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 timesepaable 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 consumptionsmoothing 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 longhoizon limit, consumption is simply popotional to intetempoal wealth. The infinitehoizon 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 infinitehoizon 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 noponzigame constaint of the fom µ T lim B T + 0 T to ensue that the consume cannot ollove 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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