# 经济代写|国际经济学代写International Economics代考|EC3016

## 经济代写|国际经济学代写International Economics代考|Returns to Scale and Convexity of the Production Set

Shape of PPF depends on how does the opportunity cost or MRT change when production of cotton textiles is increased and correspondingly the production of computers is decreased. If the opportunity cost or MRT increases, PPF is strictly concave. Opportunity cost is essentially the ratio of social marginal costs, and in absence of any production externality private and social marginal costs are the same. Thus a change in the ratio of private marginal costs, which is determined by both the returns to scale and the law of diminishing marginal productivities, governs the shape of PPF.

To explain, suppose at an initial relative price $p_o$, the economy was producing $T_o$ units of textiles and $C_o$ units of computers. Suppose also that at the initial wage-rental ratio, computers were produced by a relatively capital-intensive technique of production and textiles were produced by a relatively labour-intensive technique of production. Now consider an additional unit of production of textiles. With all resources initially fully employed, this requires a corresponding reduction in the production of computers. At the initial wage-rental ratio, producers of textiles will maintain the same technique of production while expanding their production. There will thus be a scale expansion in the textiles sector, that is, use of labour and additional capital will be increased proportionately. For similar reasons, a scale contraction will take place in the computer sector. Corresponding changes in marginal costs of production in the two sectors will therefore be governed by the returns to scale. If both the goods are produced under constant returns to scale (CRS) technology, the marginal cost of production, at initial factor prices, should remain constant in both the sectors. Hence, at initial factor prices, an expansion in the production of textiles and a corresponding contraction in the production of computers along PPF will not change the opportunity cost or MRT. If the goods are, however, produced under IRS, then the marginal cost of production should decline in the expanding textiles sector and should rise in the contracting computer sector. Note that IRS technology means that an additional unit of production of textiles requires less than one unit of additional labour and capital, and this is why average and marginal costs decline. Thus, under IRS, MRT declines at initial factor prices. Similarly, if the goods are produced by decreasing returns to scale technology, MRT at initial factor prices rises as textiles production expands.

## 经济代写|国际经济学代写International Economics代考|MEASURING THE WELFARE CHANGE

Gains from trade (GFT) is a theorem regarding pre- and post-trade welfare levels. But social welfare is not always measurable, particularly because of its ordinal utility dimension. The duality approach, however, provides us a more convenient way of measuring GFT in terms of the expenditure function. The expenditure function, which indicates the minimum expenditure to attain a particular level of welfare or utility, can be used to measure change in welfare or utility in monetary units. Two concepts are particularly relevant here. One is the compensating variation (CV) and the other is the equivalent variation (EV).

Compensating variation (CV) requires the amount of money that is to be given to a consumer to fully offset her for the loss from a price rise, that is, to allow her to remain on the same indifference curve as before the price rise. Equivalent variation (EV), on the other hand, considers what fall in money income would hurt a buyer as much as the price increase does. Figure $3.1$ illustrates these two measures for a typical consumer’s choice problem. The representative buyer allocates her given money income $M_o$ on two goods- $X$ and $Y$-and does not save. Her preference ordering satisfies all the standard axioms and assumptions, which can be represented by smooth, downward sloping, and a strictly convex set of indifference curves, with successively higher indifference curves indicating successively higher utility indices. Given a budget set $\left{M_o, P_X^o, P_Y^o\right}$, the consumer’s most preferred bundle is indicated by $C_o$ on the budget line $A B_{\mathrm{o}}$. An increase in price of good $X$, ceteris paribus, rotates her budget line to $A B_1$, and changes the most preferred bundle to $C_1$. Her utility falls to $U_1$.

Under compensating variation, the consumer must be given a money income that pushes her budget line to the one labeled $\mathrm{CV}$ to allow her to consume the bundle $C_O^*$ on the earlier higher indifference curve $U_0$. Let $e\left(p_1, U_0\right)$ and $e\left(p_1, U_1\right)$ denote respectively minimum expenditures required to attain utility level $U_n$ and $U_1$ at the new price $p_1=\left{P_X^1, P_Y^o\right}$. Hence, the CV measure of welfare change of this price increase is the difference $\left[e\left(p_1, U_0\right)-\right.$ $\left.e\left(p_1, U_1\right)\right]$

On the other hand, for measuring the welfare change by EV, we allow the consumer to buy goods at the initial prices but hypothetically tax the consumer so that her budget line shifts down to the one labeled $\mathrm{EV}$ to enable her to purchase the bundle $C_1^*$. This amount of money income that is taxed away, $\left[e\left(p_0, U_0\right)-e\left(p_0, U_1\right)\right]$, is the EV measure of the change in welfare following the price rise.

In the next two sections, we use these alternative monetary measures of welfare change to measure or estimate GFT.

## 经济代写|国际经济学代写International Economics代考|Returns to Scale and Convexity of the Production Set

PPF的形状取决于当棉纺织品的产量增加而相应的计算机产量减少时机会成本或MRT如何变化。如果机会成本或 MRT 增加，则 PPF 是严格凹的。机会成本本质上是社会边际成本的比率，在没有任何生产外部性的情况下，私人和社会边际成本是相同的。因此，由规模报酬和边际生产率递减规律决定的私人边际成本比率的变化决定了 PPF 的形状。

## 经济代写|国际经济学代写International Economics代考|MEASURING THE WELFARE CHANGE

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