经济代写|微观经济学代写Microeconomics代考|ECON1001

经济代写|微观经济学代写Microeconomics代考|From Short-Period to Long-Period Supply

The short-period $A C$ and $M C$ curves were derived for a given fixed plant and given fixed cost and rentals of inputs. We can now extend the analysis to the long period, i.e. to the analytical situation where we treat all inputs as variable, by imagining that the price-taking firm can choose among a set of different fixed plants, each one entailing a certain fixed cost; input rentals are still given; for each fixed plant the firm can derive the $A C$ and $M C$ curves which have the shapes of – Fig. 5.6. We need not assume perfect divisibility of the elements which go to form a fixed plant. For each level of $\mathrm{q}$, there will be a fixed plant which allows producing it at the lowest $A C$. The locus of these lowest long-period $A C$ ‘s as functions of $\mathrm{q}$ is the long-period average cost curve, $L A C$. The smallest of the minimum average costs associated with the several alternative fixed plants is the long-period minimum average cost, $\operatorname{Min} L A C$; indicate the associated output as $q^*$ (৫ Fig. 5.7).

The long-period average cost curve LAC is the lower envelope of the (short-period) average cost curves associated with different fixed plants, cf. – Fig. 5.8. Along the LAC curve the type of fixed plant changes. If the different alternative fixed plants are so numerous and their short-period average cost curves change so gradually as to determine a differentiable lower envelope as in – Fig. $5.8$ (traditionally this was obtained by conceiving the possible short-period fixed plants to be a continuum, corresponding to the ‘quantity of capital’ embodied in each plant, where capital was the single factor of variable ‘form’ already encountered when discussing the differentiability of production functions), then each point of the LAC curve is tangent to one short-period average cost curve, but not at the point of minimum $\mathrm{AC}$ of that short-period average cost curve, except when the MinLAC is reached. The graphical representation shows this fact clearly enough, so I omit the mathematical proof.

经济代写|微观经济学代写Microeconomics代考|The Product Exhaustion Theorem with U-Shaped LAC

In the previous section we admitted that firms might be able to reach $\operatorname{MinLAC}$ at several levels of output, or at a single one. Disagreements among economists on this issue were mentioned in Sect. $5.6 .3$ and Fn. 19, with some, e.g. Edith Penrose, arguing that in many instances firms are able to grow to enormous sizes without any increase in average cost and therefore the limits to size must be found either on the demand side or on the need for own capital or collateral; and others arguing that the general case is U-shaped $L A C$ curves because of control difficulties of management over subordinates that increase with size. In the face of the empirical evidence of more and more mammoth firms and increasing concentration in many industries, it would seem Penrose got it right.

Assuming now that maximum output is a continuous function of total cost and that the LAC curve is $\mathrm{U}$-shaped, returns to cost are increasing where $L A C$ is decreasing in $q$ and are decreasing where $L A C$ is increasing in $q$; hence $\operatorname{Min} L A C$ is reached where the returns to costs, in passing from locally increasing to locally decreasing, are locally constant. If the production function is differentiable with respect to all inputs, ${ }^{30}$ then the locally constant returns to costs at $\operatorname{Min} L A C$, that is, the equality of average and marginal cost, imply locally constant technical returns to scale. I prove this for the two-factors case. At the point of minimum average cost it is
$$M C=\frac{v_1}{M P_1}=\frac{v_2}{M P_2}=A C=\frac{v_1 x_1+v_2 x_2}{f\left(x_1, x_2\right)} ;$$
this can be rewritten $$f\left(x_1, x_2\right)=\frac{v_1 x_1+v_2 x_2}{M C}=\frac{v_1 x_1}{\frac{v_1}{M P_1}}+\frac{v_2 x_2}{\frac{v_2}{M P_2}}=M P_1 x_1+M P_2 x_2,$$
which implies that the production function is locally homogeneous of degree 1 .
This implies that, if $p=\operatorname{Min} L A C$, the payment to each factor of its marginal revenue product exhausts revenue. Thus the fact that the long-period cost curve is U-shaped entails no contradiction between assuming zero profits of competitive firms in long-period equilibrium, and assuming that each factor is paid its marginal revenue product, because in long-period equilibrium the firm produces where returns to scale are locally constant.

经济代写|微观经济学代写Microeconomics代考|The Product Exhaustion Theorem with U-Shaped LAC

$$M C=\frac{v_1}{M P_1}=\frac{v_2}{M P_2}=A C=\frac{v_1 x_1+v_2 x_2}{f\left(x_1, x_2\right)}$$

$$f\left(x_1, x_2\right)=\frac{v_1 x_1+v_2 x_2}{M C}=\frac{v_1 x_1}{\frac{v_1}{M P_1}}+\frac{v_2 x_2}{\frac{v_2}{M P_2}}=M P_1 x_1+M P_2 x_2,$$

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