Tag Archives: 曼切斯特代写

发展经济学的高级课题 Advanced Topics in Development Economics ECON61212T

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发展经济学的高级课题 Advanced Topics in Development Economics ECON61212T

The gross national product of our economy is defined as the money value of all final products (goods and services) produced in a period of time, usually a year. This product can be divided into two categories, consumption $(C)$ and investment (I). Thus we have the following definitional equation:
$$
Y=C+I
$$
where $Y$ stands for GNP.
In the production of goods and services making up the GDP, an equal amount of income is generated in the form of wages, rent, interest, and profit. All income earned is either spent for consumption or saved. Thus we have another definitional relation to state the disposition of income:
$$
Y=C+S
$$
Setting equations ( 1$)$ and ( 2 ) equal to each other, we obtain:
$$
C+S=Y=C+1
$$
and thus
$$
C+S=C+I
$$

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ECON61212T COURSE NOTES ：

$$
S=I+(X-M)
$$
In Chapter 12 we observed that the balance of trade in goods and services $(X-M)$ is equal to the change in the home country’s net creditor/debtor position relative to the rest of the world, which can also be regarded as net foreign investment. ${ }^{1}$ Consequently, the familiar identity between saving and investment still holds, with investment including both domestic and foreign investment. That is:
$$
S=I_{d}+I_{f} \text { where } I_{f}=X-M
$$
Now we are ready to explain how income is determined in an open economy. We assume that exports, like investment, are exogenous – that is, the level of exports does not depend on domestic income. Imports, on the other hand, are a function of income: an increase in income leads to an increase in imports. This gives us a relationship (an import function) such as the following:
$$
\mathrm{M}=m Y
$$
where $m$ represents the “marginal propensity to import,” the fraction of additional income that is spent for imports. That is:
$$
m=\frac{\Delta \mathrm{M}}{\Delta \mathrm{Y}}
$$









环境政策经济学 Economics of Environmental Policy ECON60782T

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经济增长 Economic Growth ECON60662T

Solve for the dynamically efficient resource allocation for $r=0,0.05$, $0.10,0.2$, and $0.5$. Note that from the exact solution we have $q_{0}=$ $\left(b Q_{\text {tot }}+r(a-c)\right) / b(2+r)$, and $q_{1}=Q_{\text {tot }}-q_{0}$.

Build a table with columns showing ” $r, “{ }^{” } q_{0} “{ }^{} q_{1}, “$ “PV of marginal profit period $0^{” *}$ and ” $P V$ of marginal profit period $1 . “$ There will be five rows, one for each ” $r^{47}$ value above. Confirm that Hotelling”s rule is satisfied in each case. Provide a brief interpretation of the impacts of rising discount rates on the dynamically optimal price and quantity path over time.

Now suppose that $Q_{\text {tot }}=70$. Build a second table like the one in 6 a above. Provide a brief narrative economic interpretation of the impact of a smaller resource stock on the dynamically efficient allocation of the resource, as well as prices and marginal profit.

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ECON60782T COURSE NOTES ：

Suppose that $X(t)$ is the stock or biomass of economically valuable fish at time $t$, and $F(X)$ is the biological growth function for the stock over time (from a calculus standpoint, think of this as the instantaneous growth rate of $X$, equal to $d X / d t)$. $F(X)$ reflects the rate of net recruitment (number of new fish entering a fishery, net of fish removed from the fishery). Suppose that for the particular species in question $F(X)$ can be described by a logistic function:
$$
F(X)=r X(l-X / k)
$$
Note that $r$ is interpreted as the rate of growth of $X$ when the stock $X$ is nearly zero. Note that $k$ is interpreted as the maximum value for the stock (carrying capacity) for a given habitat. Thus:
$$
X=k \rightarrow F(X)=0
$$









进一步的计量经济学 Further Econometrics ECON60622T

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进一步的计量经济学 Further Econometrics ECON60622T

