On the Concept of Returns to Scale: Revisited

Transcription

1 3 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X On te Concept of Returns to Scale: Revisited Parvez Azim Abstract Tis paper sows w it is tat in Economics text books and literature we invariabl consider constant returns to scale (linearl omogeneous production functions) and not increasing returns to scale or decreasing returns to scale production. It as been demonstrated in tis paper b using bot cost elasticit output approac and Euler s teorem tat te constant returns to scale production functions enable us to acieve productive efficienc and equilibrium. Production under increasing returns to scale or decreasing returns to scale are not at equilibrium. Onl under constant returns to scale tecnolog do we acieve productive efficienc and equilibrium. ewords: Returns to scale, linearl omogeneous production function, cost elasticit of output, productive efficienc JE classification: D24, O47, C23. Introduction In te literature, te terms suc as constant returns to scale (constant economies of scale), increasing returns to scale (economies of scale) and decreasing returns to scale (diseconomies of economies of scale) ave been used quite frequentl. However, it is as not been explicitl explained w we make use of constant returns to scale (linear omogeneous production function) and not increasing and decreasing returns to scale. Tis paper will sed ligt on tis, classification will be done and te relationsip between tem will be elaborated upon. Understanding of tese issues is important to understand various studies dealing wit economies and diseconomies and pedagogical purposes. 2. Metodolog To start wit, let tere be a production function of te constant elasticit of substitution (CES) form wose associated cost function is derived ere. It is trivial to obtain te ratio between te marginal cost (MC) and average cost (AC). Te ratio is represented b te following expression wic is called cost elasticit of output. It is used to measure te returns to scale. Constant returns to scale (CRS) is wen =, MC=AC, increasing returns to scale (IRS) is wen <, MC < AC and decreasing returns to scale (DRS) is wen >, MC > AC. It is important to give definition of eac of tese. CRS ( =) means tat a percent increase in output results in exactl percent increase in te total cost. IRS ( <) means tat a percent increase in output results in less tan percent increase in te total cost. DRS ( >) means tat a percent increase in output results in more tan percent increase in te total cost. Professor of Economics, GC Universit Faisalabad

2 32 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X Te relationsip between AC and output level could also be used to explain CRS, IRS and DRS. C( w, ) et AC ( ) were w is assumed to be fixed vector, and ( ) signifies te average cos t. B partial differentiation, we get C C C C 2 2 MC AC 0 wen MC AC CRS prevails if AC is constant wit an increase in output 0 wen MC AC IRS prevails if AC is falling wit an increase in output 0 wen MC AC DRS prevails if AC is rising wit an increase in output It is evident tat CRS ( ) prevails if and onl if CRS ( ) prevails. It is evident tat IRS ( ) prevails if and onl if IRS ( ) prevails. It is evident tat DRS ( ) prevails if and onl if DRS ( ) prevails. et / Tus it can be concluded tat / / / if and onl if if and onl if and if and onl if 0, 0 0. C C C C C C C( ) sows tat

3 33 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X wen MC AC, AC does not cange wit an increase in ouput. as sown in te figure below. AC AC 0 0 MC AC ( ) MC AC 0 MC AC ( ) MC lies below AC AC 0 MC AC ( ) AC lies below MC ooking at te usual U-saped AC curve, te AC curve is decreasing to a certain level of output (in wic 0 and ) and ten increases as output increases (in wic 0 and ). MP, AP Stage I MP Stage II AP Q MC, AC MC AC Productive efficienc and equilibrium is acieved onl at Q Fig IRS CRS DRS AC > MC AC = MC MC > AC

