Econ 0: Poblem Set Anses Instucto: Alexande Sollaci T.A.: Ryan Hughes Winte 08 Question (a) The fixed cost is F C = 4 and the total vaiable costs ae T CV (y) = 4y. (b) To anse this question, let x = (x,..., x n ) be the vecto of inputs used in this poduction n function. Recall that y = F (x) and that T V C(y) = p i x i, hee p i is the pice of facto i. Then i= T V C(y) = 4 [F (x,..., x n )] = n p i x i. i= F (x,..., x n ) = n p i x i and it follos that F (x) has deceasing etuns to scale. (c) T V C() = 4 + 4 = 8, T V C() = 4 + 4 = 0, T V C(4) = 4 + 4 4 = 68. See figue fo the plot. (d) AF C(Y ) = F C/y = 4/y. Theefoe, AF C() = 4, AF C() = and AF C(4) =. See figue fo the plot. (e) AV C(y) = T V C(y)/y = 4y. Theefoe, AV C() = 4, AV C() = 8 and AV C(4) = 6. (f) AT C(y) = AT C(y)/y = 4/y +4y. Theefoe AT C() = 8, AT C() = 0 and AT C(4) = 7. (g) As shon above, AT C(y) = 4/y + 4y. We can check that this is a convex function in the elevant ange of y by taking its second deivative: AT C (y) = 4 4y = AT C (y) = y > 0 y > 0. No that e kno that AT C(y) is convex, e kno that taking the FOC ill be sufficient to find a minimum. Then, i= [y] : 4 4y = 0 = y = ±. Since y 0, e can discad the y = solution above, and e ill fund that y = is the value at hich AT C(y) achieves its minimum value.
Figue : Total, Vaiable and Fixed Cost. Figue : Aveage Costs. (h) The maginal cost is found by taking the deivative of T C(y)..t. y. Hence, MC(y) = 8y. It is easy to check that maginal cost cuve cosses the aveage total cost cuve at the point hee AT C(y) is at its minimum. See figue fo the plot.
To undestand hy this is tue, it is useful to conside an example using test scoes. Imagine that eveyone in a class takes a test and e compute the aveage scoe. If a ne student joins the class and this student s scoe is loe than the class aveage, then the class aveage ill dop; convesely, if the ne student s scoe is highe the class aveage, then the class aveage ill ise. In othe ods: if MC(y) is belo AT C(y), then AT C(y) is deceasing in y; if MC(y) is above AT C(y), then AT C(y) is inceasing in y. It follos that heneve MC(y) cosses AT (y)c (and theefoe becomes lage than AT C), AT C(y) changes fom a deceasing function to an inceasing function, hich means the point hee they coss is the point hee AT C(y) eaches its smallest value. Figue : Aveage and Maginal Cost.
Question (a) See figue 4: Figue 4: Plot of the poduction set, using A =, α = /4, β = / and fixing K = Note: if e used gaphs ith the x-, y-, and z-axes, e could epeat the above execise ithout holding K constant. Gaphically, e get figue 5. (b) MP K = f(k, L) K = αakα L β MP L = f(k, L) L = βak α L β The economic intepetation of MP K is the quantity by hich output inceases hen e incease capital, holding labo fixed. Since α < and β > 0, it is clea fom the expession above that MP K is deceasing on K and inceasing on L. Similaly, the intepetation fo MP L is the quantity by hich output inceases hen e incease labo, holding capital fixed. Since β < and α > 0, it is clea that MP L is deceasing on L and inceasing on K. (c) The poduction function has inceasing etuns to scale: f(tk, tl) = A(tk) α (tl) β = t α+β Ak α L β = t α+β f(k, L). Since α + β > and f is homogeneous of degee α + β, it follos that f is homogeneous of degee >, hich means that it has inceasing etuns to scale. f does not violate the La of Diminishing Retuns. As shon in pat (b), both maginal poducts ae deceasing on thei espective facto of poduction (MP K is deceasing on K and MP L is deceasing on L). 4
Figue 5: Suface plot of the poduction function F (K, L) = K 4 L. (d) The anses to pat (c) ae not contadictoy. It is a common mistake to think that Diminishing Retuns and Retuns to Scale ae elated, but they ae vey diffeent things. Diminishing Retuns ae about hat happens to output hen e incease one facto of poduction hile keeping all othes fixed. Retuns to Scale ae about hat happens to output hen e incease all factos of poduction hile keeping thei atios constant. (e) T RS = MP L = βakα L β MP K αak α L β = β K α L. The TRS epesents the tadeoff beteen K and L hile holding output constant. Fo example, if L = L and K = K and e ant to incease L by an infinitesimal amount ε, e ill have to decease K by β α ε to be able to keep output constant. K L (f) Yes, the aveage cost must be deceasing on y. Recall that the fim s total cost is given by Then, aveage cost is T C(y) = T C(f(K, L)) = L + K. AT C(y) = T C(y) y = L + K f(k, L) The numeato of the expession above is homogeneous of degee, hile the denominato is homogeneous of degee α + β. It follos that, if e inceased costs by a facto of t, e ould incease y by a facto of t α+β. Hence, AT C(t α+β y) = t(l + K) L + K t α+β = t α β f(k, L) f(k, L) Since t α+β y > y, this poves that AT C(y) is deceasing on y. < AT C(y) t >. 5
