# 政治经济学 Political Economy ECON2016

$$\left.c=10 p^{s} / \sqrt{[}+\left(p^{s}\right)^{2}\right]$$
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 $c^{}$.