Economic Growth and Development ECGA 6470 Darryl McLeod Government and Economic Growth (single page) Spring 2012

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1 Eonomi Growth and Development ECGA 6470 Darryl MLeod Government and Eonomi Growth (sinle pae Sprin 202 One ontroversial issue is the effet of overnment spendin on eonomi rowth. Does infrastruture spendin enhanes produtivity and inrease eonomi rowth or "rowd out" private spendin reduin rowth and effiieny? The empirial evidene is mixed: some studies find publi and private investment is omplementary while others find publi onsumption spendin and even publi investment redues private investment. Some insiht into why the empirial evidene miht be mixed an be derived from a simple model of eonomi rowth proposed by Barro (990. Startin with the Robelo or "AK" endoenous rowth model, Barro suests human plus physial apital ("K" an be made more produtive by addin just the the riht amount of publi infrastruture servies ("G". Usin the Cobb-Doulas prodution funtion Y = G K ß, with onstant returns +ß = so that ß = -. The ath is that spendin on G must be finane by an inome tax on inome enerated by physial + human apital owned by private aents, K. So assumin an overnment budet that must be balaned in the lon run, G = Y where is the inome tax rate. Usin a onstant relative risk aversion utility funtion (see the "Three rowth Models" handout, the per apita rowth rate is then, = (/θ(a - ρ, where /θ is onsumers onstant IES and ρ is the rate of time preferene or disount. In the "AK" model A is just the marinal produt of apital, f k. Now addin overnment hanes the model in two ways. First private "after tax" inome from or marinal produtivity of apital is now ( - f k. Seond, havin the riht amount in infrastruture makes apital more produtive. Hene our modified rowth equation beomes, = (/θ[( - f k - ρ] Clearly an inrease in the tax rate now has two opposite effets on rowth: it redues the inome available to the private setor but over a ertain rane more publi investment raises the MPK, f k. Substitutin y into the prodution funtion (lower ase y is per apita GDP, γ is per apita infrastruture stok, et. and a little alebra yields the new f k = A* and rowth is now, = (/θ(a* - ρ where A* = (- A /(- * /(-. Clearly, A* rises with even as ( - or after tax inome falls. The net result is that for while raisin taxes and spendin in γ raises the rowth rate, but after a ertain point raisin further redues rowth. As it happens is the optimal tax rate (overnment spendin share of national inome is. Any inrease of the tax rate above, a tehnoloy parameter, redues rowth. This may explain why some authors find rowth is affeted neatively by overnment spendin while others find a positive effet (see for example Barro, 99 who finds a positive effet. A similar approah is used by Basu and MLeod (99 to inorporate terms of trade effets into an endoenous rowth model. Further Readin: Barro, R., 990, Government Spendin in a Simple Model of Endoenous Growth Journal of Politial Eonomy 98, Ot, S Barro, R.J. "Eonomi Growth in a Cross Setion of Countries" Quarterly Journal of Eonomis, 06, 99. Barro, R.J. (996 Demoray and Eonomi Growth Journal of Eonomi Growth,:-27 (Marh, 996 Basu, P. and D. MLeod (99 "Terms of Trade Flutuations and Eonomi Growth" Journal of Development Eonomis 37:. Fisher, S. (99 Growth, Maroeonomis and Development, NBER Workin Paper #3702, May. Aemolu vs. Glaeser (2007 Demoray and Eonomi Growth exhane in the Wall Street Journal.

