Since is not uncertain and since $N_{t+1}=N_t{e}^1,$ this condition simplifies to

$\frac{1}{c_t}=e^{-}{E}_t[\frac{1}{c_{t+1}}(1+r_{t+1})].$

This is the analogue of equation $(2\mathrm{.}22)$ in the Ramsey model.

rate of return, but also on their interaction. Specifically, the expectation of the product of two variables equals the product of their expectations plus their covariance. Thus $(5\mathrm{.}23)$ implies

$\frac{1}{c_t}=e^{-}\{E_t[\frac{1}{c_{t+1}}]E_t[1+r_{t+1}]+$cov$(\frac{1}{c_{t+1}},1+r_{t+1})\}.$

where Cov$(\frac{1}{c_{t+1}},1+r_{t+1})$ denotes the covariance of $\frac{1}{c_{t+1}}$ and $1+r_{t+1}.$ Suppose, for example, that when $r_{t+1}$ is high, $c_{t+1}$ is also high. In this case,

consumption.

Chapter $8$ discusses how uncertainty and the covariance between the marginal utility of consumption and asset returns affect consumption behavior in more depth.

The Tradeoff between Consumption and Labor Supply

leave expected utility unchanged.

has a utility benefit of $e^{-t}(\frac{N_t}{H})(\frac{1}{c_t})w_{t}l.$ Equating the cost and benefit gives us

$e^{-t}\frac{N_t}{H}\frac{b}{1-l_t}l=e^{-t}\frac{N_t}{H}\frac{1}{c_t}{w}_{t}l,$

or

$\frac{c_t}{1-l_t}=\frac{w_t}{b}.($a$)$Use an argument analogous to that used to derive equation $(5\mathrm{.}23)$ to show that household optimization requires b$/(1$t$)=$ e$\backslash$rho Et $[$wt $(1+$ rt $+1)$b$/$ wt $+1(1$t $+1)].$

$($b$)$Show that this condition is implied by $(5\mathrm{.}23)$ and $(5\mathrm{.}26).($Note that $[5\mathrm{.}26]$ must hold in every period.$)$