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.
$$ \left(\hat{x}{1}-\hat{x}{2}\right)={1 /|\lambda|}(\hat{L}-\hat{K}) $$ 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.
$$ \Delta \mu_{t}=\beta_{p}\left(\bar{U}^{c}-U_{t-1}^{c}\right)+\gamma\left(\bar{\mu}-\mu_{t-1}\right), $$ 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 $$ \Delta p_{t}=\beta_{p}\left(\bar{U}^{c}-U_{t-1}^{c}\right)+\gamma\left(v_{t-1}-\bar{v}\right)+\left(\Delta w_{t}-n_{x}\right) . $$
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 $$ y=\frac{n+g-i_{1}\left(i_{o}-\pi\right)}{(1-v)\left(1-i_{1}\right)+(1-c) v} $$ adds the term $$ \bar{y}=-\frac{i_{1} \beta_{r_{2}}\left(v-v_{o}\right)}{(1-v)\left(1-i_{1}\right)+(1-c) v} $$ to our former calculations – in the place of the $\beta_{w_{2}}$ term now. This term gives rise to the following additional partial derivative $$ \tilde{y}{v}=-\frac{i \beta{r_{2}}}{\left(1-v_{o}\right)\left(1-i_{1}\right)+(1-c) v_{o}} $$
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 $$ E_{t-1} V_{t+1}^{\prime}(\cdot) \frac{\partial y}{\partial x_{t-1}}=0 $$
$$ \frac{U_{1}(c, W-R)}{U_{2}(c, W-R)}=1+r $$ (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
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: $$ \begin{gathered} \frac{1}{Y^{n}} \cdot L^{n}=f\left(\frac{P}{Y^{n}}, r_{B}, r_{E}, \frac{\dot{P}}{P}, 1\right) \text { and, by rearranging } \ L^{n}=f\left(\frac{P}{Y^{n}}, r_{B}, r_{E}, \frac{\dot{P}}{P}, 1\right) \cdot Y^{n} . \end{gathered} $$
$$ n R_{t}^{(n)}=(n-1) E_{t} R_{t+1}^{(n-1)}+r_{t}+T_{t}^{(n)} $$ 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: $$ n R_{t}^{(n)}=(n-2) E_{t} R_{t+2}^{(n-2)}+E_{t}\left(r_{t+1}+r_{t}\right)+E_{t}\left(T_{t+1}^{(n-1)}+T_{t}^{(n)}\right) . $$
$$ \begin{aligned} \Delta \text { Inventories } &=G D P-A E \ &=\$ 12,000 \text { billion }-\$ 10,400 \text { billion } \ &=\$ 1,600 \text { billion. } \end{aligned} $$ 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.
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 $$ \text { Leakages }=\text { Injections } \Longrightarrow \text { Total spending }=\text { Total output } $$
$$ \left.c=10 p^{s} / \sqrt{[}+\left(p^{s}\right)^{2}\right] $$ 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.
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 $c^{}$.
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 $x^{*}=(A-v-w) / 2 N B$.
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 $$ L(\mathbf{p})=(1 / 2)(\mathbf{p}-\mathbf{c}){\mathbf{x}(\mathbf{c})-\ddot{\mathbf{x}}(\mathbf{p})} $$
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 $$ (n-1) s_{j}=1-s_{1} $$ $$ s_{1}=(1 / n)+[(n-1) / n] s_{m} $$
(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.
$$ C=P N\left(d_{1}\right)-E e^{-r t} N\left(d_{2}\right) $$ 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.
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 $$ \begin{aligned} &\left(4-\theta^{2}\right)\left(p_{1}^{N}-c_{1}\right)=\left(2+\theta-\theta^{2}\right)\left(a_{1}-c_{1}\right)-\theta\left(a_{2}-c_{2}\right) \ &\left(4-\theta^{2}\right)\left(p_{2}^{N}-c_{2}\right)=\left(2+\theta-\theta^{2}\right)\left(a_{2}-c_{2}\right)-\theta\left(a_{1}-c_{1}\right) \end{aligned} $$
$(U, U)$ : 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 $$ \left(\frac{1}{2 t}\left(t+\frac{c}{3}\right)^{2}, \frac{1}{2 t}\left(t-\frac{c}{3}\right)^{2}\right) . $$
