ac p Answers to questions for The New Introduction to Geographical Economics, 2 nd edition Chapter 3 The core model of geographical economics

Transcription

1 Answes to questions fo The New ntoduction to Geogaphical Economics, nd edition Chapte 3 The coe model of geogaphical economics Question 3. Fom intoductoy mico-economics we know that the condition fo pofit maximization fo a fim is MC = MR, that is maginal costs equals maginal evenue. Unde pefect competition this condition implies that MC = p, that is maginal cost is equal to the pice of a good (maginal cost picing. Now use Figue 3.8 to show that with the aveage cost cuve in the coe model (use (3., maginal cost picing always esults in a loss fo the fim; implying that impefect competition is the dominant maket fom. Answe 3. p m D ac mc ac p Above the pictue is dawn again; p = pice, mc = maginal costs, ac = aveage costs, D = demand cuve, m = maginal evenue associated with D. The aveage cost cuve is a declining function of x, due to the fixed costs. Because the maginal costs ae constant the ac-cuve lies eveywhee above the mc-cuve. A thought expeiment is illuminating. What happens if a fim behaves like a pofit maximizing fim in a pefect competitive maket and sets its pice equal to maginal costs, p = mc (= m in the pefect competition case? The x

2 demand cuve gives the sales associated with p = mc. The aveage cost cuve gives the aveage cost coesponding with this level of poduction. Now, because the aveage cost cuve is always above this pice level (which equals mc the gey aea gives the total loss of ou fim with this picing stategy. So, pefect competition is not consistent with this fom of inceasing etuns to scale. What maket fom is? One option could be a monopoly. Pofit maximization in this case (now the fim can influence the pice level equates m = mc ( p. Fom the demand cuve we can infe the pice that coesponds with this level of sales. The dotted aea gives total pofits fo the monopolist. So, a monopoly could be consistent with this fom of inceasing etuns to scale. Othe foms of impefect competition can also be consistent, and as we have seen monopolistic competition is a vey ewading maket fom fo the Economic Geogaphy appoach. Question 3.. Stat again fom the example in section 3., but now assume that each fim has the possibility to open a second plant in the othe egion. Each fim minimizes the combined costs of setting up a second plant and tanspotation costs. Suppose setting up a fim costs units. Decide whee to locate given the location of the othe fims. a. f all othe fims have a single fim in South, what is optimal fo ou fim? b. Suppose all fims have two plants, one in each location; what is optimal fo ou fim? Answe 3. Location and Tanspotation costs Fim in N All fims in N 0 + = (to fames in S All fims in S 4 + = 6 (to wokes and fames in S 5% fims in N 75% fims in S 3 + = 5 (to wokes and fames in S Fim in S = 8 (to wokes and fames in N = 4 (to fames in N + 4 = 5 (to wokes and fames in N The table above epeats table 3. fom the example. Fom the example we aleady leaned that if all fims ae located in the South (S it is optimal fo a fim also to locate in the South.

3 But, can it be wothwhile to stat business in the Noth (N if set-up cost of a subsidiay is? n this case it is because the fim can foego tanspot costs of 4 by investing to set up shop in N. f all fims wee located in N a fim would be indiffeent (why?. b. With an equal division of wokes ove both egions, the lagest maket is in N, because most of the fames ae located thee. That by itself makes it an attactive location. Tanspot cost to ship to S in this case ae (fames in S+(wokes in S=4 > (set up cost of a second plant in S. So ou fim also decides to have two plants, one in each location. Question 3.3* Suppose we stat with a situation of complete agglomeation of manufactuing poduction in egion, that is λ =. Without calculating the equilibium values explicitly one might suspect that W = is the equilibium value in this case. Substituting this in the income and pice equations we find: ( + δ ( Y = Y = = = T Using these values, W = is indeed an equilibium value fo wages in egion, as can be veified fom (3.4. Real wages also equal in egion. Calculate unde which condition this is always a long un equilibium no matte how lage tanspotation costs become. Hint: use the expession fo eal wages in egion fo this case and let T become abitaily lage. Show that this only happens if ( ε εδ <0. Answe 3.3 We only have to find out whethe eal wages ae smalle than one o lage. nseting the values fo income and pice indices into the eal wage equation (3.8 gives: (* w = T + δ T δ + T ε / ε This equation gives the eal wage fo egion. The fist tem on the ight hand site states that eal wages in a egion tend to be lowe the geate the distance to the impot makets. n this case egion has to impot all its manufactued goods fom egion, the manufactuing cente in this set-up. Because T>, this tem is smalle than one, which makes egion a less attactive place fo manufactuing wokes. The tem between backets is the (nominal 3

