If a firm produces many products, a straightforward generalization of the cost function c(w, y) measures the cost of producing the list or vector of outputs y when input prices are w. The production technology displays economies of joint production or economies of scope at y = y1, y2, . . . , ym) if the total cost of producing all the outputs separately is greater than the cost of producing the outputs jointly, that is,

where yj ej = (0, 0, . . . , yj , 0 . . . 0) is the output vector consisting of yj units of good j. Show that the technology displays economies of scope if the cost function is strictly sub modular in y. Assume zero fixed costs.

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