Ecnmics 11 Caltech Spring 2010 Prblem Set 4 Hmewrk Plicy: Study Yu can study the hmewrk n yur wn r with a grup f fellw students. Yu shuld feel free t cnsult ntes, text bks and s frth. The quiz will be available Wednesday at 5pm. Fllwing the Hnr cde, yu shuld find 20 minutes and d the quiz, by yurself and withut using any ntes. Paper and pen shuld be all yu need. Then turn it in by Thursday 5pm. (drp ff in bx in frnt f Baxter 133). It will include ne questin frm each sectin The answers t the whle hmewrk will be available Friday at 2pm. Definitins Please explain each term in three lines r less! Shadw value The shadw value f capital refers t the value assciated with a cnstraint, i.e. when capital cannt be adjusted in the shrt run it creates a cnstraint n the prfit. Marginal prduct The additinal utput that can be prduced by ne mre unit f a particular input while hlding all ther inputs cnstant. It is usually assumed that an input s marginal prductivity diminishes as additinal units f the input are put int use while hlding ther inputs fixed. If,, Marginal cst The cst f prducing an additinal unit f utput. Shrt run marginal cst The cst f prducing an additinal unit f utput withut changing anything yu cannt change quickly (such as building new factries). In the shrt run, sme inputs (e.g., capital) are fixed, i.e., cannt be adjusted t change utput levels. In such case (e.g., assuming that capital is nt adjustable in the shrt run), the marginal
cst given the amunt f capital used is the derivative f the ttal csts with respect t the quantity f utput. In ther wrds,,, Shrt run average cst f prductin The shrt run average cst is the shrt run ttal cst f prductin divided by quantity, where ne factr f prductin is fixed. It is just the shrt run ttal cst f prductins (i.e., hlding K fixed) divided by the utput, i.e., the average f the ttal cst in the shrt run per unit f utput: Revealed preference The preference f cnsumers can be revealed by their purchasing habits. Fixed csts Csts that d nt change as the level f utput changes in the shrt run. Fixed csts are in many respects irrelevant t the thery f shrt run price determinatin. Ttal Factr Prductivity A variable which accunts fr effects in ttal utput nt caused by inputs. Fr example, a year with unusually gd weather will tend t have higher utput, because bad weather hinders agricultural prduct. A variable like weather des nt directly relate t unit inputs, s weather is cnsidered a ttal factr prductivity variable (Wiki). Wrd prblems Please explain each questin in a few sentences. Cnsider fixed capital, what are the implicatins f the first rder cnditin f the prfit maximizatin prblem? What des the secnd rder cnditin imply? Given a fixed level f capital K, the firm chses L t maximize prfit. Frm the firm s (shrt run) prfits,, we btain the first rder cnditin: 0,,, where the left hand side f the last equality if the value f the marginal prduct f labr, and the right hand side is the wage rate. This implies that the firm will add wrkers t the prductin prcess up t the pint where a wrker just pays his salary, in the sense that the value f the marginal prduct fr that wrker equals the cst f hiring him. The secnd rder cnditin, in turn, can be written as:,.
This expressin has t be negative (r nn psitive) since, at a maximum, the slpe ges frm psitive (when the functin is increasing up t the maximum) t zer (a the maximum) t a negative number (as the variable rises past the maximum). Why d ships and taxis g at slwer speeds as a shrt run adjustment t recessins? Given that capital equipment and thus, the cst f capital is fixed in the shrt run, taxis and ships adjust t recessins by lwering variable csts. By ging at slwer speeds, the cst f gas decrease, thus lwering variable csts. Sketch the average ttal cst (ATC), average variable cst (AVC) and marginal cst (MC) f a firm (dllars vs. quantity). Why des the MC curve necessarily intersect at the minimum f the AVC and ATC curves? Maximize: Multiply thrugh by : S at the minimum f,, fllw similar steps t get intersects at the minimum f. Why des a prfit maximizing firm prduce the quantity, where price equals marginal cst, given price is as large as minimum average variable cst? Why des a prfit maximizing firm shut dwn if the price falls belw the minimum average variable cst f prductin? 0
The prfit maximizing firm maximizes prfit (shaded blue) when it prduces quantity where price equals marginal cst. The prfit maximizing firm shuts dwn if price falls belw the minimum AVC because the firm will start lsing mney. The firm s (shrt run) prfits are given by:, and thus it maximizes prfits by chsing the quantity satisfying a) 0. This is a gd strategy nly if prducing a psitive quantity is desirable, i.e. if 0,, that is, if b), where, is the average cst ignring the investment in capital equipment (recall that the cst f capital is fixed in the shrt run, and thus 0, 0). Hence, the prfit maximizing firm prduces the quantity where price equals marginal cst, prvided that p is as large as minimum average variable cst. If price falls belw minimum average variable cst, the firm shuts dwn, since it is mre cnvenient fr the firm t face the fixed csts f prducing 0 units than t prduce a psitive quantity (i.e., it lses less by prducing 0). What des 0 and 0, where, is the prductin functin and and are capital and labr, imply abut labr and capital? Prvide an example f each case and draw the respective isquants. 0 implies that and are substitutes increase in makes an increase in bst prductin less.
0 implies that and are cmplements increase in makes an increase in bst prductin mre. Suppse we have a prfit maximizing firm and is the fixed ptimum price required. What des a decline in utput price imply abut cst f prductin? What des an increase in wages imply abut labr prductivity? At the ptimum, the prfit maximizing firm prduces the quantity where the price equals the marginal cst. Ceteris paribus, a decline in price thus will require a crrespnding decrease in the marginal cst f prductins and hence a decline in the cst f prductin. Ceteris paribus, an increase in wages has t be matched by an increase in labr prductivity, since prfit maximizatin requires that,.