4 wage ate, which is consistent with zeo pofits fo each fim. Both income tems ae weighted with tanspotation costs. The income in egion is weighted by T -ε <. Fo a fim in egion this means that income in the othe egion is of less impotance due to the tanspotation disadvantage of a fim in egion elative to a fim in egion. The income in egion is weighted by T -ε- >. This is the opposite effect fo a fim in egion ; income of this egion is of moe impotance to him than to a fim based in egion, due to the location advantage of a egion fim compaed to a fim located in egion. Fo T=, i.e. if thee ae no tanspotation costs, also w =. n the fee tade equilibium location does not matte. What happens to w if tanspoation costs ise. Diffeentiating the eal wage equation above and evaluating it aound w =, and T=, we find: dw dt δ ( ε = < 0, fo small tanspotation costs. Fo high tanspotation costs the ε analysis is slightly moe complicated. nspecting (* we immediately see that the fist tem between backets becomes abitaily small, if T becomes abitaily lage. But what happens to the second tem, depends on the sign of ( ε εδ. f this tem is smalle than zeo, this tem also becomes abitaily small and complete agglomeation is always sustainable. Howeve, if ( ε εδ > 0, the second tem in equation (* becomes abitaily lage if T becomes abitaily lage. So, complete agglomeation is not sustainable fo lage values of T. But, we also established that fo elative small values of T, complete agglomeation is always sustainable. n chapte 4 we will etun to the pecise meaning of ( ε εδ > 0. Question 3.4* Suppose a monopolistic poduce located in egion can eithe sell in egion o in egion. Let p (p be the pice chaged in egion (espectively in egion, and let x (x be the demand in egion (espectively in egion. Obviously, the demand functions depend on the pice chaged in eithe egion. Poduction equies only labo as an input, which is paid wage ate W, and benefits fom intenal etuns to scale, using α fixed labo and β We abstact fom the complication that in fact complete agglomeation is always an equilibium. The discussion in the text assumes that it it possible to compae the manufactuing wages in the two egions with each othe; with complete specialization this compaison is impossible, because thee is no 'othe' wage. 4

5 vaiable labo. Finally, thee ae (icebeg tanspot costs: the fim must poduce Tx units to ensue x can be sold in egion, with T >. The fim s pofit π and demand functions x and x (ε >, while Y and Y ae constants ae given below: π = p x + p x W ( α + β x + βt x x = p Y ; x = p Y Fist, give some comments on the pofit function above. Second, substitute the demand functions in the pofit function. Thid, detemine what the optimal pices p and p ae, that is solve the pofit maximization poblem. Fouth, show that p = Tp, that is the optimal pice chaged in egion is exactly T times highe than the optimal pice chaged in egion. Answe 3.4 Fist, we note fom the specification of the pofit function that the entepeneu can independently choose the pice chaged in each egion (see also the emak at the end of this answe. Second, if we substitute the demand functions in the pofit function we get: π = p W ( α + β p Y + βt p Y + p Y Y Thid, we can diffeentiate this pofit function with espect to the pice to be chaged in eithe egion and equate the esult to zeo to get the fist ode conditions fo pofit maximization (that is to detemine the optimal pice to be chaged in both egions: π / p = 0 ( p Y + βwε p Y = π / p = 0 ( p Y + βtwε p Y = Solving these fist ode conditions detemines the optimal pices to be chaged: p ε = βw ; ε p ε = βtw ε Fouth, it immediately follows that the optimal pice chaged in egion is exactly T times highe than the pice chaged in egion. This is impotant because it implies that consumes o othe economic agents ae not involved in paallel impots into egion, as thee ae no pofits in this activity. The fact that the pice elasticity of demand is the same in the two egions is cucial in this espect. f the pice elasticity of demand is not the same in the two egions, we must eithe assume that makets ae segmented, such that the fim can 0 0 5

