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# Econ经济作业代写Economics代考|Equality of marginal rates of substitution and transformation

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## Base case|Econ经济作业代写Economics代考

Imagine two goods consumed at a marginal rate of substitution $M R S$ and produced at a marginal rate of transformation $M R T$. We now show that optimality implies $M R S=M R T$. Assume, to the contrary, that the marginal rate of substitution (for a consumer) is lower than the marginal rate of transformation (for a producer):
$M R S=\left|\frac{d x_{2}}{d x_{1}}\right|^{\text {indifference curve }}<\left|\frac{d x_{2}}{d x_{1}}\right|^{\text {transformation curve }}=M R T .$ If the producer reduces the production of good 1 by one unit, he can increase the production of good 2 by $M R T$ units. The consumer has to renounce the one unit of good 1 , and he needs at least $M R S$ units of good 2 to make up for this. By $M R T>M R S$ the additional production of good 2 (come about by producing one unit less of good 1) more than suffices to compensate the consumer. Thus, the inequality of marginal rate of substitution and marginal rate of transformation points to a Pareto-inefficient situation.

## Perfect competition|ECON经济作业代写ECONOMICS代考

4.2. Perfect competition. We want to apply the formula
$$M R S \stackrel{!}{=} M R T$$
to the case of perfect competition. For the output space, we have the condition
$$p \stackrel{!}{=} M C .$$
We have derived “price equals marginal cost” as the profit-maximizing condition on p. 240 and have discussed the welfare-theoretic implications on p. 310 .

## First-degree price discrimination|Econ经济作业代写Economics代考

In the case of a Cournot monopoly, the “price equals marginal cost” rule is violated. However, firstdegree price discrimination fulfills this rule as shown on pp. 306 .

## Cournot monopoly|Econ经济作业代写Economics代考

A trivial violation of Pareto optimality ensues if a single agent acts in a non-optimal fashion. Just consider consumer and producer as a single person. For the Cournot monopolist, the $M R S \stackrel{!}{=} M R T$ formula can be rephrased as the equality between

• the monetary marginal willingness to pay for selling – this is the marginal revenue $M R=\frac{d R}{d y}$ (see above p. 389$)$ – and

## Household optimum|ECON经济作业代写ECONOMICS代考

The exercise helps us understand that the marginal rate of transformation is the price ratio,
$$M R T=\frac{p_{1}}{p_{2}},$$
that we also know under the heading of “marginal opportunity cost”. (Alternatively, consider the transformation function $x_{2}=f\left(x_{1}\right)=\frac{m}{p_{2}}-\frac{p_{1}}{p_{2}} x_{1}$.) Seen this way, $M R S \stackrel{!}{=} M R T$ is nothing but the famous condition for household optimality dervied on pp. 137 .

## External effects and the Coase theorem.|ECON经济作业代写ECONOMICS代考

\begin{tabular}{c|c|c|}
number of steers & marginal profit & marginal crop loss \
\hline 1 & 4 & 1 \
2 & 3 & 2 \
3 & 2 & 3 \
4 & 1 & 4
\end{tabular}
The cattle raiser’s marginal profit from steers is a decreasing function of the number of steers while the marginal crop loss increases. Let us begin with the case where the cattle raiser is liable. He can pay the farmer up to 4 (thousand Euros) for allowing him to have one cattle destroy crop. Since the farmer’s compensating variation is 1 , the two can easily agree on a price of 2 or 3 .

## COURNOT MONOPOLY|ECON经济作业代写ECONOMICS代考

Consider two individuals $A$ and $B$ who consume a private good $x$ (quantities $x^{A}$ and $x^{B}$, respectively) and a public good $G$. We define $M R S^{A}$ and $M R S^{B}$ as the marginal willingness to pay for the public good in terms of the private good. The optimality condition is
\begin{aligned} & M R S^{A}+M R S^{B} \ =&\left|\frac{d x^{A}}{d G}\right|^{\text {indifference curve }}+\left|\frac{d x^{B}}{d G}\right|^{\text {indifference curve }} \ & \stackrel{!}{=}\left|\frac{d\left(x^{A}+x^{B}\right)}{d G}\right|^{\text {transformation curve }}=M R T \end{aligned}
Assume that this condition is not fulfilled. For example, let the marginal rate of transformation be smaller than the sum of the marginal rates of substitution. Then, it is a good idea to produce one additional unit of the public good. The two consumers need to forgo $M R T$ units of the private

## PERFECT COMPETITION|ECON经济作业代写ECONOMICS代考

4.2. 完美的竞争。我们想应用公式

p=!米C.

## COURNOT MONOPOLY|ECON经济作业代写ECONOMICS代考

• 为销售支付的货币边际意愿——这是边际收入米R=dRd和（见上文第 389 页)- 和

## EXTERNAL EFFECTS AND THE COASE THEOREM.|ECON经济作业代写ECONOMICS代考

\begin{tabular}{c|c|c|} 公牛数量 & 边际利润 & 边际作物损失 \ \hline 1 & 4 & 1 \ 2 & 3 & 2 \ 3 & 2 & 3 \ 4 & 1 & 4 \结束{表格}\begin{tabular}{c|c|c|} 公牛数量 & 边际利润 & 边际作物损失 \ \hline 1 & 4 & 1 \ 2 & 3 & 2 \ 3 & 2 & 3 \ 4 & 1 & 4 \结束{表格}

## COURNOT MONOPOLY|ECON经济作业代写ECONOMICS代考

matlab代写请认准UprivateTA™.