International Monetary Economics Note 1
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1 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated with wold capital makets and that takes as given a constant wold eal inteest ate >0. The consume is bon at date t =0and lives until t = T with pefeences U(c) ove the consumption vecto c =(c 0,c,,c T ) Fo simplicity, I assume that the consume discounts the futue at a geometic ate and has time-sepaable pefeences of the fom U(c) = u (c 0 )+βu(c )+β 2 u(c 2 )+ + β T u(c T ) = β t u(c t ) This consume faces a sequence of flow budget constaints, each of the fom B t+ B t = B t + y t c t i t g t The change in net foeign assets B t+ B t is the county s cuent account balance. If B t+ >B t, the county uns a cuent account suplus in date t while if B t+ <B t,the county uns a cuent account deficit. Govenment expenditue {g t } is a known exogenous sequence. The sum B t +y t is Goss National Poduct (GNP) with Goss Domestic Poduct (GDP) denoted by y t. GDP is detemined by the physical capital stock (labo is not a facto of poduction) accoding to a poduction function y t = F (k t ) Investment is the change in the capital stock net of depeciation, i t = k t+ k t δk t whee δ denotes the depeciation ate. I will assume that δ =0so that physical capital neve depeciates. This implies that i t = k t+ k t The initial capital stock k 0 > 0 is a given paamete of the model. Choosing an investment plan is equivalent to choosing a sequence of capital installations {k t+ }. Intetempoal budget constaint The sequence of flow budget constaints can be integated to give a single intetempoal (o pesent value) budget constaint. This is done by ecusive substitution. The basic idea is to continuously eliminate the futue asset tems, B t+, fom the constaints. Mechanically, B = ()B 0 + y 0 c 0 i 0 g 0 B 2 = ()B + y c i g
2 Substituting B into the second equation gives B 2 =()[( + )B 0 + y 0 c 0 i 0 g 0 ]+y c i g Now wite out an expession fo B 3 B 3 = ()B 2 + y 2 c 2 i 2 g 2 = (){( + )[( + )B 0 + y 0 c 0 i 0 g 0 ]+y c i g } + y 2 c 2 i 2 g 2 Moe geneally, fo any t B t+ =() t+ B 0 + tx ( + ) t s (y s c s i s g s ) s=0 Dividing thoughout by the common facto ( + ) t, evaluating at t = T and eaanging gives the intetempoal budget constaint µ t µ T c t + B T + =()B 0 + Intetempoal optimization The consume s poblem is to choose a consumption vecto c and an investment plan to maximize he utility function subject to the budget constaint, the poduction function, and the definition of investment. The Lagangian fo this poblem is L = = β t u(c t )+λ ( + )B 0 + β t u(c t )+λ ( + )B 0 + (y t i t g t c t ) (F (k t ) (k t+ k t ) g t c t ) µ T B T +# µ T B T +# whee λ denotes a Lagange multiplie. The fist ode conditions that chaacteize this poblem include µ t L = 0 β t u 0 (c t ) λ =0 each t c t µ t µ t+ L = 0 λ + λ F 0 (k t+) =0 each t k t+ (We can also deive the obvious conclusion that B T + =0by noting that thee is a cost to acquiing assets in the last peiod but no offsetting benefit). The optimality conditions can be eaanged to give the familia consumption-smoothing condition and the equiement that investment take place up to the point whee the maginal poduct of capital equals the given wold eal inteest ate. In this notation, u 0 (c t ) = β( + )u 0 (c t+ ) = F 0 (k t+ ) We can invet the last condition to solve fo the capital stock in tems of. When is constant, k t+ is constant at some k =(F 0 ) () too. With a constant exogenous wold eal inteest ate, capital accumulation is not detemined simultaneously with consumption. 2
3 The consumption function To solve fo consumption, we have to combine the fist ode condition u 0 (c t )=β( + )u 0 (c t+ ) with the budget constaint c t =()B 0 + (I have used the fact that B T + =0). Example. Suppose that β( + ) =so that the discount ate ρ β is equal to the wold eal inteest ate. Then u 0 (c t )=u 0 (c t+ ) implies that c t = c t+ = c each t We still need to solve fo this level c of consumption. Substituting into the budget constaint c =()B 0 + Since c is the same fo all t we can pull it outside of the sum c =()B 0 + Now evaluating the sum on the left hand side gives (fom a standad fomula fo geometic seies, P n i=0 xi = xn+ fo 0 <x<), x = T + = ( + ) (T +) µ So ou consumption function is c = ( + ) (T +) ( + )B 0 + # (Recall that k =(F 0 ) () so that eveything on the ight hand side can be witten in tems of exogenous vaiables). This is a vesion of the pemanent income hypothesis. The 3
