Costs--Where S(P) comes from

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## Costs--Where S(P) comes from

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**Costs--Where S(P) comes from**© 1998,2007 by Peter Berck**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.**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**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.**AFC(Q)**AFC Q**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.**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) = limt0{C(q+t) - C(q)}/t**C**q q+t MC: Slope of Tangent Line t C(q+t)-C(q)**C**q q+t MC: Slope of Tangent Line**U Shaped Costs**• Now let’s assume FC is not zero • AC(0) = AVC(0) + AFC(0) is unbounded • AC(infinity) = AVC(infinity) + 0 • Let’s assume MC (at least eventually) is increasing. • Fact: MC crosses AVC and AC at their minimum points**MC crosses AC at its minimum**• Whenever AC is increasing, MC is above AC. multiply by q(q+1) and simplify**U Shaped Picture**AC MC AVC $/unit Q**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.**Theorem**• Firm’s either produce nothing or produce a quantity for which MC(q) = p**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 ….)**Candidates for Optimality**p a 0 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**Discrete Approx. Algebra**• Revenue = p q • p = p q - C(q) is profit • We will show (within the limits of discrete approximation) that “flat spots” in the p(q) function occur where p = MC(q)**Making one less unit**• Now p(q*-1) - p(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***Making one more unit...**• Now p(q*+1) - p(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**Profit Max**• If q* maximizes profits then profits can not go up when one more or one less unit is produced • so, p(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*)**Picture and Talk**MC $/unit MC-P p P-MC q BIG q SMALL q*