# 国际货币经济学 International Monetary Economics ECON3021/ECON8009

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No argument that I know of, favouring fluctuating exchanges ${ }^{2}$ is free from one of the following assumptions:
(1) that the “abstract model” corresponds with a large degree of probability to empirical conditions;
(2) that monetary internationalism is of secondary importance compared to the maintenance of “internal stability” by means of “autonomous” national policies;
(3) that exceptionally great disturbances are to be faced to which adjustment can hardly be reached without more or less important modifications of monetary parities.

## ECON3021/ECON8009 COURSE NOTES ：

G gold reserves
$M$ issue of central bank money
$M_{1}$ the part of $M$ which is actually circulating
$\mathbf{M}{2}$ the part of $M$ which is held by banks a $\quad \mathbf{M}{2}$ as a percentage of $\mathbf{M}$
$\mathbf{r} \quad$ rate of legal gold reserves as percentage of $\mathbf{M}$
$\mathrm{m}$ amount by which the actual $M$ falls short of the issue that can be made on the basis of $G$, with due regard to $\mathrm{r}$.
(Thus $\mathrm{m}=\frac{\mathrm{G}}{\mathrm{r}}-\mathrm{M}$ )
$\mathbf{M}^{\prime}$ volume of demand deposits subject to cheque
$\mathbf{r}^{\prime}$ ratio of legal or customary cash reserves against demand deposits subject to cheque as a percentage.
$\mathrm{m}^{\prime}$ amount by which the actual $\mathrm{M}^{\prime}$ falls short of what it might be on the basis of $\mathrm{M}{2}$ with due regard to $\mathrm{r}^{\prime}$. might be on the basis of $M{2}$ with due regard to $r^{\prime}$.
(Thus $\mathrm{m}^{\prime}=\frac{\mathrm{M}_{2}}{\mathrm{r}^{\prime}}-\mathbf{M}^{\prime}$ )

# 战略思考：博弈论介绍 Strategic Thinking: An introduction to Game Theory ECON2141

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Proof. I first prove part a. By the definition of Nash equilibrium we have $U_{2}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \geq U_{2}\left(\alpha_{1}^{}, \alpha_{2}\right)$ for every mixed strategy $\alpha_{2}$ of player 2 or, since $U_{2}=-U_{1}$, $$U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \leq U_{1}\left(\alpha_{1}^{}, \alpha_{2}\right) \text { for every mixed strategy } \alpha_{2} \text { of player } 2 .$$
Hence
$$U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right)=\min {\alpha{2}} U_{1}\left(\alpha_{1}^{}, \alpha_{2}\right) .$$ Now, the function on the right hand side of this equality is evaluated at the specific strategy $\alpha_{1}^{}$, so that its value is not more than the maximum as we vary $\alpha_{1}$, namely $\max {a{1}} \min {a{2}} U_{1}\left(\alpha_{1}, \alpha_{2}\right)$. Thus we conclude that
$$U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \leq \max {\alpha{1}} \min {\alpha{2}} U_{1}\left(\alpha_{1}, \alpha_{2}\right) .$$

## ECON2141COURSE NOTES ：

$U_{1}\left(\alpha_{1}^{}, \alpha_{2}\right) \geq U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right)$ for every mixed strategy $\alpha_{2}$ of player 2 , 342 or $U_{2}\left(\alpha_{1}^{}, \alpha_{2}\right) \leq U_{2}\left(\alpha_{1}^{}, \alpha_{2}^{}\right)$ for every mixed strategy $\alpha_{2}$ of player 2 .
Similarly,
$$U_{2}\left(\alpha_{1}, \alpha_{2}^{}\right) \geq U_{2}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \text { for every mixed strategy } \alpha_{1} \text { of player } 1 \text {, }$$ or $$U_{1}\left(\alpha_{1}, \alpha_{2}^{}\right) \leq U_{1}\left(\alpha_{1}^{}, \alpha_{2}^{}\right) \text { for every mixed strategy } \alpha_{1} \text { of player } 1 \text {, }$$
so that $\left(\alpha_{1}^{}, \alpha_{2}^{}\right)$ is a Nash equilibrium of the game.

