# Production Function in the Long-Run. Business Economics Theory of the Firm II Production and Cost in the Long Run. Description of Technology

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1 Business Economics Theory of the Firm II Production and Cost in the ong Run Two or more variable input factors Thomas & Maurice, Chapter 9 Herbert Stocker Institute of International Studies University of Ramkhamhaeng & Department of Economics University of Innsbruck Production Function in the ong-run Production function: Again, describes the maximum output that can be obtained with any combination of inputs, given a specific technology, but now (at least) two inputs are variable Q = Q(,) Different combinations of inputs can produce the same output Input substitution: One input can be substituted for another ong Run Production Function Input Substitution: abor-intensive method: process that uses large amounts of labor relative to other inputs. Capital-intensive method: process that uses large amounts of capital equipment relative to other inputs. Description of Technology The Production Function again describes the maximum output that can be obtained with any combination of inputs. This can be shown as a table or more simply as a function Q = Q(,)

2 Isoquants Cobb-Douglas Production Function Input substitution can easily be modeled and illustrated with isoquants. Isoquants: show all combinations of inputs that produce a constant level of output. Isoquants (constant output) correspond to indifference curves (constant utility) in the theory of the household. They are just like contour lines on a map! 1 y x1 1 x 1 Output (Q) U 1 Product curves for Productionfunction Q1 Q1abor () 1 Isoquants 1 Q Capital () Isoquants Isoquants and Factor Substitution higher output Q 1 y x1 1 x 1 USA Africa USA Africa The same output (e.g. 1 km road within one year) can be produced with different factor intensities.

3 Technology Well-behaved Technologies: Monotonic: more inputs produce more output. Convex: sometimes averages produce more than extremes. We can t take monotonic transformations (like with utility functions) any more! Properties of Technology 1 Marginal Product Marginal Rate of Substitution 3 Returns to Scale Marginal Product Marginal Product: (MP1) MP1 is how much extra output you get from increasing the input of factor 1, holding factor fixed. MP Q, Q MP Diminishing marginal product: more and more of a single input produces more output, but at a decreasing rate aw of diminishing returns Important property of almost all technologies! Marginal Rate of Technical Substitution The Marginal Rate of Technical Substitution (MRTS) is a measure for how easily input factors can be substituted, holding output constant. The MRTS is the slope of an isoquant MRTS = The minus sign is added to make MRTS a positive number, since /, the slope of an isoquant, is negative. it shows how many units of are necessary to replace one unit of when output is kept constant.

4 Marginal Rate of Technical Substitution The MRTS can also be expressed as the ratio of two marginal products: Why? MRTS = = MP MP 1 Output is kept constant along an isoquant, therefore Q = Total derivative of the production function yields 3 Combining 1 and gives Q = MP +MP Q = = MP +MP = MP MP Diminishing MRTS Property of MRTS Isoquant becomes flatter, the further we move along an axis MRTS decreases the more intensively a factor is used in production: As one input is substituted for another one, the marginal product of this input diminishes Why? Notice: Difference between diminishing MP and diminishing MRTS MRTS Special Cases Diskrete changes: Infinitesimal changes: small Fixed-Proportions: inputs cannot be substituted; e.g. Q = min{,} MRTS is zero except at the edge, where it is not defined. 1 Q 1 1 Attention: not differentiable!

5 Special Cases Perfect Substitutes: inputs can perfectly be substituted; e.g. Q = + MRTS is constant! Returns to Scale Returns to Scale show what happens with output, if all inputs are doubled.? Q[(),()] Q 1 Q 1 1 Attention: Corner solution! Notice difference to Marginal Product (MP): shows what happens with output if only one factor of production increases while all other factors are held fixed. Returns to Scale Constant Returns to Scale: if all inputs are doubled output doubles too, i.e. Q[(),()] = Q Decreasing Returns to Scale: if all inputs are doubled output less than doubles, i.e. Q[(),()] < Q Increasing Returns to Scale: if all inputs are doubled output more than doubles, i.e. Q[(),()] > Q Returns to Scale Constant Returns to Scale: Q =,, For = 5 and = 5: 1 Q 1 1 Isoquants for Q = 1,,3,... Q(5,5) = 5, 5, = 5 For = 1 and = 1: Q(1,1) = 1, 1, = 1

