Solution to Problem First, the firm minimizes the cost of the inputs: min wl + rk + sf

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1 Econ 0A Poblem Set 4 Solutions ue in class on Tu 4 Novembe. No late Poblem Sets accepted, so! This Poblem set tests the knoledge that ou accumulated mainl in lectues 5 to 9. Some of the mateial ill onl be coveed on Lectue 8, but ou should be able to do most of the poblem set alead [as of Tu 29 Octobe]. The poblem set is focused on poduction. Geneal ules fo poblem sets: sho ou ok, ite don the steps that ou use to get a solution (no cedit fo ight solutions ithout explanation), ite legibl. If ou cannot solve a poblem full, ite don a patial solution. We give patial cedit fo patial solutions that ae coect. o not foget to ite ou name on the poblem set! Poblem. Poduction: cost minimization and pofit maximization ( points) In class e intoduced to diffeent chaacteizations of fim decisions: cost minimization and pofit maximization. In this execise ou ae asked to sho that the to optimization poblems ae equivalent as of solving the same poblem. Conside a mango fam in the Philippines. The fam has a poduction function f (L, K, F ), hee L is labo, K is capital, and F is fam land. The pice of labo is, the pice of capital is, the pice of land is s, ith all of the pices positive. Assume also fi 0 00 > 0 and fi,i < 0, ith i L, K, F.. Wite don the fist step of the cost minimization poblem and the fist ode conditions ith espect to L, K, and F. (2 points) 2. The fist ode conditions in step, togethe ith the constaint f (L, K, F ), allo us to solve fo L (,, s, ),K (,, s, ) and F (,, s, ). We can theefoe ite the cost function c (,, s, ) L (,, s, )+K (,, s, )+sf (,, s, ). In the second step of the cost minimization, the fim maximizes p c (,, s, ). Witedonthefist ode conditions ith espect to. ( point) 3. No, instead of going don the cost-minimization path, e go don the one-stage pofit-maximization path. Wite don the pofit-maximization poblem and the fist ode conditions ith espect to L, K, and F. (2 points) 4. You task no is to sho that the fou fist ode conditions in the cost minimization (fom points and 2) ae equivalent to the thee fist ode condition fom the pofit maximization (fom point 3). In othe ods, pofit maximization and cost minimization ae to diffeent as to do the same thing. To do this, ou ill need to use the envelope theoem to find an altenative expession fo c(,, s, ) /. [Hint: c (,, s, ) is the objective function of the minimization pogam min L + K + sf s.t. f (L, K, F ) 0.] (6points) Solution to Poblem.. Fist, the fim minimizes the cost of the inputs: min L + K + sf L,K,F s.t.f (L, K, F ) We can assume that the budget constaint is satisfied ith equalit given the assumption fi 0 > 0 fo all i. The Lagangean is L (L, K, F, λ) L + K + sf λ [f (L, K, F ) ] and the thee fist ode conditions ae L L λf L 0 (L,K,F )0 () L K λf K 0 (L,K,F )0 (2) L F s λf F 0 (L,K,F )0 (3)

2 2. Given the cost function c (,, s, ) L (,, s, )+K (,, s, )+sf (,, s, ), e can ite the second maximization poblem as max p c (,, s, ). The fist ode condition is p c 0 (,, s, )0. (4) 3. The pofit maximization poblem is ith fist ode conditions max pf (L, K, F ) L K sf L,K,F pfl 0 (L,K,F ) 0 (5) pfk 0 (L,K,F ) 0 (6) pff 0 (L,K,F ) s 0 (7) 4. Equation (4) in the cost minimization states that the pice p equals the maginal cost c 0 (,, s, ). But e can eite the maginal cost. Conside the cost-minimization poblem, of hich the cost function c (,, s, ) is the value function. B the envelope theoem, e kno that c(,, s, ) / L (L, K, F, λ) / [L + K + sf λ [f (L, K, F ) ]] / λ. The maginal cost equals the multiplie λ! A elated esult is that in a utilit maximization poblem the multiplie λ epesents the maginal utilit of mone. Equation (4) then implies p λ.if e substitute λ fo p in equations () to (3), e obtain equations (5) to (7). Theefoe, the fist-ode conditions fo the to-stage cost minimization pocess coincide ith the fist ode conditions fo the one-stage pofit maximization. We ae done! 2