# 公共部门经济学 Public Sector Economics ECON2131

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$$\varepsilon_{p_{j}}^{s_{j}} \equiv\left(\partial s_{i} / \partial p_{j}\right)\left(p_{j} / s_{i}\right)$$
and the elasticity of the total consumption of equation
$$\varepsilon_{p}^{x} \equiv(\partial x / \partial \bar{p})(\bar{p} / x)$$
To see the relation between share elasticities and total elasticity of demand define
$$w_{j}=\left(p_{j} x_{j}\right) / \bar{p} x$$
the expenditure in period $j$ relative to the cost of total consumption at the index price. Then the full elasticity can be written
$$\eta_{i j}=\varepsilon_{p_{j}}^{s_{i}}+\varepsilon_{\bar{p}}^{x} w_{j} \quad$$

## ECON2131COURSE NOTES ：

$$\mathrm{d} y_{i}=D_{i}^{o} \mathrm{~d} \mathbf{p}+\left(\mathrm{d} R-\mathbf{x}^{o} \mathrm{~d} \mathbf{p}\right) / H$$
It follows immediately that
$$\sum_{i=1}^{H} \mathrm{~d} y_{i}=\mathrm{d} R$$
Moreover, Roy’s identity implies that
$$V_{i}\left(\mathbf{p}^{o}+\mathrm{d} \mathbf{p}, \mathbf{q}, y_{i}+\mathrm{d} y_{i}\right)=V_{i}\left(\mathbf{p}^{o}, \mathbf{q}, y_{i}\right)+\left(\partial V_{i} / \partial y_{i}\right)\left(\mathrm{d} y_{i}-D_{i}^{o} \mathrm{~d} \mathbf{p}\right)$$
Since marginal utility of money $\partial V_{i} / \partial y_{i}$ is positive, and since $\mathrm{d} y_{i}-D_{i}^{o} \mathrm{~d} p$ must be positive if holds, it can be easily seen that utility
$$V_{i}\left(\mathbf{p}^{0}+\mathrm{d} \mathbf{p}, \mathbf{q}, y_{i}+\mathrm{d} y_{i}\right)>V_{i}\left(\mathbf{p}^{0}, \mathbf{q}, y_{i}\right)$$

# 经济学和金融经济学的优化 Optimisation for Economics and Financial Economics ECON2125

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When both lagged endogenous variables and serial correlation in the disturbance term appear, we need to impose additional conditions to identify a model. For instance, consider the following two equation system (Koopmans, Rubin and Leipnik, 1950):
$$y_{1 t}+\beta_{11} y_{1, t-1}+\beta_{12} y_{2, t-1}=u_{1 t} \beta_{12} y_{1 t}+y_{2 t}=u_{2 t} .$$
If $\left(u_{1 t}, u_{2 t}\right)$ are serially uncorrelated, $(6)$ is identified. If serial correlation in $\left(u_{1 t} u_{2 t}\right)$ is allowed, then
$$\begin{array}{r} y_{1 t}+\beta_{11}^{} y_{1, t-1}+\beta_{12}^{} y_{2, t-1}=u_{1 t}^{} \ \beta_{12} y_{1 t}+y_{2 t}=u_{2 t} \end{array}$$ is observationally equivalent to $(6)$, where $\beta_{11}^{}=\beta_{11}+d \beta_{21}, \beta_{12}^{}=\beta_{12}+d$, and $u_{1 t}^{}=u_{1 t}+d u_{2 t}$.