Question (a) If α + β =, then the fim has CRS; if α + β < the fim DRS. (b) It is easonable to think that a fim ith CRS ould have highe pofits, since it is moe efficient i.e. fo the same esouces, it can poduce moe, povided that both K and L ae geate than. (c) The fim solves Taking FOC s, e get π(, ) = max K,L Kα L β K L. [K] : [L] : αk α L β = βk α L β = Multiplying both sides of [K] by K and both sides of [L] by L, e get that K = αk α L β and L = βk α L β. Since α + β =, it immediately follos that K + L = K α L β, hich implies that pofits ae zeo. (d) The fim s poblem emains identical, so e can use the same FOC s fom pat (c). [K]/[L] gives us α L β K = o K = α β L. Plugging this into [L] yields Afte some algeba, e find β ( ) α α β L L β = ( α ) α L = No plugging this back into [K]/[L], e get It no follos that ( α ) α( β) K α L β = α β ( β ( α ) β K = α β ( β α β ( β ) αβ α β ( α ) αβ α β ( β ) α α β. ) β α β. ) ( α)β α β = ( α ) α α β ( β Again fom pat (c) e had that K = αk α L β and L = βk α L β, so pofits ae ( α ) α π(, ) = ( α β) Note that π(, ) > 0 because α β > 0. α β ( β ) β α β. ) β α β. The esult e found is that the pofit made by the fim ith DRS is highe than the pofit made by the fim ith CRS, contay to hat might be intuitive on pat (b). The takeaay fom this execise is that etuns to scale and efficiency/pofitability ae vey diffeent concepts, even though they may appea to measue elated qualities of the poduction function. 6
(e) This claim is a diect application of Eule s Theoem (fom Poblem Set ). Recall that a poduction function ith CRS is homogeneous of degee. Fom Eule s Theoem, if g is homogeneous of degee, then fo any (x, y). No let us look at the fim s poblem. It ants to The FOc s ae g(x, y) = xg x (x, y) + yg y (x, y). (*) max g(x, y) p xx p y y x,y [x] : [y] : g x (x, y ) = p x g y (x, y ) = p y It immediately follos fom those to equations that Theefoe, e can eite the pofit as x g x (x, y ) = x p x and y g x (x, y ) = y p y π(p x, p y ) = max x,y g(x, y) p xx p y y = g(x (p x, p y ), y (p x, p y )) p x x (p x, p y ) p y y (p x, p y ) π(p x, p y ) = g(x, y ) x g x (x, y ) y g y (x, y ), hee the aguments of x and y ee omitted to save space. Using equation ( ), it follos that π(p x, p y ) = 0. Note that this esult can easily be extended to a poduction function ith any numbe of aguments. Question 4 (a) The fim s maximization poblem is The FOC s ae max pk L K L. K,L [L] : p L K = 0 = = p L K () [K] : p K L = 0 = = p K L () Equations () and () imply Plugging back into () K L = = K = L = p L ( L ) = p L 7
) ˆL(,, p) = (A) No plugging this into the expession fo K above, K = ) ) ˆK(,, p) = (B) Finally, [ (p ) ] [ ( ] p ) ˆπ(,, p) = p ) ) [ (p ) ] ) ˆπ(,, p) = p ) ˆπ(,, p) = [ ] It ill also be useful to have ) ˆπ(,, p) =. (C) [ (p ) ] [ ( ] p ŷ(,, p) = ) ) ŷ(,, p) = (D) It is staightfoad fom (A), (B) and (C) that if p, and ae positive, then both facto demands and pofits ae positive as ell. (b) We ant to sho that ˆL(t, t, tp) = ˆL(,, p) and ˆK(t, t, tp) = ˆK(,, p) Fom (A), e have and fom (B), ˆL(t, t, tp) = ( ) tp ( (t)(t) = p ) t t = ˆL(,, p) ˆK(t, t, tp) = ( ) tp ( (t) (t) = p ) t t = ˆK(,, p) The intuition hee is a vaiant of the idea that only elative pices matte. Since multiplying all pices by the same constant does not affect elative pices, demands ae also unchanged. Anothe ay of undestanding this esult is by ealizing that the unit ith hich e measue 8
