Introduction to Modern Economic Growth where ∇x U represents the gradient vector of U with respect to its first K arguments, and ∇y U represents its gradient with respect to the second set of K argu- ments Notice that (6.21) is a functional equation in the unknown function π (·) and characterizes the optimal policy function These equations become even simpler and more transparent in the case where both x and y are scalars In this case, (6.19) becomes: (6.22) ∂U(x, y ∗ ) + βV (y ∗ ) = 0, ∂y where V the notes the derivative of the V function with respect to it single scalar argument This equation is very intuitive; it requires the sum of the marginal gain today from increasing y and the discounted marginal gain from increasing y on the value of all future returns to be equal to zero For instance, as in Example 6.1, we can think of U as decreasing in y and increasing in x; equation (6.22) would then require the current cost of increasing y to be compensated by higher values tomorrow In the context of growth, this corresponds to current cost of reducing consumption to be compensated by higher consumption tomorrow As with (6.19), the value of higher consumption in (6.22) is expressed in terms of the derivative of the value function, V (y ∗ ), which is one of the unknowns To make more progress, we use the one-dimensional version of (6.20) to find an expression for this derivative: ∂U(x, y ∗ ) ∂x Now in this one-dimensional case, combining (6.23) together with (6.22), we have (6.23) V (x) = the following very simple condition: ∂U(x, π (x)) ∂U(π (x) , π (π (x))) +β =0 ∂y ∂x where ∂x denotes the derivative with respect to the first argument and ∂y with respect to the second argument Alternatively, we could write the one-dimensional Euler equation with the time arguments as (6.24) ∂U(x∗ (t + 1) , x∗ (t + 2)) ∂U(x (t) , x∗ (t + 1)) +β = ∂x (t + 1) ∂x (t) 282