Question

#1 (2 points) What are the relationships that the aggregate production function repre- sents? As a hypothetical example, describe each part of the function as if the variables represent data from the United States. #2 (2 points) In economics, what is the difference between the short run and long run? Is the aggregate production function a short run or long run model? #3 For this question use this aggregate production function: Y = AK1/4 [3/4 Part A) (2 points) Does the above Cobb-Douglas function exhibit constant re- turns to scale? Explain why or why not. Part B) (2 points) Show how you would get MPL and MPK from the function by taking the derivative with respect to capital and then with respect to labor. You don't need to simplify your result like we did in class, just show the first step. Use our equation for MPL and MPK to then write out the simplified version given the specific a in the above function. Part 2) (2 points) Does the above production function diminishing marginal re- turns with respect to capital and labor? To show this, use A = 100 to first compute real GDP using the table below. Then compute MPL and MPK using your result- ing real GDP in each case. Use for MPL Use for MPK Capital Labor Output MPL Capital Labor Output MPK 10,000 20,000 10,000 20,000 10,000 30,000 20,000 20,000 10,000 40,000 30,000 20,000

Answer 1

1. The aggregate production function represents the relationship between the output produced using certain amount of inputs. For example: the production function Y=F(K,L)= K^1/4L^3/4

Y= Output produced (endogenous variable whose value is explained by the production function

K= capital which is exogenous variable

L= labour used( exogenous)

So the amount of output produced Y will depend on the amount of inputs i.e. capital and labour used in its production process which will be determined by the above production function.

Here the capital share in total output is 0.25 and share of labour in the output is 0.75.

2. Short run- It is usually a period of 6 or less months. Here atleast one of the inputs is fixed.

Long Run- It is usually a period greater than 6 months. All the inputs are variable as firm is able to vary its input over a long period of time( it adjusts to the new technology and change its inputs accordingly which requires time).

The main difference therefore is presence of atleast one fixed factor of production in short run whereas in the long run, all factors of production are variable.

Aggregate production function is a long run model since all inputs are variable, no input is taken as fixed.

3.

3 a Y= AK?" , 3/4 Date DELTA PONO Constant Returns to scale? A paroduction function exhibit CRS f(dt) = of(t) #t>O ity Ty GruiuY-F KIL) = AK"3" Flak, ab) = A (AK) (2) 3/4 = 1/4+3/4 AK - & F (IL) 319 hence F(AkidL) = f(KIL) . this function exhibits eRS. b MPL = Oy - margin marginal product of labour de Suppose your standard cooby Dough las function Y = Aka Lira MPL = dy 11AK ! a = (1-a) ALE JA 1-2 = x Aka dy = MPK = dy = XA Kal Ltd MPK ak

Date suice here a values DELTA PO NO are specified, x= 1/4 , 1-2 = 3/4 Qui A =100 MPL = 3A Putting A=100 in MPL & MPK equations, MPL = 7.5 kg MPK = 95 122 3/4. TABLE [Formpc]. I-63 KIL IY 1 MPL 10,000 20,000 1681800 10,000 30,000 2279500 / 56.9 10.000 40.000 2828420 53.03,

Date Calculations! DELTA PO NO = 100 k44 34 K110,000, L=20,000 __ Y= 100 (10,000" (70,000) 3/4 = 100 X 10 X 1681.8 = 168 1800 MPL = 75 (10,000 = (20,000 75x0.84 - 63 Similarly other calculations are done . MPL decreases as amount of FROM the table we see that Cab our is increased. TABLE MPK. KIL I Y I mpk 10.000 20,000 1681800 42.04 20,000 20,000 2001342 25 30,000 20,000 2213373 18:5) Calculatims! Y = 100 L 10.000) (20,000) 374 = 16 & 1800 3/4 - 42.04 haldin mot = 25 | 20.1000 " 10,000 amount the aboue table of shows MPK decreases as capital is increased.

Date Hence there - wit DELTA Pg No are dimi nishing marginal returns capital and labour. You can verify his usuig derivatius. MPL = $a (5) ose mu 20 69 141) 25 = - 3 A K14, -514 16 hauce MPL is decreasing un L MPK = A ( 4 ) 314 dapk = JA LAY (-3)* ok - 3 A 1314 F-714 16 20 hence MPK is decreasing wat ko " - me about marquial production function shows diminishing returns wit labour & capital.