(a) The isoquants for each of the three production functions are show below:

Transcription

1 Problem Set 7: Solutions ECON 0: Intermediate Microeconomics Prof. Marek Weretka Problem (Production Functions) (a) The isoquants for each of the three roduction functions are show below: f(, ) = f (f < f ) f f(, ) = f (f < f ) f f(, ) = + f f (f < f )

2 (b) The marginal roductivity of caital, MP, tells us by how many units outut would increase if caital inut were increase by one unit (machine). (Mathematically, MP is the artial derivative of the roduction function; the larger the change in caital the further the aroximation gets from actual changes in outut.) The marginal roductivity of labor, MP, tells us howe much additional outut we get from increasing labor inut by one unit (worker). (c) Marginal roductivity of caital with = : MP when f(, ) = : MP = f(, ) = (MP is increasing in ) With =, MP = : MP MP (, ) = MP when f(, ) = : MP = f(, ) = (MP is decreasing in ) With =, MP = : MP MP(, ) =

3 MP when f(, ) = + : MP = With =, MP = : f(, ) = (MP is constant in ) MP MP (, ) = (d) Marginal roductivity of labor with = : MP when f(, ) = : MP = f(, ) = (MP is constant in ) With =, MP = : MP MP (,)=

4 MP when f(, ) = : MP = f(, ) = (MP is decreasing in ) With =, MP = : MP MP (,)= MP when f(, ) = + : MP = With =, MP = : f(, ) MP = (MP is constant in ) MP (,)= (e) Returns to scale: Constant Returns to Scale (CRS), f(λ, λ) = λf(, ): The means that doubling all inuts leads to a doubling of outut (or triling inuts triles oututs, etc.) An examle might be astry making at a bakery, where twice as much of all inuts (: astry chefs, : countertos, ovens, and butter, flour, eggs, etc.) leads to twice as much outut (the astries). Also, the Varian textbook mentions data centers: A thousand times as many data centers (inuts) leads to a thousand times as many webages served (outut).

5 Decreasing Returns to Scale (DRS), f(λ, λ) < λf(, ): A doubling of inuts results in less than double the outut. As the Varian text notes, DRS is usually a short-run henomenon where in fact there is some other inut that is held fixed (otherwise a firm could at least relicate a rocess and achieve CRS). For instance, in farming, a doubling of caital equiment and labor does not lead to a doubling of outut so in that case we d say there is DRS, but this is really because one of the inuts land might actually be fixed. Increasing Returns to Scale (IRS), f(λ, λ) > λf(, ): A doubling of inuts results in more than a doubling of outut. Again, here Varian gives a nice examle: An oil ieline. If we double the diameter of a ie, we use twice as much materials, but the cross section of the ie goes u by a factor of. Thus we will likely be able to um more than twice as much oil through it [u to a certain oint]. (f) et s see whether our three roduction functions exhibit CRS, DRS, or IRS for λ > : f(, ) = : f(λ, λ) = (λ) λ = λ f(, ) > λf(, ) = IRS f(, ) = : f(λ, λ) = (λ) (λ) = λ + = λ f(, ) < λf(, ) = DRS f(, ) = + : f(λ, λ) = (λ) + λ = λ( + ) = λf(, ) = CRS 5

6 Problem (Profit Maximization in the Short Run) (a) The rofit of GMC is total revenue ( f(, )) minus cost (w ): π = f(, ) w and = 6 = π = 8 w. Since caital is fixed, we are in the short run and costs include only the variable costs w. (b) Total revenue, f(, ), and labor cost, w, are shown below for = and w = : $ TC = w TR = f(,) = TR TC (c) A well-behaved function π(x) is flat at the oint at which it attains a local maximum (increasing to the left, flat, then decreasing to the right). Since the derivative is zero when π(x) is flat, finding the x at which π (x) = 0 tells us where a local maximum is. This is what we call the first-order condition. (We can assume for the rofit functions we ll be working with that there is only one local maximum and that it is the global maximum.) Warning: A function is also flat where it attains a minimum, therefore we should check whether actually our x is not minimizing the value of the function (this is the second-order condition: π (x) > 0 means it s a minimum, π (x) < 0 means it s a maximum). This won t be an issue in our alication to maximization of rofit function though. (d) Setting the derivative of the rofit function to zero we have π = 0 = f(, ) w = 0 = MP = w. Alternatively, we can see this using the roduction function f(, ) = 8 then have π = 8 w, so π = 8 w for = 6. We 6