Technical prblems Suppse a cmpany has ttal cst given by where capital is fixed in the shrt run. What is the shrt run average ttal cst and marginal cst? Plt these curves. 2 Fr a given quantity, what level f capital minimizes ttal cst? What is the minimum average ttal cst f? Minimize ttal cst wrt : 0 2 2 2 Minimize average ttal cst wrt : 0 2 2 2 Suppse capital can be adjusted in the lng run. Des this cmpany have an increasing return t scale, decreasing returns t scale r cnstant returns t scale? S we ptimize the ttal cst with respect t and plug in: 2 2 2 2 1 2 2 S the average ttal cst is cnstant independent f s cnstant return t scale.
Prfessr Smith and Prfessr Jnes are ging t write a textbk tgether. Their prductin functin fr the bk is / /, where is the number f pages in the finished bk, is the number f wrking hurs spent by Smith, and is the number f hurs spent wrking by Jnes. Smith values his labr as $3 per wrking hur. He has spent 900 hurs preparing fr the first draft. Jnes, whse labr is valued at $12 per wrking hur, will revise Smith s draft t cmplete the bk. Hw many hurs will Jnes have t spend t prduce a finished bk f 150 pages? Of 300 pages? Of 450 pages? 3 12 Plug in 3 12 Minimize ttal cst with respect t : 312 0 2 2 But we als need t stipulate that 900. If 150, then Smith has already wrked mre than the equilibrium amunt s we just need the smallest TC even thugh it wn t be a minimum. Fr 900, 312 312/360 S 900 Similarly, if 300, and 900 S 312 150 900 25 900 300 100 900 312/90 And if 450 and 900, then 312 3 12 4 0 S 900
450 225 900 What is the marginal cst f the 150 th pages f the finished bk? Of the 300 th page? Of the 450 th page? In the previus part we shwed that Smith desn t put in any mre hurs fr 450 s 30 900 12 2700 12 2700 900 24 900 If 150, 4. If 300, 8. If 450, then 12. A lawn mwing cmpany uses tw sizes f mwers t cut lawns. The smaller mwers have a 24 inch blade and are used n lawns with many trees and bstacles. The larger mwers are exactly twice as big as the smaller mwers and are used n pen lawns where maneuverability is nt s difficult. The tw prductin functins available t the cmpany are 1 : Output (square feet) Capital input (# f 24 mwers) Labr input Larger mwers 8000 2 1 Small mwers 5000 1 1 Graph the 40000 square feet isquant fr the first prductin functin. Hw much and wuld be used if these factrs were cmbined withut waste? The isquant curves fr large mwers lk like: 8000 min, Set 40000 8000 min,, we get 10,5 Answer the previus part fr the secnd functin. 1 Nichlsn s Micrecnmic Thery
The isquant curves fr large mwers lk like: 5000 min, Set 40000 5000 min,, we get 8,8 Hw much and wuld be used withut waste if half f the 40000 square ft lawn were cut by the methd f the first prductin functin and half by the methd f the secnd? Hw much and wuld be used if 3/4 f the lawn were cut by the first methd and 1/4 were cut by the secnd? What des it mean t speak f fractins f and? Assuming we can nly use integer numbers f lawn mwers: 1) Half by first methd, half by secnd: We wuld cut 24000 with the first mwer and 20000 with the secnd mwer. First mwer: 3min, 6,3; secnd mwer: 4 min, 4,4. There wuld be a waste f 4000 square feet mwed. 2) by first methd, by secnd: We wuld cut 32000 with the first mwer and 10000 with the secnd mwer. First mwer: 4min, 8,4; secnd mwer: 2 min, 2,2. There wuld be a waste f 2000 square feet mwed. On the basis f yur bservatins in the previus part, draw a 40000 isquant fr the cmbined prductin functins. Assuming and are indivisible, t draw the 40000 isquant we wuld need t enumerate integer cmbinatins f large and small mwers that cver at least 40000 square feet. If and culd be divisible we can graph the isquant as fllws: We can get an expressin fr the cmbined and : Let be the fractin f 40000 cut by mwer 1. 10 81 82 581 83
We can get an equatin f in terms f and by substituting frm the secnd equatin int the first equatin: 82 82. Suppse a gardener is ging t g int business. She has access t bth technlgies what is the ptimal number f mwer f each type that she shuld purchase in rder t minimize csts? When she hires wrkers r rents mwers she had t d it in whle hurs the wage is $10 per hur; the rental cst f a unit f capital is $5 per hur. If we assume nly integer and the cst functin can be pltted by making a table f cmbinatins f small and large mwers that will minimize cst fr different intervals f area mwed: (square feet) Small mwer Large mwer Cst(q) 1 0 15 0 1 20 2 0 30 1 1 35 In fact past 13K the best slutin is the interger value f K/8k large mwer and fr the residual a small mwer if it is less than 5K and a large n if it larger than 5K. Hwever, if we assume and are divisible we can derive the fllwing clsed frm f the cst functin: Cst functin: 10 5 Fllwing steps similar t the previus part: 4 4 20000 20000 83 20000 4 83 40000 40000 2000 3 2 Given is a fixed cnstant, we want t minimize the cst, s substituting the abve equatin int the cst functin:
10 2000 3 2 5 10 15 200 15 Increasing capital lwers cst, s the gardener shuld nly use large mwers.