Producer Theory (weeks 1-3)

Producer Theory (weeks 1-3) - Sole Propietorships Owned/run...

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Unformatted text preview: Sole Propietorships Owned/run by single individual Owner may manage firm Partnerships Jointly owned and controlled by two or more people. Operate under partnership agreement If any partner leaves, agreement dissolves Unlimited Liability Corporation Owned by shareholders in proportion to ownership of shares Elect board of directors to run firm Board of directors hire manager Limited Liability Maximize profits =R-C Socially / environmentally / ethically conscious firms: Maximize profits Subject to certain constraints (limit pollution, etc.) View of firms as black box that transforms inputs into output Ignore questions of information / organization / coordination These agency issues are at the core of econ / finance electives [Perloff, Ch. 6, p. 172] The production function specifies the maximum amount of an output that can be produced from a particular combination of inputs. q = f(L,K) where q units of output are produced using L units of labor and K units of capital. Efficient production is necessary for profit maximization, but not sufficient. What is L? Unskilled labor Skilled labor Management May often wish to separate these, but to keep things simple, will group together What is K? Long-lived inputs Land Buildings Equipment In long run, all inputs can be adjusted “In the long run, we’re all dead.” – Keynes In short run, only one input can be adjusted Typically easier to adjust labor rather than capital Cheaper and easier to hire and fire than to buy / sell new building Will look at case where capital is fixed at K Examples: 1. q = min{ L, K } 2. q = L + K 3. q = ALaKb (Cobb-Douglas) a) q = L1/2K1/2 (A = , a = , b = ) b) q = L c) q = K Suppose q = L1/2K1/2 If K = 4, q = To produce quantity, q, LSR(q,4) = ? Suppose q = ALaKb LSR (q, K) = ? Suppose q = L + K If K = 10, q = LSR (q, 10) = Suppose q = min { L, K} If K = 10, q = LSR (q, 10) = MPL = ∂f ( L, K ) ∂L is the change in total output resulting from using an extra unit of labor, holding all other factors fixed. e.g., in discrete setting, L 1 2 3 4 q 3 6 8 9 MPL 3 3 2 1 q = L1/2K1/2; suppose K = 1. MPL APL 3.5 3 2.5 Output 2 1.5 1 0.5 0 0 1 2 3 4 5 Labor Total Product 6 7 8 9 10 3.5 3 2.5 Output 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 6 7 8 9 10 Labor Total Product 3.5 3 2.5 Output 2 1.5 1 0.5 0 0 1 2 3 4 5 Labor Total Product Average Product 3.5 3 2.5 Output 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 Labor Total Product Average Product Marginal Product q = L + 12L2 – L3 MPL APL 10 300 250 Output 200 150 100 50 0 0 1 2 3 4 5 6 7 8 5 6 7 8 Labor Total Product 60 50 Output 40 30 20 10 0 0 1 2 3 4 Labor Average Product Marginal Product If MPL > APL, then APL is rising If MPL < APL, then APL is falling If MPL = APL, then APL is staying constant Simple idea: Michael Jordan had an average of 30 points per game If he scores 35 points in the next game, increases average If he scores 25 points in the next game, decreases average If he scores 30 points in the next game, average remains same Works with your GPA In quarter where GPA is higher than earlier average, pulls up GPA In quarter where GPA is lower than earlier average, pulls down GPA In quarter where GPA is same as earlier average, no change. The Law of Diminishing Marginal Returns states: if a firm keeps increasing an input, holding all other inputs and technology fixed corresponding increases in output will become smaller eventually. MPL is decreasing in L ∂MPL <0 ∂L Now suppose that K and L are variable Provides greater flexibility Production function q = f(L, K) e.g., q = L0.5 K0.5 Specifies possible combinations of inputs that will produce a given level of output q = f(L,K) Suppose q = L0.5K0.5 = 1; what are pts on the isoquant? L = 1, K = ? L = 2, K = ? L=½,K=? L = ?, K = 3 L = ? , K = 1/4 Similar concept to indifference curves Indifference curves hold utility constant Isoquants hold production constant Farther isoquant is from origin, greater level of production Isoquants do not cross Isoquants slope downwards Isoquants are thin lines q = min {L, K} q = min { L, K } At efficient point, L= K= So slope of efficient line is / K α L β K q= K+ L L Does this technology have diminishing marginal returns? Slope of isoquant indicates how one input can be substituted for another MRTS indicates while holding output constant how many units of capital can be substituted by additional unit of labor Intuition Suppose MPL = 1, MPK = 1/4, and you were producing 6 units If you hire additional worker, Increases production by MPL = 1 So output goes up by 1 Have to decrease capital to keep output same Decrease one unit of capital, output goes down by MPK = 1 So have to decrease _ units of capital dK = − MPL dL dK MPL MRTS = =− dL MPK MPK K q= K+ L=1 L Can calculate slope using standard ways K q= K+ L=1 L MPK = , MPL = MRTS = - / q = ALa Kb MPL = MPK = MRTS = Typically assume that |MRTS| is diminishing i.e., ∂ | MRTS |< 0 ∂L This is equivalent to isoquants being convex Sufficient condition: MPL is decreasing in L and MPK is increasing in L MP tells us what happens when one input is increased holding others fixed What if all