6 Returns to Scale Decreasing: Q =,, Increasing: Q =,, 1 1 Q 1 Q Modeling decision making of firms: Optimal Factor Allocation Q( 5, 5) = 5, 5,, Q(1,1) = 1, 1, Q( 5, 5) = 5, 5,,9 Q(1,1) = 1, 1, 1 Isoquants for Q = 1,,3,... Theory of the Firm Profit is the difference between total revenue and total (opportunity) cost Total revenue is determined by the output, which is sold on markets Optimal behavior of firms depends on the market structure where output is sold A necessary condition for maximum profit is that cost are minimal Minimum cost can be derived without knowing the market structure! Theory of the Firm Therefore, a firm s decision making process can be modeled by artificially splitting it up into two steps: 1 Minimize cost for any output, given technological restrictions (production function) Cost function Maximize profits for a given market structure (market restrictions) Supply function Notice: Maximizing output for a given amount of cost delivers the same result as cost minimization! E.g., nonprofit firms (fixed budget)

7 PA PB QB QA Different Perspectives of a Firm Firms Choice Ma m & Pa: Human resources manager: Cost C = w+r Technological restrictions Accountant: Economist: P MC AC Optimization (Minimization) AVC B A AFC Decisions (Factor Demand & Cost Function) Q Cost Minimization Objective: Minimize cost Isocost Curves C = w+r Constraint: Technological Restriction Production function Q = Q(,) Cost Firms have to pay for their inputs. Total costs are defined as the sum of cost for all inputs. For simplification we restrict ourselves to two inputs only. Solving for Isocostcurve: = C r w r C = w+r each point on this curve is associated with the same cost!

8 Isocostcurve C r C = w+r = C r w r = w r Isocostcurve: = C r w r Slope: d d = w r Intercept: for = = = C r Cost Minimization C r C r Graphical Solution d d = w r Q C = w+r = C r w r What is the cheapest way to produce the given quantity Q? where the intercept is minimal and production of Q still possible! That s where the isocostline and the isoquant have the same slope! Optimal Factor Input: Example Consider a market with w = 1 Euro per working hour (wh) and r = 5 Euro per machine hour (mh) 1 Euro w r = wh 5 Euro mh = mh wh In the market, machine hours cost the same as one working hour Further, we assume that a firm produces with MRTS = = 3 In production, 3 machine hours can be replaced by one working hour without changing output Optimal Factor Input: Example We obtain w r = mh wh < 3 mh wh = When the firm uses 3 mh less she saves 15 Euro. Using one additional unit of labor (wh) leaves the output unchanged, but this wh costs only 1 Euro, therefore the firm could have saved 5 Euro! In such a situation, the firm would demand less capital (mh) and more labor (wh)!

9 Optimal Factor Input Cost Minimization with agrange This arbitrage works until w r = MRTS (1) Interpretation: Factors are substituted until the exchange ratio in production (= MRTS) is equal to the exchange ratio in the market (= relative factor prices)! Eq.(1) defines the optimal factor input ratio Analytical solution with the agrange method: Objective: min, C = w+r (Technological) Restriction Q = Q(,) Decision variables:, (Market restriction: perfect competition prices are given!) Cost Minimization Optimization Cobb-Douglas production function and isocostcurve Output (Q) Capital () Production function 1 x MRTS = d = d w r A abor () x1 Isocost wall Isoquants 1 1 Cost Minimization: agrange Problem: min, agrange Function : C = w+r s.t.: Q = Q(,) = w+r +λ[q Q(,)]

10 Cost Minimization: agrange Setting partial derivatives equal to zero: = w λ Q! = = r λ Q! = λ = Q Q(,) =! The solutions of these equations = (w,r,q), = (w,r,q) define the factor demand functions!. Cost Minimization: agrange Dividing the first two equations yields Re-writting gives Q w r = Q MP MP MP w = MP r Efficient production: Marginal products of labor and capital per price unit are identical! Cost Minimization Cost Minimization On a isoquante output is constant, therefore the change in output (dq) is zero dq = = Q Q d+ d rewritting d Q d = Q = MP MP (total derivative) Therefore w r = MP = d MP d = MRTS