6 detemine the optimal pice in each egion without fea of paallel impots, o that the pice chaged in the two egions is detemined taking into account the possibility of paallel impots. Question 3.5* n the example of section 3. we showed that some equilibia ae bette fom a welfae pespective than othe equilibia. Can you show this using (3.-(3.4 and (3.'-(3.4'? Assume that the fames ae not symmetically distibuted ove both egions. Suppose egion has /3 of all fames and egion, /3 of all fames. Can you show that fom a welfae point of view, agglomeation in egion is bette than agglomeation in egion, as might be expected because egion potentially has the lagest maket. Hint: make sue that complete agglomeation in egion and complete agglomeation in egion ae both equilibia. Use the esulting equations to show that (U indicates utility ( δ U λ = = δ + + δ T 3 3 Fo λ = we have: ( And fo λ = 0: U δ + ( δ λ = 0 ( δ = + T 3 3 A moe in depth analyses of welfae is given in Chapte 4. Answe 3.5* What can we say about welfae? n the example we see that multiple equilibia ae possible, but also that a specific equilibium might be bette than anothe, fom a welfae point of view. Fom the indiect utility function in section 3.3. we see that welfae depends on (nominal income and the consumption-based pice index. The only income in this model is labou income. Futhemoe, we have fou diffeent goups in ou economy: manufactuing wokes in both egions and Food poduces in both egions. To give an example, we can deive welfae fo each of the equilibium situations we have studied above: complete agglomeation and the symmetical equilibium. Fo each equilibium the economy-wide 6

7 welfae is simply the sum of labou income fo each goup times the elevant pice index fo that paticula goup. ( δ ( δ ( λ δw 3 3 U total = λδw Note, that the eal income of the Food poduces might also diffe between the two egions, because it also mattes fo them whethe o not they have to impot manufatues. Again we can follow the pocedue in the text and guess that fo complete agglomeation in egion, W = is indeed an equilibium (veify this using 3.4. Fo complete agglomeation in (δ + ( δ egion we then have Y =, Y =, and using 3 3 =, = T and W =: ( δ U λ = = δ + + δ T 3 3 we have: ( Similaly, fo complete agglomeation in egion we get: U λ = 0 ( δ = δ + ( δ + T 3 3 Subtacting the two expessions shows that agglomeation in egion is bette fom a welfae point of view. Question 3.6* This question was answeed in the fist edition of ou book, see below. 4.6 Nomalization analysis n this section ou claim about the small impact on the shot-un equilibium and the eal wage of changes in the paametes L, α, and β will be analyzed. Fo ease of efeence we epeat equations (4.-(4.4 below. (4. Y = λ W γl + φ ( γ L (4. (4.3 (4.4 W s w s β γl = R ρ αε /( /( λsts Ws s= / ε R ρ δ = ρβ ( ε α = = Ws s Y T ε s / ε 7