## ECON2125COURSE NOTES ：

For each person $i$, let $\left(y_{0 i}^{}, y_{1 i}^{}\right)$ denote the potential outcomes in the untreated and treated states, respectively. Then the treatment effect for individual $i$ is
$$\Delta_{i}=y_{1 i}^{}-y_{0 i}^{}$$
and the average treatment effect (ATE) is defined as
$$E\left(\Delta_{i}\right)=E\left(y_{1 i}^{}-y_{0 i}^{}\right) ;$$
see Heckman and Vytlacil (2001).
Let the treatment status be denoted by the dummy variable $d_{i}$ where $d_{i}=1$ denotes the receipt of treatment and $d_{i}=0$ denotes nonreceipt. The observed data are often in the form
$$y_{i}=d_{i} y_{1 i}^{}+\left(1-d_{i}\right) y_{0 i^{}}^{}$$ Suppose $y_{1 i}^{}=\mu_{1}\left(\mathbf{x}{i}, u{1 i}\right), \quad y_{0 i}^{}=\mu_{0}\left(\mathbf{x}{i}, u{0 i}\right)$, and $d_{i}^{}=\mu_{D}\left(\mathbf{z}{i}\right)-u{d i}$, where $d_{i}=1$ if $d_{i}^{*} \geq 0$ and 0 otherwise, $\mathbf{x}{i}$ and $\mathbf{z}{i}$ are vectors of observable exogenous variables and $\left(u_{1 i}, u_{0 \dot{b}} u_{d i}\right.$ ) are unobserved random variables. The average treatment effect and the complete structural econometric model can be identified with parametric

# 货币与银行 Money and Banking ECON2026

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$$\sum_{i} s_{i} U_{i}\left(s_{i}, s_{i}, g_{i}\right)=\sum s_{i}\left(1-q_{f}\right) b-c=[1-\delta(1-\mathrm{H})] b-c$$
$\mathrm{H}=\Sigma_{i} s_{i}^{2}$ is the Herfindahl index. If on the other hand all players adopt the version of the largest player, overall utility is equal to
$$\sum s_{i} U_{i}\left(s_{i}, 1, g_{1}\right)=b-c-\left(1-s_{1}\right) c_{m}$$
Thus the net welfare gain of switching to $g_{1}$ is equal to:
$$\delta(1-\mathrm{H}) b-\left(1-s_{1}\right) c_{m}$$
This is positive if:
$$c_{m} / b<\delta(1-\mathrm{H}) /\left(1-s_{j}\right)$$

## ECON2026COURSE NOTES ：

Formally, the cost function used here is given by
\begin{aligned} C\left(w, x^{f}, y^{r}, s, t\right)=& y_{x, t}^{r} \times\left[\sum_{i=1}^{n} \sum_{j=1}^{w} \beta_{i j}\left(w w_{i}\right){s, t}^{1 / 2}+\sum{i=1}^{n} \sum_{j=1}^{m} \gamma_{i j}\left(w x_{j}^{f}\right){s, t}\right.\ &\left.+\sum{i=1}^{n} \delta_{A}\left(w_{i}\right){s t r} \times t+\sum{i=1}^{n} \delta_{i 2}(w){s, t} \times t^{2}\right] \end{aligned} where $C$ is the real total variable cost, $s$ is the industry $(s=1$ : banking, $s=2$ : insurance), and $\beta{i g}, \gamma_{i j}, \delta_{i j}$ are unknown parameters.

Applying Shephard’s lemma to the above cost function, we derive the conditional (with respect to output and the quasi-fixed factors) demand equations for the variable labour inputs $x_{i}^{v}, i \in{U S, M S, H S}_{:-\left(x_{i}^{v}\right){s, t}}=\left(\partial \mathrm{C}(-) / \partial w{i}\right){x, i^{-}}$The division of these factor demands by real value added yields a system of input-output coefficients of the following form: $$(\pi){x, t} \equiv\left(\frac{x_{i}^{v}}{y^{r}}\right){s, t}=\sum{j=1}^{n} \beta_{i j}\left(w / w_{i}\right){s, t}^{1 / 2}+\sum{j=1}^{m} \gamma_{i j}\left(x_{j}^{f}\right){s, t}+\delta{A} \times t+\delta_{j 2} \times t^{2}$$