7 and setting this equal to zero (our first order condition), we get π = 0 = 8 w = 0 = w =. () Since the marginal roduct of labor is MP = f(,) found the condition that MP = w. =, in equation () we in fact The intuition is that a firm should hire as long as marginal benefits (additions to outut MP ) are greater than the marginal costs of doing so (real wage w ), u to the oint where additional benefits and costs are exactly equal (MP = w ). Past this oint, a firm shouldn t hire any more labor since MP < w (since MP is always decreasing). (e) To find the otimal level of labor, we can use the condition we found in art (d) in equation (): MP = w or = w. Solving for we get the labor demand curve: ( D = w ) For =, w = 8, we have = ( ) 8 = For =, w =, we have = ( ) = For =, w =, we have = ( ) = These oints are shown in the grah below: w 8 D / 7

8 (f) We know from art (a) what rofit is associated with any, w, and : π = 8 w, so we have: For =, w = 8, =, we have π = 8()( ) (8)( ) = For =, w =, =, we have π = 8()() ()() = For =, w =, =, we have π = 8()() ()() = 8 Problem (abor Market) (a) ate s (erfectly inelastic) labor suly, S = is shown below: w S (b) We had that labor demand was given by D = rate by equating S = D and solving for w : ( S = D = = w ( ) = w = w ). We get the equilibrium wage ( ).5 8

9 w S.5 D (c) At a hyothetical wage above what we found in art (b), the hours of labor demanded is less than the suly at that wage of S =. This excess suly is unemloyment. Since there is willingness to work at lower wages, the wage offered would fall, bringing the excess suly (unemloyment) to zero. (Same is true at a oint below the equilibrium wage we found: There would be excess demand, so to attract more workers the wages would be bid u to the oint where there is no excess demand.) (d) Now with S =, equating S = D and solving for w we get: ( S = D = 8 = w ) = w =. w S..5 D (e) At this rice floor of w =, we have that S = 8 (unchanged) but now D = ( ) =. The unemloyment rate is now S D S 8 ( w ) = = 8 8 =.5 or 50% unemloyment. (The unemloyment rate was reviously zero: S D S =0 since in the market we have S = D = 8. 9

10 w S unem.. D 8 Problem (The ong Run) (a) To determine the returns to scale, we must comare f(λ, λ) to λf(, ) for any number λ > : f(λ, λ) = (λ) (λ) = λ f(, ) < λf(, ) = DRS So since f(λ, λ) < λf(, ), this function exhibits decreasing returns to scale. (b) The rofit function in terms of and is given by: With =, w =, and w =, π = f(, ) (w + w ). π = f(, ) ( + ). (c) First, we ll find the otimal combination of inuts and. From our rofit function above, setting the artial derivatives with resect to and, we get secrets of hainess MP = w and MP = w and substituting in the marginal roductivities of caital and labor as well as rices, this is equivalent to = and =. () Dividing the first equation by the second, we get = = = = =, 0

11 so we will be using and such that =. Plugging = into the first equation in (), we have ) () = = = ( = 08 and so = = = 08 = 5. Given these two values, the otimal level of outut is y = f(, ) = ( ) ( ) = and the rofit associated with this level of outut and the rices given is π = f(, ) (w + w ) = 8 ( ) = 5. (d) The condition for cost minimization is T RS = w w Since for the technical rate of substitution T RS we have T RS = MP MP = = = /5 /08 = and w = w = we indeed are satisfied the cost minimization condition T RS = w w.