There is still another motivation available for the imperfect price level adjustment we are assuming. For reasons of simplicity, we here consider the case of a Cobb-Douglas production function, given by $Y=K^{\alpha} L^{1-\alpha}$. According to the above we have
$$
p=w / F_{L}\left(K, L^{p}\right)=w /\left[(1-\alpha) K^{\alpha}\left(L^{p}\right)^{-\alpha}\right]
$$
which for given wages and prices defines potential employment. Similarly, we define competitive prices as the level of prices $p_{c}$ such that
$$
p_{c}=w / F_{L}\left(K, L^{d}\right)=w /\left[(1-\alpha) K^{\alpha}\left(L^{d}\right)^{-\alpha}\right]
$$
From these definitions we get the relationship
$$
\frac{p}{p_{c}}=\frac{(1-\alpha) K^{\alpha}\left(L^{d}\right)^{-\alpha}}{(1-\alpha) K^{\alpha}\left(L^{p}\right)^{-\alpha}}=\left(L^{p} / L^{d}\right)^{\alpha}
$$

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ECON60622T COURSE NOTES ：

As far as consumption is concerned we assume Kaldorian differentiated saving habits of the classical type $\left(s_{w}=1-c_{w}=1-c \geq 0, s_{c}=1\right)$, i.e., real consumption is given by:
$$
C=c v Y=c \omega L^{d}, \quad v=\omega / x, \omega=w / p \text { the real wage }
$$
and thus solely dependent on the wage share $v$ and economic activity $Y$. For the investment behavior of firms we assume
$$
\begin{aligned}
\frac{I}{K} &=i_{1}((1-v) y-(i-\pi))+n \
y &=\frac{Y}{K}, \quad n=\hat{L}+\hat{x}=n+n_{x} \text { trend growth. }
\end{aligned}
$$








应用经济学专题 Topics in Applied Economics ECON60482T

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应用经济学专题 Topics in Applied Economics ECON60482T

$\left(K_{t+h} / L_{t}\right)\left(L_{t} / L_{t+h}\right)$, this model can again be reduced to the state variable $k$ now given by $k_{t}=K_{t} / L_{t}$ and gives then rise to:
$$
k_{t+h}=\left(k_{t}+\operatorname{sh} f\left(k_{t}\right)\right) /(1+n h)
$$
At first sight, this law of motion of the period version of the Solow model looks quite different compared to the one in continuous time
$$
\dot{k}=s F(k, 1)-n k=s f(k)-n k
$$
and its discretization by way of difference quotients
$$
k_{t+h}=k_{t}+h\left(s f\left(k_{t}\right)-n k_{t}\right)=k_{t}+s h f\left(k_{t}\right)-n h k_{t}
$$

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ECON60482T COURSE NOTES ：

The Phillips curve of this approach to inflation dynamics is indeed given by
$$
\pi_{t}=f\left(U_{t}\right)+\alpha \pi_{t}^{\epsilon}, \quad 0<\alpha \leq 1, f^{\prime}<0 $$ and inflationary expectations $\pi_{t}^{e}$ are adjusted adaptively according to $$ \pi_{t+1}^{e}=\pi_{t}^{e}+c\left(\pi_{t}-\pi_{t}^{e}\right), \quad 00
$$








金融计量经济学 Healthcare Economics ECON60432T

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金融计量经济学 Healthcare Economics ECON60432T

Consider the following simple regression model:
$$
Y_{i}=a+b X_{i}+\varepsilon_{i},
$$
where the subscript $i$ refers to the $i$ th observation. The random error term $\varepsilon_{i}$ (epsilon) captures all the variation in the dependent variable $Y_{i}$ that is not explained by the $X_{i}$ (independent) variables.

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ECON60432T COURSE NOTES ：

These coefficients give a quantitative account of the relationship between a dependent variable and one or more independent variables in a regression equation. In the regression equation
$$
Y_{i}=a+b X_{i}+\varepsilon_{i},
$$
$b$ is a regression coefficient. See Multiple (Linear) Regression.