4 34 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X It is also to be noted tat MC curve intersects te AC curve from below wen te AC curve is at its lowest point. It means tat it is important to restrict te concept of returns to scale to a certain range of output levels. Terefore, te productive efficienc is acieved at te output level tat corresponds to te minimum point of AC curve were MC=AC. An output level tat is less tan or more tan * is not a productive efficienc point. It is evident tat bot IRS wen 0 and, and DRS wen 0 and, don t correspond to te point were productive efficienc is acieved. Tis appears to be te most plausible reason tat w in te literature we use te concept of CRS and not IRS or DRS. For example, te linearl omogeneous Cobb-Douglas production function and constant elasticit of substitution (CES) production function. Wit reference to te above given figure; if a firm/industr is operating to te left of te minimum AC point, ten it can increase profits b increasing production until it reaces te minimum AC point. Similarl, if a firm/industr is operating to te rigt of te minimum AC point, ten it can increase profits b decreasing production until it reaces minimum AC point wic is * in te accompanied figure. It can be observed tat wen te MC curve is below an average cost curve te AC curve is falling. Tis relation olds true regardless of weter te MC curve is falling or rising. Wen te MC curve is above an AC curve te AC is rising. Te MC curve intersects an AC curve at its minimum. It is important to mention tat wen average product (AP) is rising AC is falling and wen AP is falling, AC is rising. It is also to be noted tat wen marginal product (MP) starts to fall, MC starts to rise. Tere is inverse relationsip between AP and AC and MP and MC as sown in te figure. A rational producer will not produce in stage I were MP lies above AP. Te production will take place in te economic region in stage II. Te downward sloping portion of te long run average cost (RAC) curve corresponds to IRS (economies of scale). Te orizontal portion of te RAC curve corresponds to CRS (neiter economies nor diseconomies of scale). An upward sloping portion of te RAC curve corresponds to DRS (diseconomies of scale). In a long-run perfectl competitive (in te input markets) environment, te productivel efficient and equilibrium level of output corresponds to te minimum efficient scale marked as * in te figure. Tis is due to te zero profit requirement of a perfectl competitive equilibrium. Tis result implies production is at a level corresponding to te lowest possible AC. In fact all points along te RAC are productivel efficient b definition, but not all are equilibrium points in a long-run perfectl competitive environment. To elaborate on te point, let us consider a Constant Elasticit of Substitution (CES) production function and derive its associated cost function. Te cost function will give us a relationsip between MC and AC. It is evident as sown below tat MC < AC wen we ave Increasing Returns to Scale, MC < AC wen we ave Decreasing Returns to Scale, and MC =AC wen we ave Constant Returns to Scale,

5 35 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X et us consider te CES production function A wose associated cost function is C w r and its A CY cost elasticit of output is were is te degree of omogeneit Y C of te given CES production function. Derivation of te cost function associated wit te given CES production function From A we get ( ) MP w A ( ) MP r A dividing tem we get w r transposition leads to r r () i w w r ( ii) w substitution of ( ii) into te given production function ields r A factoring out gives us w r r A ( iii) w A w

6 36 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X substituting ( iii) into ( i) gives us r r ( iv) A w w Substitution of ( iii) and ( iv) into te cost equation C w r gives us w C A r r w r w w r w r w C A w w cancelling out te denominators we end up wit C w r QED A C ( ii) Te cost elasticit of output w r C A A C C C C C or C A C A AC MC AC MC were is te degree of omogenit of te given CES production function. Wen we ave CRS ( MC AC) production function Wen we ave IRS ( MC AC) production function Wen we ave DRS ( MC AC) production function Te concept of returns to scale could also be explained wit te elp of te Euler ' s teorem as explained below.

7 37 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X According to te Euler ' s Teorem MP MP Q were ' ' is te degree of omogeneit of te production function. Multipling trougout b P gives us PMP PMP ( PQ) were P is te price of output and PQ is total value of output. wage of capital ( VMP) r wage of labour ( VMP) w PMP PMP PQ total cost of input total cost of input Total value of output or revenue Total cost of producing Q If te firm spends wole revenue on inputs. If te firm spends more on inputs tan revenue, so incurring loss If te firm spends less tan revenue, so te firm makes a positive profit. Wen, it is termed as constant returns to scale ( CRS), increasing returns to scale ( IRS) and decreasing returns to scale ( DRS). It is obvious tat costs will increase equal to, more tan, or less tan linearl wit output as '' is equal to, greater tan, or less tan one. It is so because te CES tecnolog exibits constant, increasing or decreasing returns to scale depending on te value of ' '. 3. Conclusion In te production teor, linearl omogeneous production functions (also called constant returns to scale) are used because te depict constant returns to scale tecnolog wic sows productive efficienc and equilibrium point sown b te lowest AC curve point * in te above given figure. Output less tan or more tan tat point does not give us equilibrium point of production. Productive efficienc plus equilibrium is acieved onl at te lowest AC curve were MC is cutting AC curve from below. Compliance wit Etical Standards: Te autor declares tat e as no conflict of interest, financiall, non - financiall, directl or indirectl related to te work. References Basu, Susanto and Jon G. Fernall. Returns to Scale in U.S. Production. Estimations and Implications, Te Journal of Political Econom. Vol.05,No.2 (april,997), Hanoc,G.975. Te Elasticit of Scale and te Sapes of Average Costs. American Economic Review 65 (June): Separd,R.W Cost and Production Functions. Princeton, N.J. Princeton Universit Press. Separd,R.W Teor of Cost and Production Functions. Princeton, N.J. Princeton Universit Press. Silberberg, E Te Structure of Economics.2 nd ed. New York: McGraw-Hill. Takaama, A Matematical Economics. 2d. ed. New York: Cambridge Universit Press.

8 38 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN X Takaama, A Analtical Metods in Economics. st.ed. Hertfordsire: Harvester Weatseaf, USA. Uzawa,H Production Functions wit Constant Elasticities of Substitution. Review of Economic Studies 29 (October):