11 Expansion Path Short- and ong-run Expansion Path Optimality requires: Slope: d d Slope: w r MRTS = factor price ratio dq= dc= Expansion path: Each point represents an efficient (least-cost) input combination for each level of output at constant input prices Curve along which a firm expands output in the long-run Indicates how input usage changes when output or cost changes Expansionpath long run Expansionpath short run Expansionpath Q Q1 Assume the firm wants to expand production from Q1 to Q. In the short run the firm can only get additional labor to expand production. In the long run she can substitute labor for capital to reduce cost! Restructuring Short-Run Costs Factor Demand Because managers have greatest flexibility to choose inputs in the long run, costs are lower in the long run than in the short run for all output levels except that for which the fixed input is at its optimal level Short-run costs can be reduced by adjusting fixed inputs to their optimal long-run levels when the opportunity arises To derive the factor demand functions (the function that shows how demand for an input depends on the prices of inputs and on output) we have to consider input substitution possibilities, e.g. = (w,r,q) Input substitution depends mainly on Technology, i.e. the shape of the isoquant. Prices of Inputs. The following graph shows how this analysis can be applied to derive the factor demand equation.

12 Factor Demand w Q = 1 = w,5 r,5 Q Factor Demand: min, C = w+r s.t.: Q = Q(,) = MRTS: (assumption: r = 1) d d =! = w r ösung: Q = = [(w/r)] = w,5 r,5 Q = ; = 1; =,5; =,1 1 ong-run Cost Functions The cheapest way to produce a given Output ong-run Cost Functions: Recap... Cost minimization problem: min, C = w+r s.t. Q = Q(,) Geometric solution: slope of isoquant equals slope of isocost curve w/r = MP1/MP These optimal choices of factors are the conditional factor demand functions: = (w,r,q) = (w,r,q) ong-run Cost Functions Inserting the conditional factor demand functions in the equation of the isocostcurve gives the ong-run Cost Function (TC): C = w (w,r,q)+r (w,r,q) = C(w,r,Q) The ong-run Cost Function shows for given factor prices the cheapest possibility to produce any given output. Its shape is exclusively determined by the technology and input prices!

13 ong-run Average and Marginal Cost ong-run average total cost AC(Q;w,r) = TC Q Cost of a typical output unit ong-run marginal cost MC(Q;w,r) = TC Q Cost of an additional output unit As in the short-run, the MC-curve intersects the AC-curve in it s minimum (Dis-)Economies of Scale Properties of cost functions Economies of scale: ong-run average total cost (AC) decline as output increases Happens when returns to scale are increasing Diseconomies of scale: ong-run average total cost (AC) increase as output increases Happens when returns to scale are decreasing Why? (Dis-)Economies of Scale Economies of scale and returns to scale Doubling all inputs also doubles TC However: If returns to scale are increasing (decreasing), output more (less) than doubles AC = TC/Q decrease (increase)! Constant returns to scale are associated with constant economies of scale, i.e., the AC-curve is horizontal Be careful: Economies of scale applies to the cost function, returns to scale to the production function AC-Curve and Economies of Scale AC Economies of Scale MES (Approx.) Constant returns to scale Diseconomies of Scale AC Minimum Efficient Scale (MES): Point where economies of scale are exhausted Q

14 Occurrence of (Dis-)Economies of Scale Economies of scale: Expansion of scale usually allows... to exploit advantages from specialization and division labor, to reduce the cost of acquiring and establishing machines, and to introduce automation devices Diseconomies of scale Information asymmetries between management and employees typically increase with firm size Cost (and unit cost) of management increases Market Entry Barriers AC 1 MES MES High market entry barriers MES Q AC 1 MES ow market entry barriers Rule of thumb: Average cost at half of the MES indicates technological market entry barriers Important for intensity of competition, M&A... MES MES Q Economies to Scope Economies to Scope: Exist for a multi-product firm when producing two or more goods jointly is less expensive than producing them separately: C(QX)+C(QY) > C(QX,QY) Reason: Inputs can be jointly used to produce goods Overheads (e.g., R&D, marketing) Production of one good results in the production of another one with little or no extra cost (e.g., beef and leather) Example: Short- medium- and long run cost curves

