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Costs--Where S(P) comes from © 1998,2007, 2010 by Peter Berck

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The Cost Function C(q) Output. Product firm sells Input. Goods and services bought by firm and used to make output. includes: capital, labor, materials, energy C(q) is the least amount of money needed to buy inputs that will produce output q.

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Fixed Costs FC are fixed costs, the costs incurred even if there is no production. FC = C(0). Firm already owns capital and must pay for it Firm has rented space and must pay rent

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Average and Variable Costs VC(q) are variable costs. VC(q) = C(q) - FC. AC(q) is average cost. AC(q) = C(q)/q. AVC is average variable cost. AVC(q) = VC(q)/q. AFC is average fixed cost. AFC(q) = FC/q. limits: AFC(0) infinity and AFC(inf.) is zero.

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AFC(Q) AFC Q

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Marginal Cost MC(q) is marginal cost. It is the cost of making the next unit given that q units have already been produced MC(q) is approximately C(q+1) - C(q). Put the other way, C(q+1) is approximately C(q) + MC(q). The cost of making q+1 units is the cost of making q units plus marginal cost at q.

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C, AC and MC in a Chart Q

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C(Q) = Q 2. A Diagram

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Towards a better definition of MC Per unit cost of an additional small number of units Let t be the number of additional units could be less than 1 MC(q) approximately {C(q+t) - C(q)}/t MC(q) = lim t 0 {C(q+t) - C(q)}/t

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MC: Slope of Tangent Line q q+t C C(q+t)-C(q) t

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MC: Slope of Tangent Line q q+t C

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U Shaped Costs Now let’s assume FC is not zero AC(0) = AVC(0) + AFC(0) is unbounded AC(infinity) = AVC(infinity) + 0 SO AC and AVC get close together with large q. Let’s assume MC (at least eventually) is increasing. Fact: MC crosses AVC and AC at their minimum points

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MC crosses AC at its minimum Whenever AC is increasing, MC is above AC. multiply by q(q+1) and simplify

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AC(q+1)-AC(q)=(1/q+1) MC-AC AC increasing means MC above AC AC decreasing means MC below AC So AC constant means MC = AC

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U Shaped Picture AC AVC MC Q $/unit

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Profit Profit = P q – C(q) – = Revenue - Cost

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Firm’s Output Choice Firm Behavior assumption: Firm’s choose output, q, to maximize their profits. Pure Competition assumption: Firm’s accept the market price as given and don’t believe their individual action will change it.

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Theorem Firm’s either produce nothing or produce a quantity for which MC(q) = p

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Candidates for Optimality 0 a b Profits could be maximal at zero or at a “flat place” like a or b. Thus finding a flat place is not enough to ensure one has found a profit maximum

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Necessary and Sufficient When Profits are maximized at a non zero q, P = MC(q) P = MC(q) is necessary for profit maximization P = MC(q) is not sufficient for profit maximization (Is marijuana use necessary or sufficient for heroin use? Is milk necessary ….)

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Discrete Approx. Algebra Revenue = p q = p q - C(q) is profit We will show (within the limits of discrete approximation) that “flat spots” in the (q) function occur where p = MC(q)

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Making one less unit Now (q*-1) (q*) = { p (q*-1) - c(q*-1)}- { pq* - c(q*) } = -p + [ c(q*) - c(q*-1) ] = - p + mc(q*-1) so -p + mc(q*-1) is the profit lost by making one unit less than q*

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Making one more unit... Now (q*+1) (q*) = { p (q*+1) - c(q*+1)}-[pq* - c(q*)] = p + [ c(q*) - c(q*+1) ] = p - mc(q*) so p - mc(q*) is the profit made by making one more unit

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Profit Max If q* maximizes profits then profits can not go up when one more or one less unit is produced so, (q) must be “flat” at q* No profit from one more: p - mc(q*) 0 No profit from one less: - p + mc(q*-1) 0 p- mc(q*-1) 0 p - mc(q*) since mc increasing, p-mc must = 0 between q*-1 and q* (actually happens at q*, but need calculus to show that)

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q*q SMALL q BIG p MC Picture and Talk P-MC MC-P $/unit

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