fixed point (x*,y*,p*) with y*^ e Y^., implying that it corresponds to a Walrasian equilibrium.

(c) Given an input level v^, the maximum output level of good i is attained by using v^/J^ units for each of the J^ firms (by 0 < ¡3^ ± 1), yielding the aggregate production level

Hence the aggregate production set of sector i is given by this aggregate production function f^^- It thus exhibits increasing, constant, and decreasing returns to scale, respectively if and only if p. > i, p^ +

= 1, and p^ < i. While there exists an equilibrium in the cases of constant and decreasing returns to scale, there is no equilibrium in the case of increasing returns to scale, because unboundedly large profits can be made at any (positive) output prices.

(d) If the externally of sector t is internalized and the aggregate production set exhibits increasing returns to scale {that is, p^ + > 1), then the sector can attain an arbitrarily large profit. Hence a Walrasian equilibrium does not exist.

(e) We assume that p^ > 0. Then, by - 1, the individual production function exhibits constant returns to scale and the aggregate production function exhibits increasing returns to scale. Because of the quasilinearity, there exists a normative representative consumer (Example 4.D.2) and his (direct) utility function (Exercise 4.D.4) is also quasilinear. Let it be uiXj.) + x^. We assume that u( *) is differentiable and strictly concave. Let v* > 0 be the equilibrium level of labor input and v** > 0 be the socially optimal level of labor input. They can be graphically illustrated as follows. Note that v* < v**.

Figure 17.BB.8

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