The Economics of Production Make or Buy Decisions Capacity Expansion Learning-Curves Break-even Analysis Production Functions

Make or Buy? Let c 1 = unit purchase price c 2 = unit production cost (c 2 < c 1 ) K = fixed cost of production x = number of units required Produce if K + c 2 x  c 1 x orx  K / (c 1 - c 2 )

Make or Buy Example It costs the Maker Bi Company $20 a unit to purchase a critical part used in the manufacture of their primary product line – a thing-um-a-jig It is estimated that the part could be produced internally at a unit cost of $16 after incurring a fixed cost of $20,000 for the necessary equipment. What to do? a thing-um-a-jig

What to do? Let c 1 = $20 c 2 = $16 K = $200,000 x = number of units required Produce if x  K / (c 1 - c 2 ) = 20,000 /(20 – 16) = 5,000

Nonlinear Cost Function Let c 1 = unit purchase price c 2 = K + ax b where K, a, b > 0 x = number of units required If c 1 = 20 and c 2 = 20, x.7, then

More on that Nonlinear Cost Function xMakeBuy 1900$39,730$38, $39,911$38, $40,092$39, $40,272$39, $40,451$40, $40,630$40, $40,701$40, $40,808$41, $40,985$41, $41,162$42, $41,338$42, $41,513$43, $41,688$43, $41,862$44, $42,036$44, $42,209$45, $42,381$45,500

Strategic Decisions Capacity Expansion Capacity Growth Planning –when to construct new facilities –where to locate facilities –how large to size a facility Economies of scale –advantage of expanding existing facilities –share plant, equipment, support personnel –avoid duplication at separate locations

Capacity Expansion competing objectives: maximize market share maximize capacity utilization time number units demand capacity leads demand time number units demand capacity lags demand

We need a model let D = annual increase in demand x = time interval (in yrs) between capacity increases r = annual discount rate, compounded continuously f(y) = cost of expansion of capacity y assume y = xD, then cost = C(x) = f(xD) [1 + e -rx + (e -2rx ) + (e -3rx ) + …] = f(xD) [1 + e -rx + (e -rx ) 2 + (e -rx ) 3 + …] = f(xD) / [1 – e -rx ] assume f(y) = ky a, then find the x that minimizes C(x)

A Diversion - the Geometric Series You see? It does converge.

Discounting – another diversion Consider the time value of money $1.00 today is worth more than a $1.00 next year How much more is it worth? If r = annual interest rate, then it is worth (1+r) $1.00 After two years, it is worth (1+r) 2 $1.00 (compounded) Compounded quarterly for 1 yr = Compounded continuously for one year = After t years =

More diversionary discounting A stream of costs: C 1, C 2, …, C n incurred at times t 1, t 2,…, t n has a present value of: Why can’t you show us an example? For an infinite planning horizon where x is the time between expansions:

The Example Chemical firm expanding at a cost ($M) of –where y is in tons per year. Demand is growing at the rate of D = 5,000 tons per year and future costs are discounted at a rate of r = 16 percent Find x that minimizes

Capacity Expansion Solution alternately set C’(x) = 0 solve for x.

Learning Curves Based upon the observation that unit labor hours or costs decrease for each additional unit produced Units produced Direct labor hrs per unit

Why does this happen? Employee learning reduced set-up times better routing and scheduling of material (WIP) improved tool design more efficient material handling equip. (MHE) reduced lead-times improved (simplified) product design production smoothing quality assurance revised plant layout increased machine utilization

Learning Curve (experience curves) Y(u) = labor hours to produce the u th unit assumeY(u) = au -b a = hours to produce the first unit b = rate at which production hours decline labor coming to work

Learning Curves Assume hours to produce unit 2n is a fixed percentage of the hours to produce unit n Then for an 80 percent learning curve: Observe the simple formula

Learning Curves least-squares analysis UnitDirect Labor Number - xHours -Y(x) Fit Y(x) = ax -b using Excel Y(x) = 74x =.845 or a 84.5% learning curve

Learning Curves Cumulative Cost hours to produce i th unit cumulative direct labor hrs to produce x units average unit hours to produce x units

Learning Curves Approximate Cumulative Cost

Example Y(x) = 74x =.845 or a 84.5% learning curve

Break-Even Analysis Let x = number of units produced and sold x = S -1 (unit selling price) S(x) = unit selling price F = fixed cost g(x) = variable cost to produce x units then break-even point occurs when revenue = cost; or S(x) x = F + g(x) and profit = revenue – cost or P(x) = S(x)x – [F + g(x)] Sam Even on a break