2 ECGA 6470 Growth and Development Sprin 2008 Barro s model of overnment spendin and taxation Barro (990 suests a simple endoenous rowth model with overnment. Departin from the standard haraterization of overnment onsumption finaned by distortionary taxes as in Easterly (2005 in Barro s model publi investment (roads, ports, sanitation, shools, et. omplements private investment. Government spendin is finane by a straiht inome tax. Sine publi investment raises the produtivity of private investment, hiher taxes an be assoiated with an inrease or a derease in overall rowth. But beause the private setor inores the additional tax revenues and publi investment enerated by its private investment, it tends to invest too little. Beause it is a lose ousin of the AK model this model provides a very tratable framework for many problems suh as imported investment oods in Basu and MLeod (992. Barro s starts with the AK model and adds publi investment or apital : y = Ak - where = y, where is overnment spendin per apita is determined by the tax rate on inome per apita y sine the budet is always balaned. Usin the overnment budet onstraint to substitute for in (A we obtain, y = A(y k - brinin y over to the LHS, y - = Ay k - raisin both sides to - yields y= A k = A*( k where the new MPK in brakets A*( plays a role analoous the A in the AK model, exept that it depends on the tax rate. One miht think the rowth rate γ = /θ [A*- ρ] where aain intertemporal elastiity of substitution is /θ and ρ is the disount rate. But reall that the private setor only reeives after tax inome on apital investment, or (-(-A*(. Note that overnment has two effets on private inome, reduin after tax inome (- but also raisin the marinal produt of apital A*( due the ontribution of to total output for any iven private apital stok k. At some point these two effets balane out and we obtain the rowth maximizin tax rate whih is, the optimal share of implied by our Cobb-Doulas prodution funtion (analoous to the olden rule savins rate s =. The rowth maximization problem has another twist however. The private setor does not take into aount that the taxes it pays on investment inome inrease (or perhaps they don t believe it it will under save/invest and the rowth rate will be too low. The soially optimal rowth rate is (-A*( > (-(-A*( so the deentralized rowth rate is lower that the optimal or entral planners rowth rate. Note also that if = 0, = 0 and we have a publi investment poverty trap as the private MPK = 0 and the rowth rate is neative. Of ourse if the overnment takes the whole national produt so that = the rowth rate also oes to zero sine there is no private investment. The optimal tax rate is obviously somewhere in between 0 and. To solve the households (and planners optimal rowth problem more expliitly, assume infinitely lived households own firms and produe output usin a prodution funtion that benefits from per apita

3 infrastruture investment (roads, ports, dams, et. Households maximize a onstant relative risk aversion or CRRA utility u( t, θ t ρt max e dt subjet to k t = ( yt t where θ 0 y = Ak and 0 < <. t t t Per apita publi spendin on infrastruture is finaned by a proportional tax on inome so = y. The overnment does not borrow so the budet is always balaned. All variables are per apita but note that labor is not required to produe y but we assume k inludes human as well as physial apital. To solve maximization problem of a representative, infinitely lived private aent who ontrols k and onsumes takin and as iven (set by the overnment we an use a present value Hamiltonian solvin for the present value of onsumption subjet to the aumulation onstraint in brakets: θ t H = e + μ[ ( y ] where y = Ak θ ρt t t t t Sine the private setor only ontrols k, we differentiate H with respet k t and t only to obtain the first order onditions (FOCs: θ ρt H : t e μ = 0 ( H k : μ[ ( ( Ak ] = μ (2 H : ( y = k where aain, y = Ak μ [ ] t t t t The transversality ondition is lim [ ( t k( t ] = 0 t. μ whih the dynami version of the Inada onditions. To solve for the rowth rate of onsumption first take lo of ( to obtain: θ *ln ρ* t = lnμ. Takin the time derivative of this equation yields: μ θ ρ = μ t Dividin both side of (2 by μ ives [ ( Ak ] (3 μ ( =. Usin this expression to replae the μ ( ( Ak = θ ρ RHS of (3 yields: [ ] Solvin for we et the deentralized ompetitive solution: ( ( Ak ρ or ( A* ρ where A* ( ( Ak = θ = = θ = MPK (4 This expression says the rowth in onsumption depends on (i the ap between the marinal produt of apital and the rate of time preferene, ρ and (ii on the IES /θ. The hiher the θ, the more value