金融计量经济学 Financial Econometrics ECON60332T

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微观经济计量学 Microeconometrics ECON60052T

The distribution of $z_{t}$ is the mixture, with equal weights, of the distributions of $\mathrm{w}{t}$ and $\mathrm{u}{t}$. Ordinary properties of mixtures lead to
$$
\begin{aligned}
E\left(\mathrm{z}{l} \otimes \mathrm{z}{l} \otimes \mathrm{z}{l}\right)=& \frac{1}{2} E\left(\mathrm{w}{t} \otimes \mathrm{w}{l} \otimes \mathrm{w}{l}\right)+\frac{1}{2} E\left(\mathrm{u}{l} \otimes \mathrm{u}{t} \otimes \mathrm{u}{t}\right) \ &=\frac{1}{2}\left(\mathrm{~g}{1}+\mathrm{g}{2}\right) \end{aligned} $$ Hence the multivariate skewness of the vector $z{t}$ can be represented as follows:
$$
S\left(z_{t}\right)=\frac{1}{4}\left(g_{1}+g_{2}\right)^{T} \Gamma\left(g_{1}+g_{2}\right)
$$
The distribution of $w_{t}$ equals that of $-u_{t}$ only when $\gamma=0$. By assumption $\gamma \neq 0$, so that
$$
\mathrm{g}{1}+\mathrm{g}{2} \neq 0 \Rightarrow S\left(z_{l}\right)=\frac{1}{4}\left(\mathrm{~g}{1}+\mathrm{g}{2}\right)^{\mathrm{T}} \Gamma\left(\mathrm{g}{1}+\mathrm{g}{2}\right)>0
$$

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ECON60332T COURSE NOTES ：

$$
X_{l}=W_{t} \sqrt{a+a_{0} X_{l}^{2}+\sum_{i=1}^{p} a_{i} X_{l-i}^{2}}
$$
When the implicit ARCH is fitted to the data, the following residuals ensue:
$$
\hat{W}{t}=\frac{X{t}}{\sqrt{\hat{a}+\hat{a}{0} X{l}^{2}+\sum_{i=1}^{p} \hat{a}{i} X{l-i}^{2}}}
$$








国际宏观经济学 International Macroeconomics ECON60132T

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微观经济计量学 Microeconometrics ECON60052T

$$
1+i_{t}=\left(1+i_{t}^{}\right) \frac{F_{t}}{S_{t}} . $$ When (1.2) is violated a riskless arbitrage profit opportunity is available and the market is not in equilibrium. For example, suppose there are no transactions costs, and you get the following 12-month eurocurrency, forward exchange rate and spot exchange rate quotations $$ i_{t}=0.0678, \quad i_{t}^{}=0.0422, \quad F_{t}=0.9961, \quad S_{t}=1.0200 .
$$

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ECON60132T COURSE NOTES ：

It will be the case that $i_{a}>i_{b}, i_{a}^{}>i_{b}^{}, S_{a}>S_{b}$, and $F_{a}>F_{b}$. An arbitrage that shorts the dollar begins by borrowing a dollar at the gross rate $1+i_{a}$, selling the dollar for $1 / S_{a}$ pounds which are invested at the gross rate $1+i_{b}^{}$ and covered forward at the price $F_{b}$. The per dollar profit is $$ \left(1+i_{b}^{}\right) \frac{F_{b}}{S_{a}}-\left(1+i_{a}\right) .
$$
Using the analogous reasoning, it follows that the per pound profit that shorts the pound is
$$
\left(1+i_{b}\right) \frac{S_{b}}{F_{a}}-\left(1+i_{a}^{*}\right) .
$$








微观经济计量学 Microeconometrics ECON60052T

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微观经济计量学 Microeconometrics ECON60052T

which under our assumptions implies
$$
h(x, z)=F_{v}^{-1}(P(x, z)) .
$$
Following the analysis in Matzkin (1992), we can recover both $h(x, z)$ and $F_{v}()$ nonparametrically up to normalization.