3 Poblem 2. Poduction: pefect competition (42 points) In this execise, e chaacteize the shot-un solution to the Philippino fames poblem fo the case of pefect competition. We assume that the fim has the poduction function AL K F γ. In the shot-un, hoeve, the quantit of land famed is fixed to F, so thee effectivel ae onl to factos of poduction ith espect to hich the fim maximizes.. Wite don the cost minimization poblem ith espect to L and K and the fist ode conditions ith espect to L and K (2 points) 2. Solve fo L,, s,, F and K,, s,, F. (3 points) 3. Ho does L va as inceases? Compute L / using the solution in point 2. oes it make sense? Ho about if thee is technological pogess and A inceases? What happens to L? (3 points) 4. Wite don the cost function c,, s,, F and deive the expessions fo the maginal cost c 0,, s,, F and the aveage cost c,, s,, F /. (4 points) 5. Assume + <and plot the maginal cost c 0,, s,, F and the aveage cost c,, s,, F / as a function of. [I am inteested in the shape of the cuves, not in exact daings]. a the suppl cuve fo the fim, that is,,, s, F. Is the suppl cuve (eakl) inceasing in p? (5 points) 6. Assume + and plot the maginal cost c 0,, s,, F and the aveage cost c,, s,, F / as a function of. Ho does the suppl cuve fo the fim look like? Wite it don analticall. Wh does the fim neve ant to poduce a positive and finite quantit of output despite constant etuns to scale in labo and capital? (5 points) 7. Assume + >and plot the maginal cost c 0,, s,, F and the aveage cost c,, s,, F / as a function of. Ho does the suppl cuve fo the fim look like? (3 points) 8. We no conside maket suppl and demand fo the onl ell-behaved case, that is, fo + <. Fist, let s look at aggegation of the suppl side. Thee ae poduces ith poduction function as above. Each one of them, theefoe, has a suppl function as at point 5. Assuming that e ae on the inceasing potion of the suppl cuve, e can ite the equation fo the suppl function as j () p (8) + hee We can invet expession (8) to get! " µ ( + ) p j + µ (). This is the fim suppl function povided that the pice p is high enough (e ae not checking this, enough algeba!). No, ou tun. Aggegate ove the fims to geneate the indust suppl function Y S. (2 points) 9. No, the consume side. Assume that consumes have Cobb-ouglas pefeences fo mangoes (good ith pice p) and all the othe goods (good m ith pice ). Thei utilit function is u (, m) m. Thee ae consumes, each ith income,,...,. Fom the man execises e did on Cobb- ouglas utilit it follos that in equilibium j p. This is the individual demand function fo good. Compute no the aggegate demand function Y fo the hole econom. Notice that in this special case aggegate demand does not depend on the distibution of income. (3 points) 3

4 0. Finall, the maket equilibium! Equate the demand Y and suppl Y s to solve fo the equilibium pice p M. What is p M? Plug back the value of p M into ou expession fo Y to get the maket level of poduction Y (4 points). You should get the folloing solutions " P p M M () " () j ( + ) µ + µ and X Y () A F " µ γ ( + ) + µ (). etemine no the sign of the folloing changes on p M and Y. Povide intuition. Also, make sue that the esults squae ith ou gaphical intuition about moving suppl and demand function. (8 points) (a) an incease in total income P ; (b) an incease in poductivit A. Solution to Poblem 2.. The cost minimization poblem is min L + K + sf L,K s.t.al K F γ h hee e can assume that the constaint is satisfied ith equalit since AL K F i γ / L > 0 and h AL K F i γ / K > 0. The Lagangean is L (L, K, F, λ) L + K + sf λ hal K F i γ and the fist ode conditions ith espect to the inputs L and K ae (emembe that land is fixed) λal K F γ λal K F γ We can easil tansfom the fist ode conditions in the usual a to get K L o K L. (9) We can substitute this expession fo K into the constaint to get µ A (L ) γ L F o! µ L A F γ 4

5 Using expession (9), one similal finds K! µ! µ 3. It is staightfoad to compute the compaative statics: L! µ + µ < 0. Unsupisingl, as the age inceases, the demand fo labo L deceasesasthefim substitutes aa fom labo into capital. Similal, L µ! A + µ A 2 F γ < 0. As the fim becomes moe poductive, the fim needs fee inputs to poduce the same quantit of outputs. 4. The cost function equals c (,, s, ) L (,, s, )+K (,, s, )+sf! µ µ A F γ + + sf and the maginal cost equals c 0 (,, s, ) () +! " µ + µ and the aveage cost equals c (,, s, ) / ()! " µ + µ + s F 5. Fo + <the maginal cost is inceasing in, and the aveage cost is fist deceasing and then inceasing. The suppl function is the pat of the maginal cost function above the aveage cost function, and is otheise zeo. See pictue. 6. Fo + the maginal cost is constant, hile the aveage cost is alas deceasing and highe than the maginal cost. The suppl function is degeneate: it is zeo fo pices belo p, ith p! " µ + µ and fo p>p. See pictue. This happens because in the shot-un the choice of land is fixed, and theefoe suboptimal. This geneates a fixed cost sf that the fim has to bea, and that induces the fim not to ant to poduce fo p p. 5

6 7. Fo > the maginal cost is deceasing in, and the aveage cost is deceasing as ell and alas lies above the maginal cost. The suppl function is equal to fo an pice p>0. See pictue. Given the inceasing etuns to scale, the fim ants to poduce moe and moe to achieve a moe efficient scale. efinetel, this is not compatible ith pefect competition! (unde pefect competition each fim ought to poduce a small quantit). 8. The maket suppl function is simpl the sum of the suppl functions of the fims: Y S X ( + ) p () ( + ) p () 9. The maket demand function Y is just obtained b summing ove the individual demand functions P X Y j. (0) p This aggegate demand function does not depend on the distibution of income. This depends on the fact that ith Cobb-ouglas pefeences each agent spends a shae of income on mangoes. 0. In ode to find the maket-cleaing pice, e equate demand Y and suppl Y S to obtain o o p M and, using (0), Y Y S Y X ( + ) Y S pm p () M () P p M P M j ( + ) " P M () j ( + ) " P M () " j ( + ) () () P X " M ( ()) " j () A F " γ ( + ) µ µ () µ Y + µ + µ + µ () (). Hee ae the compaative statics: (a) An incease in total income P inceases both the equilibium pice and the equilibium quantit poduced. Highe income coesponds to a positive demand shock. This inceases demand and, since the suppl cuve is upad sloping, inceases pice as ell. (b) AninceaseinpoductivitA deceases the equilibium pice and inceases the equilibium quantit. Highe poductivit induces a positive suppl shock. This inceases demand and, since the demand cuve is donad sloping, deceases pice as ell. Think at the effect of inceased poductivit of computes: qualit of computes goes up and pice goes don. 6

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