Introduction to Modern Economic Growth which requires that the discounted market value of the capital stock in the very far future is equal to This “market value” version of the transversality condition is sometimes more convenient to work with We can derive further results on ³ the consumption ´ behavior of households In Rt particular, notice that the term exp − r (s) ds is a present-value factor that converts a unit of income at time t to a unit of income at time In the special case where r (s) = r, this factor would be exactly equal to exp (−rt) But more generally, we can define an average interest rate between dates and t as Z t r (s) ds (8.17) r¯ (t) = t In that case, we can express the conversion factor between dates and t as exp (−¯ r (t) t) , and the transversality condition can be written as r (t) − n) t) a (t)] = lim [exp (− (¯ (8.18) t→∞ Now recalling that the solution to the differential equation y˙ (t) = b (t) y (t) is y (t) = y (0) exp µZ t we can integrate (8.14), to obtain c (t) = c (0) exp àZ t ả b (s) ds , ¶ r (s) − ρ ds εu (c (s)) as the consumption function Once we determine c (0), the initial level of consumption, the path of consumption can be exactly solved out In the special case where εu (c (s)) is constant, for example, εu (c (s)) = , this equation simplifies to àà ả ả r (t) − ρ c (t) = c (0) exp t , θ and moreover, the lifetime budget constraint simplifies to Z ∞ Z ∞ c (t) exp (− (¯ r (t) − n) t) dt = a (0) + w (t) exp (− (¯ r (t) − n) t) dt, 0 382