Next, take the conditional (on $X, Z$ ) expectation of the outcome for the treated group
$$
E(Y \mid X=x, Z=z, D=1)=g_{1}(x)+E\left(\varepsilon_{1} \mid X=x, Z=z, D=1\right)
$$
We can write the last term as
$$
E\left(\varepsilon_{1} \mid X=x, Z=z, D=1\right)=E\left(\varepsilon_{1} \mid v<h(x, z)\right)=E\left(\varepsilon_{1} \mid v<F_{v}^{-1}(P(x, z))\right)
$$
That is, we can write it as a function of the known $h(x, z)$ or, equivalently, as a function of the probability of selection $P(x, z)$,
$$
E(Y \mid X=x, Z=z, D=1)=g_{1}(x)+K_{1}(P(x, z))
$$

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ECON60052T COURSE NOTES ：

$$
R(\theta, \delta)=E_{\theta}[L(\theta, \delta(Z, U))]=\int_{0}^{1} \int L(\theta, \delta(z, u)) d P_{\theta}(z) d u .
$$
A rule $\delta$ is admissible if there exists no other rule $\delta^{\prime}$ with
$$
R\left(\theta, \delta^{\prime}\right) \leq R(\theta, \delta), \quad \forall \theta \in \Theta,
$$
and
$$
R\left(\theta, \delta^{\prime}\right)<(\theta, \delta) \text { for some } \theta \text {. }
$$








发展微观经济学 Development Microeconomics ECON60022T

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发展微观经济学 Development Microeconomics ECON60022T

$$
Y=W(1-\alpha) e+F(\alpha A, \alpha e, \theta)-R \alpha A \leq \alpha F(A, e, \theta)
$$
as $R \alpha A \geq W(1-\alpha) e$
and the landlord’s income by
$$
\begin{aligned}
\pi=& R \alpha A+F[(1-\alpha) A,(1-\alpha) e, \theta]-W(1-\alpha) e \
& \geq(1-\alpha) F(A, e, \theta), \text { as } R \alpha A \geq W(1-\alpha) e
\end{aligned}
$$

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ECON32202T COURSE NOTES ：

$$
Y=F(e, x)-\beta \mathrm{pz}+p(z-x)
$$
and
$$
\pi=(1-\alpha) F(e, x)-(1-\beta) p z
$$
The tenant chooses $e$ and $x$, given $\alpha, \beta$, and $z$. Taking this behaviour into account the landlord decides on $\alpha, \beta$, and $z$. We can rewrite (10) and (11) as
$$
Y=\alpha \theta F(e, x)-\mathrm{px}+K
$$
and
$$
\pi=(1-\alpha) F(e, x)-K
$$








卫生经济学 Health Economics ECON32202T

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卫生经济学 Health Economics ECON32202T

The general form of Bayes’ theorem (sometimes called Bayes’ rule) is:
$$
p(A \mid X)=\frac{p(X) \times p(A)}{p(X \mid A) \times p(A)+p(X \mid \sim A) \times p(\sim A)}
$$
where
$p(A)$ is the prior (our prior knowledge, for example, of the prevalence of cancer in the population as a whole);
$p(A \mid X)$ is the posterior probability (a revised estimate of the probability of $A$, given $X$, in our example, of there being cancer, given that the test result was positive);
$p(X \mid A)$ is the conditional probability of $X$, given $A$ (in our example, of a positive test when a patient has cancer); $p(X \mid \sim A)$ is the conditional probability of $X$, given not- $A$ (in our example, of a positive test when a patient does not have cancer).

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ECON32202T COURSE NOTES ：

$\chi^{2}$ (chi-squared or ‘chi-square’ – statisticians are not agreed) is a statistical test based on a comparison between a test statistic and a critical value from a chi-squared distribution. A chi-squared variable can be regarded as the sum of a number of squared independent normal variables, each with zero mean and unit variance. The number of such squared terms is the number of degrees of freedom of the $\chi^{2}$ distribution. A chi-squared test can be used to test the null hypothesis that two or more population distributions do not differ. When comparing observed values with those expected under the null hypothesis, it is the sum of the ratio of the squared differences between observed $(O)$ and expected $(E)$ values to the expected value:
$$
\chi^{2}=\sum \frac{\left[O_{i}-E_{i}\right]^{2}}{E_{i}}
$$
There are two well-known versions, the Pearson $\chi^{2}$ test and the MantelHaenszel test. See Statistical Significance.