inputs are increased in a proportional way? E.g., if you double the # of inputs, do you increase output by more than 2? (Increasing Returns to Scale) increase output by exactly 2? (Constant Returns to Scale) increase output by less than 2? (Decreasing Returns to Scale) Suppose >1 Decreasing Returns to Scale: f( L, K) < f(L,K) Increasing Returns to Scale: f( L, K) > f(L,K) Constant Returns to Scale: f( L, K) > f(L,K) Suppose f( L, K) = x f(L,K), for all > 1. a) When would f have CRS? b) When would f have Decreasing RS? c) When would f have Incr. RS? Theory of Production Isoquants: Combination of inputs necessary to make a fixed level of output Slope of isoquant = Assume Diminishing MRTS Economies of Scale Explicit Costs Opportunity Costs Direct out-of-pocket payments for inputs Value of best alternative use of resource Wages, Salaries If input is owned, opportunity cost can be measured by price at which input could be sold Cost of capital, depreciation Total Cost = Fixed Cost + Variable Cost Fixed Cost Cost of inputs whose use does not vary with output level Price of land, large machinery Variable Cost Production expense that varies with output Cost of labor and materials Example: Taxicab service JetBlue: New planes, leather seats, live satellite TV Bring “humanity back to air travel”; instead of packing them like sardines. Costs Variable Costs: Labor: Flight attendants, ground crew, ticket counter staff, pilots Jet fuel Fixed Costs: 10 year lease with Port Authority of New York for Terminal 6 at JFK Salary contracts for CEO and managers Cost of planes (unless planes are leased with opt-out clauses) Assume capital is fixed C(q) = VC(q) + F Marginal cost (MC) is cost change if firm produces one more unit of output Average cost (AC) is total cost per unit Average variable cost (AVC) is variable cost per unit Average fixed cost (AFC) is fixed cost per unit F = 100, VC(q) = q2 + 10q MC: AVC: AFC: AC: F = 450, VC(q) = 100q -4q2 + 0.2q3 MC: AVC: AFC: AC: Cost of using L units of labor = wL Suppose capital is fixed at K. VC(q) = wL(q,K) MC(q) = Cost of using L units of labor = wL Suppose capital is fixed at K. VC(q) = wL(q,K) MC(q) = It takes 5 hours of prep time to prepare one more lecture, so MP of 1 hour of labor = 1/5 Wage = $10 per hour MC of one more lecture = $10 / (1/5) = $ 50 V (q ) wL AVC (q) = = q q Suppose MPL is strictly decreasing in L (Diminishing Marginal Returns). Then APL is strictly decreasing in L. Then MC and AVC are strictly increasing in q, but AC may still decr. in q. Short Run: Capital is fixed at K MC(q) = w / MPL AVC(q) = wL / q = w/APL Long Run Capital and Labor are flexible Develop concept of isocost line Similar to budget line from consumer theory Tangency of indiff curve and budget line = optimal for consumer Tangency of isoquant and isocost line = optimal for producer In long-run, all costs are adjustable Fixed cost of land: can sell off some land Can fire managers / CEO Can sell planes As before, firms have variety of input choices Firm will determine optimal mix based on price of inputs Useful analytical tool: Isocost line Specifies total cost of using combination of inputs Similar to budget curve from consumer theory r is rental rate of capital Even if firm owns capital, can think about implicit rental rate Slope of isocost curve = - w / r Slope of Isoquant = Slope of isocost line But what do we need for an interior solution? q = L1/2K1/2, w = 4, r = 2 q = L + K, w = 4, r = 2 q = min{L,K}, w = 4, r = 2 MinL,K wL + rK subject to q = f(L,K) Langrangian Approach MinL,K wL + rK subject to q = f(L,K) Langrangian Approach MinL,K wL + rK subject to q = f(L,K) Alternative Approach MinK wL(q,K) + rK FOC: Talked about minimizing cost given a fixed output How about maximizing output given a cost? MaxL,K f(L,K) subject to wL + rK C MaxL,K f(L,K) subject to wL + rK C Suppose f(L,K) = (L2 + K2)1/2, w = 1, r = 2 Could look at tangency, but this will give exactly the wrong answer. AC is U-shaped as one increases output AC is downward sloping initially because AC is upward sloping eventually because Hint: MC is going up because AC in long-run different from AC in short-run No fixed costs in long-run Not diminishing marginal returns since both inputs can be increased Action is from economies of scale C(q) has economies of scale if AC(q) decreases as q increases C(q) has no economies of scale if AC(q) is constant as q increases C(q) has diseconomies of scale if AC(q) increases as q increases Returns to Scale Economies of Scale Returns to scale is about production, economies of scale about costs However, Returns to scale equivalent to economies of scale (book is incorrect) Increasing returns to scale Economies of Scale Constant returns to scale No Economies of Scale Decreasing returns to scale Diseconomies of Scale Factors that drive scale effects Claim: If f(L,K) is CRS, then for all q, C(q) = qC(1) MC(q) = AC(q) = C(1) Pf: Argue by contradiction: suppose for some q’>1 1. 2. C(q’) > q’C(1): Will lead to contradiction when scaling up from 1 to q’ C(q’) < q’C(1): Will lead to contradiction when scaling down from q’ to 1 Practice makes perfect ...
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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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