in the study of systems of equations. Topics covered may include matrix algebra, determinants, vector spaces, and linear transformations.
3. Optimization: Optimization is a central theme in mathematical economics, and students will learn how to use calculus and linear algebra to solve optimization problems. Topics covered may include constrained optimization, Lagrange multipliers, and Kuhn-Tucker conditions.
4. Game Theory: Game theory is a mathematical framework for analyzing strategic interactions between individuals or groups. Topics covered may include the concept of Nash equilibrium, dominant strategies, and the prisoner’s dilemma.
5. Dynamic Optimization: Dynamic optimization involves studying how economic variables evolve over time, and how decisions made at one point in time affect outcomes in the future. Topics covered may include difference equations, differential equations, and optimal control theory.
6. Probability and Statistics: Probability and statistics are important tools for understanding uncertainty in economic models. Topics covered may include probability distributions, hypothesis testing, and regression analysis.

These topics will provide students with a strong foundation in mathematical economics and prepare them for more advanced coursework in the field.

Given a national income model as follows: \begin{aligned} & \mathrm{Y}=\mathrm{C}+\mathrm{I}_0+\mathrm{G}_0 \\ & \mathrm{C}=\mathrm{C}_0+\mathrm{b} \mathrm{Y}_{\mathrm{d}} \\ & \mathrm{T}=\mathrm{T}_0+\mathrm{tY}, \end{aligned} Where $\mathrm{Y}=$ income; $\mathrm{Y}_{\mathrm{d}}=$ disposable income; $\mathrm{C}=$ consumption; $\mathrm{C}_0=$ autonomous consumption; $\mathrm{I}_0=$ autonomous investment; $\mathrm{G}_0=$ autonomous government expenditure; $\mathrm{T}=\operatorname{tax} ; \mathrm{T}_0=$ autonomous tax; $\mathrm{b}$ and $t$ are the coefficients. a) Solve the model for the equilibrium national income ( $\left.\mathrm{Y}^*\right)$

a) Starting with the first equation, we can substitute in the expressions for consumption and taxes:

\begin{aligned} & \mathrm{Y}=\mathrm{C}0+\mathrm{b} \mathrm{Y}{\mathrm{d}}+\mathrm{I}_0+\mathrm{G}_0 \ & \mathrm{Y}=\mathrm{C}_0+\mathrm{b}\left(\mathrm{Y}-\mathrm{T}_0\right)+\mathrm{I}_0+\mathrm{G}_0\end{aligned}

Expanding and rearranging:

\begin{aligned} & \mathrm{Y}-\mathrm{bY}+\mathrm{bT} \mathrm{T}_0=\mathrm{C}_0+\mathrm{I}_0+\mathrm{G}_0 \ & (1-\mathrm{b}) \mathrm{Y}=\mathrm{C}_0+\mathrm{I}_0+\mathrm{G}_0-\mathrm{bT}_0 \ & \mathrm{Y}=\frac{\mathrm{C}_0+\mathrm{I}_0+\mathrm{G}_0-\mathrm{bT}}{1-\mathrm{b}}\end{aligned}

Thus, the equilibrium national income is:

$\mathrm{Y}^*=\frac{150+200+350-0.65 \times 180}{1-0.65}=\frac{570}{0.35}=1628.57$

b) Using a), determine the government expenditure multiplier and explain its meaning.

The government expenditure multiplier is given by:

$\frac{\Delta \mathrm{Y}^*}{\Delta \mathrm{G}_0}=\frac{1}{1-\mathrm{b}}$

In this case, the multiplier is:

$\frac{\Delta \mathrm{Y}^*}{\Delta \mathrm{G}_0}=\frac{1}{1-0.65}=2.857$

This means that for every dollar increase in autonomous government expenditure, the equilibrium national income will increase by $2.857. 问题 3. c) Now, given the following information that$\mathrm{b}=0.65 ; \mathrm{t}=0.25 ; \mathrm{C}_0=150 ; \mathrm{I}_0=200 ; \mathrm{G}_0=350$; and$\mathrm{T}_0=180$, calculate the equilibrium level of national income$\left(\mathrm{Y}^*\right)$, consumption$\left(\mathrm{C}^*\right)$, and taxation$\left(\mathrm{T}^*\right)$. 证明 . Using the equation for equilibrium national income derived in part a), we can calculate the equilibrium levels of consumption and taxation:$\frac{\Delta \mathrm{Y}^*}{\Delta \mathrm{G}_0}=\frac{1}{1-0.65}=2.857$Substituting in the given values and solving for consumption:$\begin{aligned} & 1628.57=\frac{\mathrm{C}^+200+350+\mathrm{T}^}{1-0.65} \ & \mathrm{C}^=\frac{(1-0.65) \times 1628.57-200-350-\mathrm{T}^}{1} \ & \mathrm{C}^=0.35 \times 1628.57-150-0.65 \times \mathrm{T}^=616.43-0.65 \times \mathrm{T}^*\end{aligned}$Next, using the equation for taxes:$\mathrm{T}^=\mathrm{T}_0+\mathrm{tY}^=180+0.25 \times 1628.57=571.43$Therefore, the equilibrium levels of national income, consumption, and taxation are:$\begin{aligned} & \mathrm{Y}^=1628.57 \ & \mathrm{C}^=616.43-0.65 \times 571.43=236.43 \ & \mathrm{~T}^*=571.43\end{aligned}\$