8 Fist, suppose the labo foce L inceases by some multiplicative facto, say θ, taking the distibution of the labo foce as given. Assume that the wage W does not change. Fom equation (4. it then follows that income in each egion changes by the same facto θ, while equation (4. shows that the pice index in each egion inceases by the facto θ /(. Using these two esults in equation (4.3 shows indeed that the wage in each egion does not change. The eal wage in each egion theefoe changes equipopotionally by the facto θ δ /(, see equation (4.4, such that the distibution of elative eal wages is not affected. Recalling the one-to-one elationship between the size of the manufactuing wok foce in a egion and the numbe of vaieties poduced in that egion, see equation (3.5, it is obvious that an equipopotional incease in the manufactuing wok foce in all egions allows fo a lage ange of vaieties to be poduced in each egion. This dives down the pice index in all egions equipopotionally, which in tun inceases the eal wages equipopotionally though a love-of-vaiety effect. Second, suppose the fixed cost of poduction α incease by a multiplicative facto θ fo all egions. Assume, fo the sake of agument, that the wage does not change. Fom equation (4. it follows that income does not change, and fom equation (4. that the pice index inceases by the facto θ /(. Using these two esults in equation (4.3 shows that the wage in each egion indeed does not change. The eal wage in each egion theefoe changes equipopotionally by the facto θ δ /(, see equation (4.4, such that the distibution of elative eal wages is not affected. The intuitive easoning is vey simila to the fist case descibed above. Just like a ise in the labo foce, a fall in the fixed costs of poduction in a egion allows fo an incease in the numbe of vaieties poduced in that egion, see equation (3.5. f this holds fo all egions it dives the pice index down in all egions equipopotionally, thus inceasing the eal wages equipopotionally though a love-of-vaiety effect. The tem "distibution of elative eal wages" is moe appopiate in a multi-egion setting than in a - egion setting. Poposition 4. below, howeve, applies fo an abitay numbe of egions. 8

9 Thid, suppose the maginal cost of poduction β incease by a multiplicative facto θ fo all egions. Assume, fo the sake of agument, that the wage W does not change. Fom equation (4. it follows that income in each egion does not change, and fom equation (4. that the pice index inceases by the facto θ. Using these two esults in equation (4.3 shows again that the wage in each egion indeed does not change. The eal wage in each egion theefoe changes equipopotionally by the facto δ θ, see equation (4.4, such that the distibution of elative eal wages is not affected. Quite simila to the above two cases, the eal wage ises if the maginal labo cost of poduction fall by loweing pices and the pice index, see equations (3. and (3.7. The above easoning poves Poposition Poposition 4.. Suppose that (Y,,W,w solves equations (4.-(4.4. Then a change in the size of the population L o the manufactuing cost function paametes α and β by a facto θ changes this solution to (θy, θ (Y,θ,W, δ θ /(,W, θ δ /( w, (Y, θ /(,W, θ δ /( w, and w, espectively. The equipopotional change in the eal wage implies that the paametes L, α and β essentially do not influence the dynamics and stability of the model. These paametes do, howeve, influence the eal wage (=welfae level. Table 4. Paamete nomalization γ = δ L = β = ρ α = γl/ε Now that it has been established how the shot-un equilibium eacts to changes in the paametes α, β and L, and that these paametes affect the eal wage in all egions equipopotionally, thus essentially not influencing the dynamics of ou model, we ae in a position to suitably choose these paametes to simplify notation following Fujita, Kugman and Venables (999, see Table 4.. Note that we have taken the libety to 3 See Neay (00 fo a discussion. 9

10 nomalize γ by setting it equal to δ (see also poblem 4.. Thee ae two justifications fo this. Fist, as demonstated in Figue 4., panel c, this paamete has vitually no impact on the elative eal wage. Second, as the eade may wish to veify, by setting γ = δ the eal wage fo the mobile wok foce is the same as the eal wage of the immobile wokfoce in the speading equilibium. Obviously, that agument only has some weight if the speading equilibium is established in the long-un. To the extent that γ has some impact on the elative eal wage, and thus the dynamics of the system, its nomalization is questionable. Note, finally, that with this nomalization, using ε = /( ρ, equations (4.-(4.3 simplify to (4.'-(4.3'. (4.' Y = δλ W + ( δ φ (4.' (4.3' W s = R s= λ T s s R = YT = W ε s /( s / ε Question 3.7* Following the instuctions of the question exactly and compae you steps to the pocedue used in Technical Note 3.8 should lead you to the stated esult. f you ae having difficulties bush up on you diffeentiation skills. 0