Question

Question 7 Consider the following production function: Q=AL' K• Assume A > 0. Further assume 0 <a <1, and 0 <b<1. 1. What is the Marginal Product of Labor (MPL)? Is it diminishing as L increases? What is the Marginal Product of Capital (MPK)? Is it diminishing as K increases? 2. What is the Average Product of Labor (APL)? What is the Average Product of Capital (AP)? 3. What is the TRSL,K? Is the absolute value of TRSL K diminishing in K? Is it diminishing in L? 4. Are there constant, decreasing, or increasing returns to scale? How does this depend on the parameters?

Answer 1

Q= ALK Part (1) MPK = ALK-1 дК = Ab(6-1) LºKº-2 MPK diminishes as K increases MPL = Aala- 2 = Aala - 1) K'La-2 MP diminshes as L increases aL

Part (2) APL = AL-K APK = AK-11"

Part (3) MPL ak TRS(L,K) = K bL ƏT RS(L, K)) a OKb. TRS(L,K) increases as K increases TRS(L,K) aK AL 612 TRSL,K) diminishes as L increases

Part (4) Returns to scale of production function depends on whether value of (a + b) is greater , lesser or equal to 1

PRODUCTION FUNCTION Q = ALKb ocaciocbc - Axo MARGINAL PRODUCT OF LIABOR IS INCREMENTAL INCREASE IN TOTAL OUTPUT THAT RESULTS FROM ONE UNIT INCREASE IN LABOR WHILE 1. KEEPING KEEPING ALL ALL OTHER INPUTS CONSTANT mp = dQ = Aakba-1 OL NOW WE HAVE TO CHECK WHETHER me is DIMINISHING AS INOCREASES DIFFERENTIATING MP WITH RESPECT TO L OMP = AaCa-1 kb, a-1-1= AaCa-Dkbla-2 OL NOW AXO ato kb >o La-2 (a-l) co because coast ocac) ASAWA WOWOT : ampe=Give) IL AND mp DIMINISHES INCREASES AS AS L INCREASES L DIMINISHES 8 MARGINAL PRODUCT OF CAPITAL 15 INCREMENTAL INCREASE in TOTAL OUTPUT THAT RESULTS FROM ONE UNIT INCREASE IN CAPITAL WHILE KEEPING ALL OTHER INPUTS CONSTANT Abkbola de Scanned with OK CamScanner mpa Q

NOW WE HAVE TO CHECK WHETHER MPC is DIMINISHING AS I INCREASES * = Ab(b-1) 29 Kb-1- Ab (B-1) Lakb-2 ak NOW AXO bro caso KB->o (6-1)<o: because Ocb cl impi DimINISHES ASK INCREASES AND INCREASES AS K FALLS AVERAGE PROOULT OF LABOR CAP ις το14L PROρυκτζουτρυτ) ρινιρεο 39 ΤΗe TOTAL QUANTITY OF LABOR APL = Q = Ala Kb Ala-ikb AVERAGE PRODUCT OF CAPITAL CIA PA IS TOTAL PRODUCT COUTPUT) DIVIDED BY THE TOTAL QUANTITY OF CAPITAL APR = Q = Alakb = Alakb TRS,K (TOTAL TECHNICAL RATE OF SUBSTITUTION) MEASURES ADDITIONAL AMOUNT OF CAPITAL Twin IS NEEDED TO REPLACE ONE UNIT OF LABOR WHILE KEEPING LEVEL OF OUTPUT CONSTANT TRSCL,K)= - ma ei ole - TRSCL, K) = - en = -() (5) CS TRS Chip k) = -B) KLY CamScanner

NOW WE HAVE TO CHECK WHETHER TRSCL, K) CABSOLUTE VALUE) FALLS OR RISES WITH RESPECT TO L AND I TRSIL, K) = a(a KL ƏTRSC 4, 6) = (-1)(A) KL-'~= (-1)(3)k2 OL OTRSCU, K) – (-019) ) IL OTES CL, k) = Live) อ - BECAUSE a70 570 l) >0 K70. [>0.2270 :: OTRSCG,K)= (-1) (%)(-) = (-1) C+ive) ƏTRS (L,K) = -ive aL al . TRSCL, K) DIMINISHES AS L INCREAS AND INCREASES As L FALLS 7 Scanned with CamScanner ki

SESSIS WA TRSCL,K)=() K2= 17) 15) ames, Cu, K)= (6)(+) =(tive) BECAUSE ano b>0 [>o :*(+)>0 TRS (LI) INCREASES Ask INCREASES AND FALLS AS K DIMINISHES HTTF A NOW WE HAVE TO CHECK WHETHER THE GIVEN PRODUCTIOW FONCTION EXHIBHITS INCREASING, DECREASING OR CONSTANT RETURNS TO SCALE TO DETERMINE THIS, WE STUDY How OUTPUT CHANGES WHEN WE MODIFY SCALE E THAT IS WHEN WE mULTIPLY INPUTS BY A CONSTANT . LETS MULTIPLY INPUTS BY FACTOR CWHERE >ID AND SEE WHETHER OUTPUT Q=f(L,K) = AL"Kb CHANGES BY SAME FACTOR OR NOT f(XL, XK) = A (XL)" CAK) ² = A xatb Lakb +(x4XK) = xatbAlako = nato f(L, R) RETURNS TO SCALE EXHIBHITED BY PRODOLTION FONCTION DEPENDS ON THE CS PARAMETERS CamScanner

TA IF a+b >I caso, b>0) BUT ocaci AND Ocbc) THEN patb >> ITETTUTTO atb :: FAL, dk) = jatof(L,K) > XXL, K) IF atb>I THEN, ABOVE PRODUCTION FUNCTION SHOWS INCREASING RETURNS TO SCALE IF atbal THEN xatbax ... ILAL, AK) = xatbf(L,K) < f(L, K) PRODUCTION FONCTION SHOWS DECREASING RETURNS TO SCALE IF atb=) f(XL, XK) = atb FCL, K)= XJ(L, R) PRODUCTION FUNCTION SHOWS CONSTANT SRETURINS TO SCALE: th CamScanner