notes on returns to scale and economies of scale

# notes on returns to scale and economies of scale - q...

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Note about the relation between returns to scale and economies of scale Prof. Watson, Winter 2011 Suppose that, for a ﬁrm, the relation between output q and labor and capital inputs L and K is given by q = f ( L,K ). For any two output levels q and q , denote by L and K the optimal (cost minimizing) input amounts to produce q , and let L and K be the optimal (cost minimizing) input amounts to produce q . Let c be the cost function derived from f for given input prices. Claim: If f exhibits increasing returns to scale then c exhibits economies of scale. Proof: Consider any q and q such that q > q and let λ q/q . By increasing returns to scale, we know that f ( λL ,λK ) > λf ( L ,K ) = q . That is, scaling up the inputs by λ causes output to exceed q . Since this higher output level is produced at cost λc ( q ), we know that q can be produced for less than this amount. Thus, c ( q ) < λc ( q ). Using the deﬁnition of λ , this becomes c ( q ) < ( q/q ) c ( q ), which simpliﬁes to c ( q ) / q < c ( q ) /q . This condition means that c exhibits economies of scale (average costs decrease as
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Unformatted text preview: q increases). Claim: If f exhibits decreasing returns to scale then c exhibits diseconomies of scale. Proof: Consider any q and q such that q > q and let λ ≡ q/q . By decreasing returns to scale, we know that f ( L, K ) < λf ( L/λ, K/λ ). The left side of this inequality is q . Dividing both sides by λ and simplifying, this inequality becomes q < f ( L/λ, K/λ ), which means that input levels L/λ and K/λ are more than enough to produce q . As a result, the minimal cost of producing q is strictly less than ( w L + r K ) λ , which itself is c ( q ) /λ . So we have that c ( q ) < c ( q ) /λ . Using the deﬁnition of λ , this becomes c ( q ) < ( q / q ) c ( q ), which simpliﬁes to c ( q ) /q < c ( q ) / q . This condition means that c exhibits diseconomies of scale (average costs increase as q increases). 1...
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## This note was uploaded on 11/30/2011 for the course ECON 311 taught by Professor Zambrano during the Fall '08 term at Cal Poly.

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