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values of $X_{1}$ and $X_{2}$ in both countries once the appropriate substitutions are made. Setting $e=\phi e_{0}$ and $b=\phi b_{0}$ in both equations yields:
$$X_{1}(\phi)=\frac{\frac{r}{\phi}+b_{0}+e_{0}}{2 e_{0} P} \quad X_{2}(\phi)=\frac{e_{0}}{b_{0}+e_{0}}\left[L-{X,(\phi)}^{2}\right]$$
Thus, $X_{1}$ is decreasing in $\phi$ and $X_{2}$ is increasing in $\phi$. It follows that the country with higher turnover has a relative supply curve which is further to the right than its counterpart’s. As a result, the autarkic price of the search-sector good is lower in the country with the higher turnover.

## ECON3016 COURSE NOTES ：

$$\left(\hat{x}{1}-\hat{x}{2}\right)={1 /|\lambda|}(\hat{L}-\hat{K})$$
An increase in the relative endowment of labor compared with capital raises by a magnified amount the relative output of the first commodity. In more detail, if the endowment of labor increases relative to that of capital, with commodity prices constant,
$$\hat{x}{1}>\hat{L}>\hat{K}>\hat{x}{2}$$
The Rybczynski result refers to the fall in $x_{2}$ ‘s output if $\hat{K}$ is assumed to be zero.

# 高级经济学理论 Adv. Econometric Theory ECON3012

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$$\Delta \mu_{t}=\beta_{p}\left(\bar{U}^{c}-U_{t-1}^{c}\right)+\gamma\left(\bar{\mu}-\mu_{t-1}\right),$$
where $\gamma>0$. Inserted into the formula for price inflation this in sum gives:
$$\Delta p_{t}=\beta_{p}\left(\bar{U}^{c}-U_{t-1}^{c}\right)+\gamma\left(\bar{\mu}-\mu_{t-1}\right)+\left(\Delta w_{t}-n_{x}\right) .$$
In terms of the logged wage share $v_{t}=-\mu_{t}$ we get
$$\Delta p_{t}=\beta_{p}\left(\bar{U}^{c}-U_{t-1}^{c}\right)+\gamma\left(v_{t-1}-\bar{v}\right)+\left(\Delta w_{t}-n_{x}\right) .$$

## ECON3012 COURSE NOTES ：

Inserting the Taylor rule
$$i=\rho_{o}+\pi+\beta_{r_{1}}(\pi-\bar{\pi})+\beta_{r_{2}}\left(v-v_{o}\right), \quad \beta_{r_{1}}, \beta_{r_{2}}>0$$
into the effective demand equation
$$y=\frac{n+g-i_{1}\left(i_{o}-\pi\right)}{(1-v)\left(1-i_{1}\right)+(1-c) v}$$
$$\bar{y}=-\frac{i_{1} \beta_{r_{2}}\left(v-v_{o}\right)}{(1-v)\left(1-i_{1}\right)+(1-c) v}$$
to our former calculations – in the place of the $\beta_{w_{2}}$ term now. This term gives rise to the following additional partial derivative
$$\tilde{y}{v}=-\frac{i \beta{r_{2}}}{\left(1-v_{o}\right)\left(1-i_{1}\right)+(1-c) v_{o}}$$

0

with complementary slackness between $\lambda_{t}$ and $\left(A_{t}+y_{t}-c_{y}\right)$. By the envelope property, $x_{t-1}$ will satisfy
$$E_{t-1} \frac{d V_{t}(\cdot)}{d x_{t-1}}=E_{t-1} U^{\prime}\left(c_{t}\right) \frac{\partial y}{\partial x_{t-1}}=0$$
Substituting for $u^{\prime}(c)$, we have
$$E_{t-1}\left[\beta(1+r) V_{t+1}^{\prime}(\cdot)+\lambda_{t}\right] \frac{\partial y}{\partial x_{t-1}}=0$$
So if $\lambda_{t}=0$ in all states of period $t$, so that the individual knows that the liquidity constraint will not bind in period $t$, then $x_{p-1}$ is chosen so that
$$E_{t-1} V_{t+1}^{\prime}(\cdot) \frac{\partial y}{\partial x_{t-1}}=0$$

