We are going to delve into some economics today. Specifically we are going to talk about production and returns to scale.


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1 Firms and Production We are going to delve into some economics today. Secifically we are going to talk aout roduction and returns to scale. firm  an organization that converts inuts such as laor, materials, and caital into oututs, the goods and services that it sells. efficient roduction  a firm s roduction is efficient if it cannot roduce its current level of outut with fewer inuts given the existing state of knowledge aout technology and the organization of roduction. roduction function the roduction function is a function that returns the maximum level of oututs for a given level of inuts. Examle: Q min( K, L)  caital and laor are erfect comliments. They must e used in eual uantities for efficient roduction. Examle: Q K +L  caital and laor are erfect sustitutes. You can relace one unit of caital with a unit of laor and not lose any outut. Examle: Q a K L  CoDouglas roduction function. Returns to scale refers to how much additional outut can e otained when we change all inuts roortionately. Decreasing returns to scale when we doule all inuts, outut is less than douled. 2 Q< F(2 k,2 L) a A concrete examle is the CoDouglas roduction function ( Q K L ) < 1. To see this multile oth inuts y t ( ) ( ) ( ) ( ) a a t K t L t K L < t Q with 1
2 Constant returns to scale when we doule all inuts, outut is exactly douled. 2 Q F( 2 k,2 L) A concrete examle is the CoDouglas roduction function Q K α L β withα + β < 1. ( ) What aout erfect sustitutes? What aout erfect comliments? Increasing returns to scale when we doule all inuts, outut is more than douled What roduction function that we have already talked aout exhiits increasing returns to scale? 2
3 Profit Maximization and Duality What is the ojective of most firms? How to firms go aout meeting this ojective? Stes to rofit maximization 1) The firm must determine how to roduce outut in an efficient manor in the sense that they are roducing the maximum level of oututs for a given level of inuts roduction engineers hel with this. Basically they have to find their roduction function. 2) Figure out how to roduce in a cost efficient way Given inut rices what cominations of different inuts should we emloy to roduce a given level of outut. 3) Figure out how much outut to roduce. Lets consider ste 2) Ste 2) says the firm has to find the cost function Cost Function C ( ) a function that returns the least cost way of roducing units of outut. You lug in a uantity and get ack the least cost way of roducing that uantity. The cost function can e found y solving the following rogram KL, [ w L+ r K] [ F K L ] min suject to the constraint (, ) In English  minimize the cost of roducing units of outut. The values of K and Lthat solve the cost minimization rolem are called conditional factor demands. They are conditional ecause they will generally deend on the level of oututs that need to e roduced. Note that these conditional factor demands will generally e a function of the inut rices (r and w) and the uantity to e roduced (). Thus, K K( : w, r) L L( : w, r) If we want the cost function we just multile these conditional factor demands y their rices and sum. That is C ( ) wl ( : wr, ) + r K ( : wr, ) 3
4 a Examle: For the CoDouglas roduction function with two inuts K L * r L a w K * a w r a 1 1 a 1 1 * * r a w C ( ) wl r K w r + + a w r ( )( ) a 1 a C ( ) a r w 4
5 Proerties of the Costs Functions 1) The cost function is increasing in factor rices. If I increase w or r costs have to increase. 2) If I increase oth w an r roortionately then cost increase y that factor of roortionality. For examle If I doule inut rices cost doule. We will use this eventually. Links: Cost and Returns to Scale It should e ovious that there are links etween (technology) and roduction functions. Why? In articular, there are links etween returns to scale and cost functions. What are these links? If a technology exhiits decreasing returns to scale then average cost will e decreasing in outut. If a technology exhiits constant returns to scale then average cost will e constant in outut. If a technology exhiits increasing return to scale then average cost will e decreasing in outut Examle: CoDouglas ( )( ) a 1 a C ( ) a r w C ( ) AC( ) ( a + )( a ) r w a 1 ( ) a Is there decreasing, constant, or increasing returns to scale? It deends If <1 there are decreasing returns to scale ecause average cost increase when outut increases If 1 there are constant returns to scale ecause average cost stay the same when outut increases If >1 there are increasing returns to scale ecause average cost decrease when outut increases. Because of these relationshis we can rewrite the CoDouglas cost function as a ( ) φ a 1 φ φ φ C ( ) φ a r w where φ is known as the scale arameter. 5
6 Why Care Aout Returns to Scale? The iggest thing is that when there are increasing returns to scale a firm s average cost of roduction is decreasing. There are imortant imlications of this. If one firm with increasing returns to scale is caale of roducing enough outut for the entire market then there is a arrier to entry ecause this one firm could roduce a level of outut that would satisfy the market at less cost than any other firm. If one firm with increasing returns to scale is caale of roducing enough outut for the entire market then it may e etter to have just one firm in the market and to regulate it in some way. Such a firm is called a natural monooly and can e regulated in a numer of ways. $ 0.5 6
7 Nerlove s Model Nerlove s asic setu is just CoDouglas Technology with 3 inuts (caital, laor, and fuel). Consider the CoDouglas roduction with three inuts: (A1) A K L F a c where A is a constant, L is laor, K is caital and F is fuel. In class it was shown that the cost function dual to this CoDouglas roduction function is (A2) 1 a c φ φ φ φ f Crw (,,, ) h r w f where φ + c is the returns to scale arameter and the log of this euation and adding an error term ( ) i Mark Nerlove (1963) a c h φ A a c φ. Taking e yields the euation estimated y (A3) * ln( Ci ) β0 β ln( i ) βk ln( ki) βl ln( li) e i where C * i i ki li C r i w i fi fi fi β0 ln( h) β 1 φ β a k φ β l φ In this model we know that if returns to scale is going to deend on the value of φ + c. If φ < 1 decreasing returns to scale If φ 1 constant returns to scale If φ > 1 increasing returns to scale The nice feature of this model is that the coefficient on ln( i ) in the aove regression is the inverse of the returns to scale arameter. Thus, when we estimate the model we get an estimate of returns to scale. 7
8 Elasticity elasticity is a measure of the resonsiveness of one variale with resect to another variale. The elasticity of y with resect to x is measures as the ercentage change in y induced y a 1ercent change in x. $ Anytime deendent variale and indeendent variale are in natural logs the coefficient on the logged indeendent variale is the elasticity of the unlogged deendent variale with resect to the unlogged indeendent variale. β is the elasticity of costs with Examle: In Nerlov s ase secification (A3) the resect to outut. If outut changes y 1ercent, then costs changes y Below are estimates of (A3) β ercent. reg lnc ln lnl lnk Source SS df MS Numer of os F( 3, 141) Model Pro > F Residual Rsuared Adj Rsuared Total Root MSE lnc Coef. Std. Err. t P> t [95% Conf. Interval] ln lnl lnk _cons What can we say from the estimates? Our estimate of the elasticity of cost with resect to outut is If outut increases y 1ercent, cost will increase y an estimated 0.72 ercent. Our estimate of the return to scale arameter is increasing returns 0.72 to scale. Our estimate of the elasticity of cost with resect to the rice of laor is If the rice of laor increases y 1ecent, costs will increase y an estimated 0.59 ercent. Our estimate of the elasticity of cost with resect to the rice of caital is If the rice of caital increases y 1ercent, cost will decrease y an estimated ercent. We can also recover an estimate of the elasticity of cost with resect to the rice of fuel. If the rice of fuel increases y 1ercent, cost will increase y 1 β lnk β lnl If the rice of fuel increases y 1ercent, costs will increase y an estimated
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