15 Short- vs. long run Example in the short run only few factors are variable, e.g. labor. in the medium run more factors are variable, e.g. capital. in the long run all factors are variable, e.g. buildings. Example with Cobb Douglas Production function: Q = G Output: Q; Inputs:, and G. Short-run: only is variable, and G are fixed, Medumium-run: and are variable, G is still fixed, ong-run:, and G are variable. Example Short-run Cost Function Q =.5.5 G.5 Returns to scale: Short-run: (t).5.5ḡ.5 = t.5 Q decreasing returns to scale in Medium-run: (t).5 (t).5ḡ.5 = t.75 Q decreasing returns to scale in and ong-run: (t).5 (t).5 (tg).5 = t 1 Q constant returns to scale in, and G min w+r +mḡ Ḡ 1 s.t. Q = 1 1 ( ) Inserting Q = in the equation of the 1 Ḡ 1 isocostcurve: C k = w +r +mḡ = w Q } {{ Ḡ +r } +mḡ {{ } } Fixed cost Variable C.

16 Short-run MC and AC Medium-run Cost Curves AC MC MC k 1 AC k 1 ( = 5) AC k ( = 1) MC k MC k 3 MC k AC k 3 ( = 15) AC k ( = ) Q In the medium-run the firm can optimize the input of and. min, w+r +mḡ s.t. Q = 1 1 Ḡ 1 this can be solved e.g. by the angrange method: [ ] m = w+r +mḡ +λ Q 1 1 Ḡ 1 Medium-run Cost Curves Solution: conditional factor demands: ( ) = 3 w 3 r 3 Ḡ 1 3 Q 3 ( ) = w 3 r 1 3 Ḡ 1 3 Q 3 Inserting these in the isocostequation gives the following medium-run cost function: C m = w ( ) +r +mḡ 3 = w 3 r 3 Ḡ 1 3 Q 3 +m Ḡ Short- and medium-run AC AC MC 15 AC k 1 ( = 5) AC k ( = 1) AC k 3 ( = 15) AC k ( = ) 1 medium-run AC m 5 (optimal &, G = 5) Q

17 Short- and medium-run AC AC AC k 1 AC k AC k 3 AC k AC m 1 (G = 5) Medium-run average cost funcion for G = 5 (scales changed!) Q Medium-run AC and MC MC Different firm sizes MC m 1 MC m MC m 3 AC m 1 (G = 5) AC m (G = 1) AC m 3 (G = 15) Q ong-run Cost Functions In the long-run all inputs are variable! Additionally we can suspect, that most firms produce with approximately constant returns to scale. Solution: min,,g w+r +mg s.t. Q = 1 1 G 1 C l = w +r +mg = w r m Q Medium- and long-run AC and MC AC MC AC m (G = 1) AC l = MC l long-run averageand marginal cost function AC m 1 (G = 5) AC m (G = 1) AC m 3 (G = 15) Q

18 ong Run Average Cost Curve ong Run Average Cost Curve Though the ong Run Average Cost Curve is much flatter than the Short Run Average Cost Curve it might still be U-shaped, because it is an increasing cost industry (decreasing returns to scale). factors are scarce and become more expensive with increasing demand. firms are differently efficient or own unique factors. In this case market price will reflect the minimum average cost of the marginal supplier. Other Factors Influencing the ong Run Average Cost Curve: earning by doing: reflects drop in unit costs as total cumulative production increases because workers become more efficient as they learn their tasks. Transportation costs: if these increase with large-scale production, the RAC has lower optimum scale of operation than a curve without these costs. Factor prices might increase with increasing demand. Any questions? Thanks!

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