## ECON3010 COURSE NOTES ：

$$\frac{U_{1}(c, W-R)}{U_{2}(c, W-R)}=1+r$$
(10)
This suggests a further interpretation of the equilibrium implicit contract. Suppose that the landlord charges peasants an interest rate on consumption loans of $r$ plus an ‘entry fee’ of $f$ for the privilege of borrowing at this rate. A peasant will optimally choose a consumption credit, $c$, that satisfies the optimal intertemporal consumption condition,
$$\frac{U_{1}(c, W-(1+r) c-f)}{U_{2}(c, W-(1+r) c-f)}=1+r$$
If $f=\mathrm{R}^{\prime}-(1+r) i$, then the peasant will optimally choose a credit of $\iota=i$. Thus, the equilibrium implicit contract effects an outcome equivalent to a two-part

# 高级货币经济学 Adv. Monetary Economics ECON3008

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If market agents do not suffer from money illusion, a change in $P$ and a change in $Y^{n}$ by the multiple $\lambda$ will change the demand for money by the same amount (homogeneity hypothesis of money demand):
$$\lambda \cdot L^{n}=f\left(\lambda \cdot P, r_{B}, r_{E}, \frac{\dot{P}}{P}, \lambda \cdot Y^{n}\right) .$$
If holds true for any arbitrary realisation of the parameter $\lambda$, we can, for instance, define $\lambda=1 / Y^{\pi}$. Substituting this expression in yields:
$$\begin{gathered} \frac{1}{Y^{n}} \cdot L^{n}=f\left(\frac{P}{Y^{n}}, r_{B}, r_{E}, \frac{\dot{P}}{P}, 1\right) \text { and, by rearranging } \ L^{n}=f\left(\frac{P}{Y^{n}}, r_{B}, r_{E}, \frac{\dot{P}}{P}, 1\right) \cdot Y^{n} . \end{gathered}$$

## ECON3008 COURSE NOTES ：

$$n R_{t}^{(n)}=(n-1) E_{t} R_{t+1}^{(n-1)}+r_{t}+T_{t}^{(n)}$$
one period yields:
$$(n-1) R_{t+1}^{(n-1)}=(n-2) E_{t+1} R_{t+2}^{(n-2)}+r_{t+1}+T_{t+1}^{(n-1)}$$
When we take expectations of by writing $E_{t} E_{t+1}=E_{t}$ and substituting in yields:
$$n R_{t}^{(n)}=(n-2) E_{t} R_{t+2}^{(n-2)}+E_{t}\left(r_{t+1}+r_{t}\right)+E_{t}\left(T_{t+1}^{(n-1)}+T_{t}^{(n)}\right) .$$