pices should not affect ou decisions. Fo example, if e ee to use cents (instead of dollas) to measue all pices (so t = 00), ould you expect the demands fo inputs of fims to change? No! nothing in the eal economy has changed, so demands shouldn t change. This same idea also caies ove to exchange ates: it does not matte if pices ae in dollas, pounds o pesos; fims should alays make the same decisions. (c) Just take deivatives: ˆL(,, p) ˆK(,, p) ) = < 0 ) = < 0 Both the expessions above ae stictly negative, since (,, p) 0. This esult is analogous to the substitution effect fom consume theoy: if the pice of one facto inceases, then fims ill use less of that facto in poduction. (d) Fom (C), e have ˆπ(t, t, tp) = ( ) tp ( (t)(t) = p ) t t = tˆπ(,, p). The intuition hee is simila to pat (b): suppose e stated measuing pices in cents instead of in dollas. We d bette also measue pofits in cents! (e) We have the pofit function in (C). Taking deivatives, ˆπ(,, p) ˆπ(,, p) = ) ) = = ˆL(,, p), fom (A). = ) ) = = ˆK(,, p), fom (B). ˆπ(,, p) p = p ) = p ( ) ) = = ŷ(,, p), fom (D). (f) Hotelling s Lemma is a diect application of the Envelope Theoem, hich e poved in Poblem Set (except this is even an easie case, since the maximization has no constaints). Question 5 (a) The fim solves The Lagangian is Taking FOC s min K,L K + L s.t y = K L L = K + L + λ[y K L ]. Fo any vecto x, x 0 means that evey enty in the vecto is stictly positive. 9
[L] : λl K = 0 = = λ L K () [K] : λk L = 0 = = λ K L (4) [λ] : y K L = 0 = y = K L (5) ()/(4) = K L = K = L (6) (6) into (5) yields y = ( L ) L L (,, y) = y ( ) Plugging this into (6), e get K (,, y) = y ( ) and C (,, y) = y ( ) ( + y ) (b) Plugging L and K into () e get ( = λ y C (,, y) = y (). (7) ( ) ) ( y ( ) ) ( = λy ) ( y ) 6 And fom (7), λ (,, y) = y (8) C (,, y) y = y () = λ (,, y) Thus, the Lagange multiplie equals the maginal cost. Again, this is a diect application of the Envelope Theoem of the fim s poblem in pat (a). 0
(c) The fims no chooses y to maximize its pofit, π = py C (,, y): max {py y () } y FOC: SOC: [y] : p y () = 0 [yy] : y () < 0 Note fom the FOC that e have p = y () }{{} C y =λ (9) Fom pats (a) and (b), e had that Plugging this into (9), e get C y = λ = y (). λ (,, y) = p, hich immediately implies that () () and () (4) that is, the cost minimization poblem and the pofit maximization poblem geneate the same fist ode conditions. Question 6 (a) The poblem ith this poduction function is that it is not smooth it has a kink at ak = bl, hich means that the deivative of F at that point does not exist and theefoe the FOC s ae not ell defined. To solve this poblem, e instead use the same intuition fo pefectly complementay goods in consume theoy. Fist, note that the fim ill alays optimally choose K and L such that ak = bl. To see this, suppose that a fim chose instead ˆK and ˆL such that a ˆK > bˆl. The fim s pofit ould be: min{a ˆK, bˆl} ˆK ˆL = bˆl ˆK ˆL Clealy, this fim could educe its demand fo capital and incease it s pofits. In fact, it could do so up to the point hee a ˆK = bˆl; if it educed moe, then min{a ˆK, bˆl} = a ˆK and it ould have the same poblem ith hiing too much labo. Since ak = bl, it also follos that y = min{ak, bl} = ak = bl, hich implies that the conditional facto demands ae K = y/a and L = y/b. (b) Technically speaking, thee is nothing ong ith this poduction function meaning that all of its deivatives ae ell defined. Hoeve, ou usual techniques (taking deivatives and equating them to zeo) ae not helpful. The poblem in this case is that one of the fist ode conditions ill in geneal not be equal to zeo. Instead, e can use economic intuition just like in the pevious case. The main issue hee is that capital and labo ae pefectly substitutable in poducing output. Hence, the fim ill Fo anyone inteested, e can solve this poblem using linea pogamming o the Kuhn-Tucke Conditions instead.
alays choose the input that is elatively cheape. The thought pocess goes like this: to poduce unit of output, y, the fim has to hie /a units of capital o /b units of labo. In tun, /a units of capital costs /a dollas and /b units of labo costs /b dollas. It follos that if /a < /b, then it cheape to poduce each unit of output using only capital; convesely, if /b < /a, then it is cheape to poduce output using only labo. If /b = /a, then the mix does not matte. Hence, the conditional facto demands ae y/a if a < b K(, ) = [0, y/a] if a = b 0 if a > b y/b if a > b and L(, ) = [0, y/b] if a = b 0 if a < b