Break-Even Analysis x $ F Break even pt loss profit Max profit Revenue curve Cost curve loss Diminishing returns

Break-Even Analysis Demand Curve x S(x) S(x) = d + e x + f x 2 (quadratic) d, e, and f are constants to be determined

Break-Even Analysis Demand Curve x S(x) S(x) = d + e x + f x 2 d, e, and f are constants to be determined Approximate as linear S(x) = d + e x

Break-Even Analysis Unit Cost LetM = direct material unit cost ($/unit) L = direct labor rate ($/hour) B = factory burden rate Y(x) = direct labor hours to produce unit x C(x) = cost to produce unit x C(x) = M + L Y(x) + L B Y(x) = M + (1+B) L Y(x) = M + (1+B) L a x -b Learning curve effect

The Factory Burden Diversion Manufacturing Costs Factory burdenDirect costs Direct labor Direct material Indirect material Indirect labor Indirect expense -Supervision -Engineering -Maintenance -Heating -Lighting -Depreciation -Rent & Taxes -Office & janitorial supplies -Paint

Factory Burden - example Categoryannual cost Indirect material$ 6,120 Indirect labor 42,800 Indirect expenses 22,900 total$71,820 Product annual productionlabor hours ratewages A100, $9/hr$9,000 B140, $7/hr 9,800 C 80, $7/hr 11,200 total4000 $30,000 burden rate = 71,820 /30,000 = per direct labor $

Manufacturing Costs General Overhead Costs Profit Selling Price S(x) Administrative Costs Marketing Costs Development Costs Demands

Cumulative Cost g(x) = M x + L (1+B) T(x) = M x + L (1+B) [a x 1-b / (1-b)] total cost = F + M x + L (1+B) a x 1-b / (1-b) where F is a fixed cost to produce product x Unit cost: C(x) = M + (1+B) L a x -b Learning curve

Break-Even Analysis -Profit Profit = P(x) = S(x) x - [F + g(x)] letting S(x) = d + ex, e < 0 P(x) = (d + e x) x - F - M x - L (1+B) a x 1-b /(1-b) = d x + e x 2 - F - M x - g x 1-b where g = L (1+B) a /(1-b)

More Break-Even Analysis P(x) = (d - M) x + e x 2 - g x 1-b - F break-even: set P(x) = 0 and solve for x maximize profit: set and solve for x for e < 0, a max point can exist

Break-Even Analysis - example P(x) = d x + e x 2 - F - M x - g x -b+1 where g = (1+B) L a /(1-b) Data: d = 100 e = -.01 F = $100,000 M = $4 B =.5 L = $20 / hr a = 10 b =.60 P(x) = 100 x -.01 x 2 – 100, x – (1+.5) (20) (10) x.4 /.4 = 96x -.01x 2 –750 x.4 –100, = 66%

The Math

The Graph x = 1382 x = 4706

Production Functions A production function expresses the relationship between an organization's inputs and its outputs. It indicates, in mathematical or graphical form, what outputs can be obtained from various amounts and combinations of factor inputs. In its most general mathematical form, a production function is expressed as: Q = f(X 1,X 2,X 3,...,X n ) where: Q = quantity of output and X 1,X 2,X 3,...,X n = factor inputs (such as capital, labor, raw materials, land, technology, or management)

Production Functions There are several ways of specifying this function. One is as an additive production function: Q = a + bX 1 + cX 2 + dX 3,... where a,b,c, and d are parameters that are determined empirically. Another is as a Cobb-Douglas production functionCobb-Douglas Q = f(L,K,M) = A * (L alpha ) * (K beta ) * (M gamma ) where L = labor, K = capital, M = materials and supplies, and Q = units of product.

Cobb-Douglas Production Function Q = f(L,K,M) = A * (L alpha ) * (K beta ) * (M gamma ) Properties of the Cobb-Douglas production function: Decreasing returns to scale: alpha + beta + gamma < 1 Increasing returns to scale: alpha + beta + gamma > 1 Let C L, C K, and C M = the unit cost of labor, capital, and material, then C(L,K,M) = C L L + C K K + C M M is the total cost function

A Little Production Problem An interesting problem: Given a monthly budget of $B, how should the money be spent to obtain a specified output Q? Find L, K, and M where L = dollars spent on labor, K = dollars spent on facilities and equipment, and M = dollars spent on material I know I can work this one.

The Inevitable Example laborcapitalmaterial Aalphabetagamma LKMQ $8,333$66,667$25,000794,700RHS budget ,

Stop the madness. Optimize your production system! profits homework: turn-in breakeven problem text: Chapter 1- 29, 30, 31, 32, 34, 35 36, 37,38, 43, 44