The Supply Side of the Economy. 1 The Production Function: What Determines the Total Production

Transcription

1 Lecture Notes 3 The Suly Side of the Economy (Mankiw, Cht. 3) 1 The Production Function: What Determines the Total Production of Goods and Services? An economy's outut of goods and services - its GDP - deends on (1) its quantity of inuts, called the factors of roduction, and (2) its ability to turn inuts into outut, as reresented by the roduction function. Factors of roduction are the inuts used to roduce goods and services. The two most imortant factors of roduction are caital and labor. Caital is the set of tools that workers use (the construction worker's crane, the accountant's calculator, and your ersonal comuter). Labor is the time (the number of hours) eole send working. We use the symbol K to denote the amount of caital and the symbol L to denote the amount of labor. Assume that the economy has a xed amount of caital and a 1

2 xed amount of labor. We write K = K L = L The overbar means that each variable is xed at some level. We also assume here that the factors of roduction are fully utilized-that is, that no resources are wasted. Technology determines how much outut is roduced from given amounts of caital and labor. Economists exress the available technology using a roduction function. Letting Y denote the amount of outut, we write the roduction function as Y = F (K L) This equation states that outut is a function of the amount of caital and the amount of labor. Examle 1 (linear roduction function) F (K L) =K + L For examle, suose K =2andL =5,thenY = F (K L) = 2+5=7. This roduction has the roerty that caital and labor are erfect substitutes. For examle, if one reduces K by 1 unit but increases L by 1 unit, then outut remains the same. 2

3 Examle 2 F (K L) =K L This roduction function exhibits comlementarity btween caital and labor. To see this, suose the initial caital and labor are K and L. Suose now one additional unit of labor becomes available. How much more outut is roduced because of the additional labor? Answer: K (L +1); K L = K 1 which deends ositively on K. That is, a higher stock of caital makes the additional amout of labor more roductive. (Does this exlain the higher labor roductivity in the US than in Mexico?) Examle 3 (A Cobb-Douglas Production Functions) F (K L) =A K L 1; where A and are constants, A>0 and 0 <<1. For examle, A =0:5, =1=2. Let K =4andK =9,then Y =0:5 4 9=3. Examle 4 (A Leontie roduction function) F (K L) = minfl Kg This roduction function describes a technology with which in order to roduce one unit of out, exactly (no more and no less) one unit of caital and one unit of labor is required. For examle if L =5<K= 6, then Y = 5, the extra unit of caital is useless. 3

4 The roduction function reects the current technology for turning caital and labor into outut. If someone invents a better way to roduce a good, the result is more outut from the same amounts of caital and labor. Thus, technological change alters the roduction function. Suose today's roduction function is Y = F (K L), suose tomorrow's new technology is such that for any given amounts of caital and labour, outut is doubled. Then tomorrow's roduction function is Y =2 F (K L) Suose today's roduction function is Y = K + L. suose tomorrow's new technology makes tomorrow's caital twice as roductive as today's caital. Then tomorrow's roduction function is Y =2 K + L 4

5 2 Constant Returns to Scale A roduction function has constant returns to scale if an increase of an equal ercentage in all factors of roduction causes an increase in outut of the same ercentage. If the roduction function has constant returns to scale, then we get 10 ercent more outut when we increase both caital and labor by 10 ercent. Mathematically, a roduction function has constant returns to scale if zf(k L) =F (zk zl) for any ositive number z: This equation says that if we multily both the amount of caital and the amount of labor by some number z, outut is also multilied by z. Examle 1 F (K L) =K + L Examle 3 F (K L) =AK L 1; Examle 4 F (K L) = minfk Lg 5

6 3 The marginal roduct of Labor The Marginal Product of Labor (MPL) is the extra amount of outut that the economy gets from one extra unit of labor, holding the amount of caital xed. MPL(K L) =F (K L +1); F (K L) That is, MPL is the dierence between the amount of outut roduced using K units of caital and (L + 1) units of labor and the amount of outut roduced using K units of caital and L units of labor. Imortant to note: MPL deends on (K L), the levels of caital and labor at which MPL is comuted. Let F (K L) =K + L: Then MPL(K L) =F (K L+1);F (K L) =(K+L+1);(K+L) =1 Thus MPL is constant at one for all combinations of K and L. Let F (K L) =KL. Then MPL(K L) =K(L +1); KL = K: In this examle, MPL deends on K : it is higher when K is higher. 6

7 Let F (K L) =K 1=2 L 1=2 = KL. Then MPL(K L) = r K(L +1); KL = K L +1; L For examle, MPL(1 0) = 1[ 0+1; 0] = 1 MPL(1 1) = 1[ 1+1; 1] = 0:414 MPL(1 2) = 1[ 2+1; 2] = 1:73 ; 1:41 = 0:32 Notice that here M(1 L) decreases as L increases. 7

8 4 The Law of Diminishing Marginal Product Many roduction functions have the roerty of Diminishing Marginal Product: Holding the amount of caital xed, MPL decreases as L increases. Consider the roduction of bread at a bakery. Asabakery hires more labor, it roduces more bread. As more workers are added to a xed amount of caital, however, the MPL falls. Fewer additional loaves of bread are roduced because workers are less roductivewhen the kitchen is more crowded. In other words, holding the size of the kitchen xed, each additional worker adds fewer loaves of bread to the bakery's outut. Consider a farmer's roduction of otatoes. He has a xed amount of land that's his caital K. Labor inut L here is the number of hours he sends working on his land. Outut is higher as the farmer uts in more eort, but the roductivity (additional outut) assoaciated with each additional hour the farmer uts in declines. Professor Wang has a comuter that he uses as his caital. Professor Wang roduces research by sending time with the comuter. He is very roductive at 9:00 in the morning, he is slower at 1:00m, and he feels his brain is useless at 6:30 in the afternoon. 8

9 5 The Marginal Product of Caital The Marginal Product of Caital (MPK) is the extra amount of outut that the economy gets from one extra unit of caital, holding the amount of labor xed. MPL(K L) =F (K +1 L) ; F (K L) That is, MPK is the dierence between the amount of outut roduced using K + 1 units of caital instead of K units of caital. Let F (K L) =K + L: Then MPL(K L) =F (K+1 L);F (K L) =(K+1+L);(K+L) =1 Thus MPK is constant at one for all combinations of K and L. Let F (K L) =K L. Then MPL(K L) =(K +1)L ; KL = L: In this examle, MPK deends on L :itishigherwhenl is higher. 9

10 6 MPL and the demand for Labor Consider a rm which has a xed amount of caital and wants to determine how manyworkers to hire. Let P denote rice and W be the nominal wage (number of dollars aid to a worker). Suose P MPL(K L) >W Then the rm would want to hire more workers. Suose P MPL(K L) <W Then the rm would want to hire less workers. The rm's equilibrium L is setermined by or P MPL(K L) =W MPL(K L) = W P where W=P is called the real wage. Figure Case Studies We know that given K, the rm's otimal amount of labor, L, is determined by the following equation 10

11 MPL(K L) = W P Suose it is now given that MPL(K L) =K ; L where K = 10 is xed. Suose also W =10,P = 2. Then L satises 10 ; L = 10 2 or L =5 Suose now the rm has just made some new caital investment and caital stock has increasesd from 10 to 15. Then the new L satises 15 ; L = 10 2 and L = 10. Suose now the economy is heading into a recession and the rm has decided to cut caital stock from 15 to 5. Then the new L satises 5 ; L = 10 2 and L = 0. That is, the rm is shut down. 11