# 宏观经济学原理 Principles of Macroeconomics ECON2018

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\begin{aligned} \Delta \text { Inventories } &=G D P-A E \ &=\ 12,000 \text { billion }-\ 10,400 \text { billion } \ &=\ 1,600 \text { billion. } \end{aligned}
When GDP is equal to $\$ 4,000$billion, aggregate expenditure is equal to$\$5,600$ billion, so that the change in inventories is
\begin{aligned} \Delta \text { Inventories } &=G D P-A E \ &=\ 4,000 \text { billion }-\ 5,600 \text { billion } \ &=-\ 1,600 \text { billion. } \end{aligned}
Notice the negative sign in front of the $\$ 1,600$billion; if output is$\$4,000$ billion, then inventory stocks will shrink by $\$ 1,600$billion over the year. ## ECON2018 COURSE NOTES ： Here’s another way to see the logic behind Say’s law, with some simple equations. Because the loanable funds market clears, we know that the interest rate-the price in this market-will rise or fall until the quantity of funds supplied (savings,$S$) is equal to the quantity of funds demanded (planned investment plus the deficit, or$\left.I^{p}+(G-T)\right)$: Rearranging this equation by moving$T$to the left side, we have: Loanable funds market clears$\Longrightarrow \underbrace{S+T}{\text {Leakages }}=\underbrace{I^{p}+G}{\text {Injections }}$So now, we know that as long as the loanable funds market clears, leakages equal injections. Finally, remember that $$\text { Leakages }=\text { Injections } \Longrightarrow \text { Total spending }=\text { Total output }$$ # 政治经济学 Political Economy ECON2016 0 这是一份nottingham诺丁汉大学ECON2016作业代写的成功案例 $$\left.c=10 p^{s} / \sqrt{[}+\left(p^{s}\right)^{2}\right]$$ Along the supply curve, the quantity of cheese,$c$, increases steadily with$p^{S}$, from 0 when$\mathrm{p}^{\mathrm{S}}=0$, to$\sqrt{50}$when$\mathrm{p}^{\mathrm{S}}=1$, to 10 when$\mathrm{p}^{\mathrm{S}}$rises to infinity. With$\mathrm{c}^{}=\mathrm{b}^{}=\sqrt{50}$and with$\theta$set equal to$\frac{1}{2}$, the value of$\mathrm{u}$in equation (4) become$(50)^{1 / 4}$which is the highest attainable utility consistent with the production possibility curve in equation (6). Employing the utility function at those values of$\theta$and$u$to eliminate$\mathrm{b}$from the demand curve in equation (5), it is easily shown that the demand curve for cheese, the relation between the quantity demanded and the demand price of cheese, becomes $$c=4 \sqrt{(50)} /\left(1+\mathrm{p}^{\mathrm{D}}\right)^{2}$$ Along the demand curve, the quantity of cheese,$c$, decreases steadily with$\mathrm{p}^{\mathrm{D}}$, from$4 \sqrt{5} 0$when$\mathrm{p}^{\mathrm{D}}=0$, to$\sqrt{50}$when$\mathrm{p}^{\mathrm{D}}=1$, to 0 when$\mathrm{p}^{\mathrm{D}}$rises to infinity. These demand and supply curves are illustrated on the bottom left-hand portion of figure 4.4. They cross at$\mathrm{c}=\mathrm{c}^{*}=\sqrt{5} 0$where people are as well off as possible with the technology at their command. ## ECON2016 COURSE NOTES ： How is the burden of taxation divided between producers and consumers? As shown in figure$4.2$above, a tax on cheese of t loaves per pound reduces production and consumption of cheese from$c^{}$to$c^{ }$pounds, raises the demand price of cheese from$p^{}$to$p^{D}\left(c^{* }\right)$and lowers the supply price of cheese from$p^{}$to$p^{S}\left(c^{* }\right)$. Denote the fall in the output of cheese by$\Delta \mathrm{c}$, the rise in the price of cheese by$\Delta \mathrm{p}^{D}$and the fall in the price of cheese by$\Delta \mathrm{p}^{\mathrm{S}}$where, by construction, $$\Delta \mathrm{c}=\mathrm{c}^{}-\mathrm{c}^{* }, \quad \Delta \mathrm{p}^{\mathrm{D}}=\Delta \mathrm{p}^{\mathrm{D}}\left(\mathrm{c}^{ }\right)-\mathrm{p}^{} \quad \text { and } \quad \Delta \mathrm{p}^{\mathrm{S}}=\mathrm{p}^{}-\mathrm{p}^{\mathrm{S}}\left(\mathrm{c}^{ }\right)$$ and $$\Delta \mathrm{p}^{\mathrm{D}}+\Delta \mathrm{p}^{\mathrm{D}}=\mathrm{t}$$ Thus the consumers’ share of the burden of the tax becomes$\Delta \mathrm{p}^{\mathrm{D}} / \mathrm{t}$and the producers’ share becomes$\Delta \mathrm{p}^{\mathrm{S}} / \mathrm{t}$. It is easily shown that consumers’ and producers’ shares depend on elasticities of demand and supply. Specifically, $$\Delta \mathrm{p}^{\mathrm{D}} / \mathrm{t}=\epsilon^{\mathrm{S}} /\left(\epsilon^{\mathrm{S}}+\epsilon^{\mathrm{D}}\right) \text { and } \Delta \mathrm{p}^{\mathrm{S}} / \mathrm{t}=\epsilon^{\mathrm{D}} /\left(\epsilon^{\mathrm{S}}+\epsilon^{\mathrm{D}}\right)$$ where$\epsilon^{D}$and$\epsilon^{S}$are the arc elasticities of demand and supply over the range from$c^{ }$to$c^{}$. # 环境与资源经济学 Environmental & Resource Econ ECON2015 0 这是一份nottingham诺丁汉大学ECON2015作业代写的成功案例 Maximize total surplus:$\quad-\sum_{i} c_{i} f_{i}+\sum_{i} b_{i} f_{i}$subject to: Balance of flow:$\sum_{i \in S_{k}} f_{k}=\sum_{i \in E_{j}} f_{j} \quad(\forall$nodes$j$) Conveyance capacity:$\quad d_{i} \leq f_{i} \leq u_{i}(\forall \operatorname{arcs} i)$## ECON2015 COURSE NOTES ： tion effort$\left(x_{i, j} / X_{t}\right)$. Hence, user$i$s pay-off in period$t$equals: $$\pi_{i, l}^{C P R}\left(x_{i, l}, X_{l}\right)=w\left[e-x_{i, l}\right]+\frac{x_{i, l}}{X_{l}}\left[A X_{l}-B X_{l}^{2}\right]-v x_{i, l}$$ with$A-v-w>0$. The socially optimal extraction effort level is the one that maximizes the unweighted sum of the pay-offs of all$N$users in the group as defined in (6.1). Therefore, the equitable socially optimal extraction effort level is$x^{*}=(A-v-w) / 2 N B. # 公共部门经济学 Public Sector Economics ECON2014 0 这是一份nottingham诺丁汉大学ECON2014作业代写的成功案例 At prices p, the welfare loss as compared to the first best solution amounts to \begin{aligned} L(\mathbf{p}) &=W(\mathbf{c})-W(\mathbf{p}) \ &=(1 / 2)(\mathbf{p}-\mathbf{c}) \mathbf{A}(\mathbf{p}-\mathbf{c}) \end{aligned} Since $$x(\mathbf{c})-x(\mathbf{p})=\mathbf{A}(\mathbf{p}-\mathbf{c})$$ the loss can also be written as $$L(\mathbf{p})=(1 / 2)(\mathbf{p}-\mathbf{c}){\mathbf{x}(\mathbf{c})-\ddot{\mathbf{x}}(\mathbf{p})}$$ ## ECON2014 COURSE NOTES ： In this case all firms including firm I show Cournot behaviour. Thus\mathrm{d} x / \mathrm{d} x_{1}=1$. Then$\mathrm{d} W_{m} / \mathrm{d} x_{1}=0$implies $$\left(p-C_{1}^{\prime}\right) / p=-(1 / \varepsilon)\left(s_{1}-s_{m}\right)$$ whereas for the$n-1$profit maximizing firms $$\left(p-C_{j}^{s}\right) / p=-\left(s_{j} / \varepsilon\right)$$ holds. To elaborate on this textbook case of oligopoly theory, let us consider$C_{1}^{\prime}=C_{j}^{\prime}=$constant. Because there is only one market price,$p$, the right hand sides of have to be equal, implying $$s_{1}=s_{m}+s_{j}$$ Furthermore, by definition of the market shares $$(n-1) s_{j}=1-s_{1}$$ $$s_{1}=(1 / n)+[(n-1) / n] s_{m}$$ # 金融经济学 Financial Economics ECON2012 0 这是一份nottingham诺丁汉大学ECON2012作业代写的成功案例 (a) We have$P=1000, i=0.06 / 12=0.005$, and$n=5 \times 12=60$. From equation (2.12), $$S=P(1+i)^{n}=1000(1.005)^{60}=\ 1348.85$$ The compound interest is$S-P=\$348.85$.
(b) We have $P=1000, i=0.15 / 12=0.0125$, and $n=30 \times 12=360$. From equation (2.12),
$$S=1000(1.0125)^{360}=\ 87,541.00$$
The compound interest is $S-P=\$ 86,541.00$, which is more than 86 times the original investment of$\$1000$. If the investment had been at 15 per cent simple interest, the interest earned would have been only
$$I=1000(0.15)(30)=\ 4500$$
This illustrates the power of compound interest. A high rate of interest for a long period of time generates far more than receiving only simple interest.