4 individuals plae on smoothin onsumption. The hiher θ is, the hiher return to apital (MPK it takes to enourae investment whih is what raises the rowth rate of onsumption (and output y. It appears that the blue MPK term in (4 depends on k, but it does and should not sine this an AK lass endoenous rowth. To see this reall that = Y so that = Ak ( Ak, solvin for yields, = = ( A k whih implies that = ( A k. Substitutin into the blue MPK term above auses the k terms to drop out leavin, MPK = ( ( A = A*( Hene the after-tax marinal produt of apital alulated above, A*(, plays the same role as the A in the AK model (exept that it depends on the tax rate. The expression onsumption rowth equation (4 shows that overnment affets the marinal produt of apital throuh two hannels: inreases in raise the MPK up to a point, but taxes always redue the private return to apital, the task of a ood overnment is balane these two effets: MPK = ( ( Ak neative effet of taxation positive effet of publi servies. (5 To solve the model from the point of view of a benevolent rowth maximizin overnment (entral planner we know that the planner satisfies the ondition δy/δ = ( marinal produt of overnment spendin or MPG. The overnment should provide publi investment or servies until the MPG equals one. So the MPG must equal overnment spendin : Ak = MPG spendin Solvin for we find that / = (y/= /, so that = when MPG =. The key differene between the deentralized and the entralized solution, is that the firm responds to the the after-tax private marinal produt MPK= (- y/ k while the planner takes into aount the soial marinal produt ( y/ k: additional private inome also raises taxes and as well as k. To et the soial planner s rowth rate, eliminate the (- from (5 to find the planner s MPK and we et that MPK = ( A where substitutin = yields MPK = ( A So the planner should set the tax rate to maximize rowth rate and the red MPK, = θ (( A MPK ρ. (6

5 Note the MPK in the deentralized solution is (-( y/ k, it is smaller than the soial marinal produt y/ k, beause of the tax rate. This ap between soial and private returns leads to a lower rowth rate in the deentralized solution summarized by (4 ompared to the soially optimal rowth shown in eq. (6. Note however, that whether the relevant rowth rate is ditated by (4 or (6, the rowth maximizin tax rate is =. For the deentralized solution, a benevolent overnment would want to maximize the utility of individuals with a ertain size of the overnment,. To find this, we differentiate the followin rowth rate (ombination of 4 and 5 with respet to : = ( ( MPK A ρ θ Differentiatin: + = ( ( ( / / ( A A =0 Simplifyin: = ( ( ( A A, ( ( =, whih an only hold if =. The ondition that = orresponds to the olden rule savins rate in the Solow model (reall that s = in problem set # here the tax rate is the savins rate for publi owned apital, so is the optimal tax rate in both solutions. Graphially, the optimal tax rate = is /

6 A note on Demoray and Eonomi Growth There is lare (inonlusive literature on whether demoraies have hiher rowth rates. Assumin taxpayers vote and payin taxes is painful (distortionary the optimal tax rate for a demoray is likely to be a bit lower than with no fritions (lower than in the above example. If voters (taxpayers have no influene on the other hand, and authoritarian rulers benefit diretly from hiher tax revenues (e.. hiher rents or salaries the optimal tax rate for an authoritarian reime is likely to be hiher than. The impliation of these aruments is summarized in the diaram below, the demorati and authoritarian reime and may have similar rowth rates, but the overnment setor is likely to be smaller with a demorati reime. This would be an interestin hypothesis to test, all else equal. Further Readin: Barro, R. (990 Government Spendin in a Simple Model of Endoenous Growth Journal of Politial Eonomy 98, Otober, S03-S25. Barro, R.J. (99 "Eonomi Growth in a Cross Setion of Countries" Quarterly Journal of Eonomis, 06, 99. Basu, P. and D. MLeod (992 "Terms of Trade Flutuations and Eonomi Growth in Developin Eonomies" Journal of Development Eonomis 37:. Fisher, S. (99 Growth, Maroeonomis and Development, NBER Workin Paper #3702, May. Easterly, W. et. al. (994 "Good Poliy or Good Luk? Country Growth Performane and Temporary Shoks" Journal of Monetary Eonomis.