Note that the value function of the representative agent is restricted to a given path of
ha,t. The corresponding Bellman equation is given by:
V (kt, ht; At, ha,t) ≡ sup {ln ct + βEt [V (kt+1, ht+1; At+1, ha,t+1)]} . (27)
ct,ut
Taking the derivatives with respect to the two controls and inserting the market-clearing
factor prices (8) gives us the following first-order necessary conditions:
ct
ɪ = βEt ïdVM ,
c* L∂kt+ι J ,
(28)
ut
„ *
ut
^t[≤¾1] (1-α)τw AtV
= l Et [∂⅛] B )
kthα,t
ht
(29)
where Vt+1 is a shortcut for V (kt+1 , ht+1 ; At+1 , ha,t+1 ). Equation (28) is very standard
and characterizes the effect of shifting one unit of today’s output from consumption
to investment. Today’s marginal change in utility should equal the expected discounted
marginal change in tomorrow’s wealth with respect to tomorrow’s capital stock. Equation
(29) considers the shifting of a marginal unit of human capital from the goods production
sector to the schooling sector, or vice versa. The condition states that the marginal
change in goods production due to this shifting, weighted by the expected shadow price
of physical capital, should equal the marginal change in the schooling sector weighted by
the expected shadow price of human capital. Using the envelope property of the optimal
decision rules:
ct* = c (kt, ht; At, ha,t) and ut* = u (kt, ht; At, ha,t) ,
leads us to the following envelope conditions:
(30)
∂Vt — E I"dVt+1 1 ατryt „„d ∂Vt __ OE "dVt+1 1 B
∂kt = βEt [∂kt+lj ~kΓ, and ∂ht = βEt [∂ht+1J B∙
These conditions together with the above first-order necessary conditions along the op-
timal consumption path (28) and for the optimal allocation of human capital (29) imply
the following Euler equations:
1
ct
βEt Г ɪ y ,
t ct+1 kt+1 ,
(31)
ut
(E Г 1 ατryt + 1 ^] ∖ α
tlct+ι kt+ι J At I
E Г 1 yt+ι 1 B
t Lct+1 ut+1ht+1 J /
kthaα,t
ht
(32)
The two Euler equations (31) and (32) are necessary for a policy to attain the optimum.
Together with the following transversality conditions they are also sufficient:
lim βτEt "ɪ τrαyτkT 1 = 0 and lim βTEt "ɪ τw (1-α)yT hT 1 = 0 (33)
T→∞ t cT kT T T→∞ t cT uThT T
Note that the first fraction in both conditions is the derivative of the utility function and
the second fraction is the derivative of the goods sector production function with respect
to the inputs of physical and human capital. To be more precise, the last derivative is
taken with respect to the fraction of human capital that is allocated to the goods sector,
i.e. utht . These derivatives are multiplied by the respective state variable. The transver-
sality conditions tell us that the expected discounted marginal utility of an additional
unit of the capital stocks in the “last period” is equal to zero. These requirements rule
out that the agent plays Ponzi games.