## ECON2012 COURSE NOTES ：

$$C=P N\left(d_{1}\right)-E e^{-r t} N\left(d_{2}\right)$$
where $C=$ the price of a call option, $P=$ the current price of the underlying, $E=$ the exercise price, $e=$ the base of natural logarithms, $r=$ the continuously compounded interest rate, and $t=$ the remaining life of the call option.
$N\left(d_{1}\right)$ and $N\left(d_{2}\right)$ are the cumulative probabilities from the normal distribution of getting the values $d_{1}$ and $d_{2}$, where $d_{1}$ and $d_{2}$ are as follows:
\begin{aligned} &d_{1}=\frac{\ln (P / E)+\left(r+0.5 \sigma^{2}\right) t}{\sigma \sqrt{t}} \ &d_{2}=d_{1}-\sigma \sqrt{t} \end{aligned}
where $\sigma=$ the standard deviation of the continuously compounded rate of return on the underlying asset.

# 工业经济学 Industrial Economics ECON2010

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If demand for both varieties is non-negative, ${ }^{9}$ the quantity demanded of variety 1 satisfies
$$b\left(1-\theta^{2}\right) q_{1}=\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right)+\theta\left(p_{2}-c_{2}\right)-\left(p_{1}-c_{1}\right)$$
with an analogous expression for variety 2 . Firm 1’s profit then satisfies
$$b\left(1-\theta^{2}\right) \pi_{1}=\left(p_{1}-c_{1}\right)\left[\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right)+\theta\left(p_{2}-c_{2}\right)-\left(p_{1}-c_{1}\right)\right]$$
The first-order condition for maximizing $\pi_{1}$ with respect to $p_{1}$ gives the equation of firm 1’s price reaction function
$$2\left(p_{1}-c_{1}\right)=\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right)+\theta\left(p_{2}-c_{2}\right)$$
Solving the equations of the two reaction functions gives static Nash equilibrium prices, which satisfy
\begin{aligned} &\left(4-\theta^{2}\right)\left(p_{1}^{N}-c_{1}\right)=\left(2+\theta-\theta^{2}\right)\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right) \ &\left(4-\theta^{2}\right)\left(p_{2}^{N}-c_{2}\right)=\left(2+\theta-\theta^{2}\right)\left(a_{2}-c_{2}\right)-\theta\left(a_{1}-c_{1}\right) \end{aligned}

## ECON2010 COURSE NOTES ：

$(U, U)$ : The market boundary for the two firms is given by the location $\vec{x}$ of the consumer who is indifferent between buying from either firm: $p_{1}+t \bar{x}=p_{2}+t(1-\bar{x})$, from which it immediately follows that $\bar{x}=\left(p_{2}-p_{1}+t\right) / 2 t$. As consumers are distributed with a unit density, profits of firm 1 are given by $\pi_{1}=p_{1} \bar{x}$ and those of firm 2 by $\pi_{2}=\left(p_{2}-c\right)(1-\bar{x})$. The unique pair of equilibrium prices is obtained from the first-order conditions as
$$\left(t+\frac{c}{3}, t+\frac{2 c}{3}\right),$$
yielding market areas
$$\left(\frac{1}{2}+\frac{c}{6 t}, \frac{1}{2}-\frac{c}{6 t}\right)$$
and equilibrium profits
$$\left(\frac{1}{2 t}\left(t+\frac{c}{3}\right)^{2}, \frac{1}{2 t}\left(t-\frac{c}{3}\right)^{2}\right) .$$