# 管理经济学 Managerial Economics ECON2014/ECON6014

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In this calculation, $\pi_{i}$ is the profit or return associated with the $i$ th outcome; $p_{i}$ is the probability that the $i$ th outcome will occur; and $E(\pi)$, the expected value, is a weighted average of the various possible outcomes, each weighted by the probability of its occurrence.
The deviation of possible outcomes from the expected value must then be derived:
$$\text { Deviation }{i}=\pi{i}-E(\pi)$$
The squared value of each deviation is then multiplied by the relevant probability and summed. This arithmetic mean of the squared deviations is the variance of the probability distribution:$$\text { Variance }=\sigma^{2}=\sum_{i=1}^{n}\left[\pi_{i}-E(\pi)\right]^{2} p_{i}$$
The standard deviation is found by obtaining the square root of the variance:
$$\text { Standard Deviation }=\sigma=1 \sqrt{\sum_{i=1}^{n}\left[\pi_{i}-E(\pi)\right]^{2} p_{i}}$$
The standard deviation of profit for project $A$ can be calculated to illustrate this procedure:

## ECON2014/ECON6014COURSE NOTES ：

A variant of NPV analysis that is often used in complex capital budgeting situations is called the profitability index (PI), or the benefit/cost ratio method. The profitability index is calculated as follows:
$$P I=\frac{\text { PV of Cash Inflows }}{\text { PV of Cash Outflows }}=\frac{\sum_{t=1}^{n}\left[E\left(C F_{i t}\right) /\left(1+k_{i}\right)^{t}\right]}{\sum_{t=1}^{n}\left[C_{i t} /\left(1+k_{i}\right)^{t}\right]}$$
The PI shows the relative profitability of any project, or the present value of benefits per dollar of cost.

In the SVCC example described in Table 15.4, NPV >0 implies a desirable investment project and $P I>1$. To see that this is indeed the case, we can use the profitability index formula, given in Equation 15.5, and the present value of cash inflows and outflows from the project, given in Equation 15.4. The profitability index for the SVCC project is
\begin{aligned} P I &=\frac{P V \text { of Cash Inflows }}{P V \text { of Outflows }} \ &=\frac{\ 27,987,141}{\ 20,254,820} \ &=1.38 \end{aligned}

# 微观经济学 Microeconomics  ECON1101

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Let us return to the case of a village with a monopolistic, omniscient moneylender. He knows which villagers have access to which type of land. As before, we assume that his opportunity cost of funds is $\varrho$. His problem is to set an interest rate $i_{3}(t)$ for each type of borrower to solve:
$\underset{i(t)}{\operatorname{Max}} \pi(t) i(t)$
subject to
$$\pi(t)(R(t)-i(t)) \geq W$$
and
$$\pi(t) i(t) \geq \rho$$
As long as $\mathrm{R}-\varrho \geq W$, the equilibrium will involve lending to each type of borrower at interest rates $i_{3}(t)=(\mathrm{R}-W) /$ $\pi(t)=i(t)$. Each type of borrower achieves an expected utility of $W$, and the lender earns an expected return of $\mathrm{R}-\varrho \geq$ $W$ on each loan.

## ECON1101 COURSE NOTES ：

$$U_{i}=\sum_{t=1}^{T} \beta^{t} \sum_{s=1}^{S} \pi_{s} u_{i}\left(c_{\text {ist }}\right)$$
where $u()$ is twice continuously differentiable with $u^{\prime}>0, u^{\prime \prime}<0$ and $\operatorname{Lim}{x \rightarrow e^{\prime}} u^{\prime}(x)=+\infty .^{49} \mathrm{~A}$ Pareto-efficient allocation of risk within the village can be found by maximizing the weighted sum of the utilities of each of the $N$ households, where the weight of household $i$ in the Pareto programme is $\lambda, 0<\lambda{i}<1, \Sigma \lambda_{i}=1$ :
$$\operatorname{Max}{C{\text {iht }}} \sum_{i=1}^{N} \lambda_{i} U_{i}$$subject to the resources available in the village at each point in time in each state of nature:
$$\sum_{i=1}^{N} c_{\text {ist }}=\sum_{i=1}^{N} y_{\text {ist }} \forall s, t$$
$$c_{\mathrm{ist}} \geq 0 \forall i, s, t$$