4 main deteminant of consumption is intetempoal wealth (o pemanent income). The maginal popensity to consume out of wealth depends on T As T become lage, ( c = lim ( + )B T ( + ) (T +) 0 + #) = ( + )B 0 + # Thus in the long-hoizon limit, consumption is simply popotional to intetempoal wealth. The infinite-hoizon model In this case, the consume s pefeences ae odeed by U(c) = u (c 0 )+βu(c )+β 2 u(c 2 )+ = β t u(c t ) which is well defined if 0 <β<and the peiod utility function is eithe i) bounded, o ii) such that consumption does not gow too fast. The natual infinite-hoizon budget constaint is µ t µ T c t + lim B T + =()B 0 + T In ode to make this well defined, it is standad pactice to impose a no-ponzi-game constaint of the fom µ T lim B T + 0 T to ensue that the consume cannot oll-ove debt continuously. This leads to the equiement that the pesent value of consumption satisfy c t ( + )B 0 + (Of couse, if u(c t ) is stictly inceasing in c t this will always hold with equality). The same fist ode conditions can be obtained, namely, u 0 (c t ) = β( + )u 0 (c t+ ) = F 0 (k t+ ) 4
5 Example 2. Now suppose that peiod utility has the isoelastic fom u(c) = c σ σ whee σ>0denotes the constant intetempoal elasticity of substitution of the consume. Then the maginal utility of consumption at date t is u 0 (c t )=c σ t so that the consumption smoothing condition can be witten o c σ t = β( + )c σ t+ c t+ = β σ ( + ) σ c t If β( + ) =we again have that c t+ = c t. Moe geneally, we have c t =[β σ ( + ) σ ] t c 0 so that consumption at any date is a scaled up o down vesion of consumption at date zeo. As befoe, if the consume is elatively patient so that she discounts less than the wold inteest ate she has a gowing consumption path, while if the consume is elatively impatient she has a shinking consumption path. Now combine the fomula c t =[β σ ( + ) σ ] t c 0 with the intetempoal budget constaint c t =()B 0 + to detemine the initial consumption c 0. Obviously, X c 0 β σ ( + ) σ t =()B 0 + But β σ ( + ) σ t = β σ ( + ) = σ β σ ( + ) σ (Assuming that 0 <β σ ( + ) σ < ). Hence c 0 = + v ( + )B 0 + # whee the numbe v is v β σ ( + ) σ v summaizes the influence of σ and of β( + ) 6=. If β( + ) =,wehavethesame consumption function as in Example with v =0. 5
6 Dynamics of the cuent account Let me intoduce some notation which is helpful fo discussing pesent value budget constaints. Fo any vaiable x, let x t denote the pemanent value of x at date t. This is the solution to µ s t x t = µ s t x s Fo a given wold eal inteest ate >0, this is a mapping fom the sequence {x s } to the single numbe x t.specifically, x t = µ s t x s (Using P i=0 zi =( z) fo 0 <z< and eaanging). Hence the pemanent value is a measue of the cental tendency of the sequence {x s } weighted by the discount factos. Now suppose that β( + ) =as in Example. Then as in that example, the consumption function is c 0 = ( + )B 0 + O at any initial date t, c t = ( + )B t + # µ s t (y s i s g s )# (This follows using the changes of vaiable 0 7 t and t 7 s t). In tems of pemanent values, this is just c t = B t +ỹ t ĩ t g t Now ecall the flow budget constaint B t+ B t = B t + y t c t i t g t and eliminate consumption using c t = B t +ỹ t ĩ t g t.thisgives B t+ B t =(y t ỹ t ) (i t ĩ t ) (g t g t ) In this example, the cuent account B t+ B t isthesumoftheetems,eachthediffeence between a vaiable and its pemanent value. If y t is elatively high, so that y t > ỹ t, thee will (ceteis paibus) be a cuent account suplus, B t+ >B t. Similaly, if i t o g t is elatively, high, thee will be a cuent account deficit. Ove time, of couse, the pesent value of cuent accounts must be zeo. Chis Edmond, August
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