文档介绍：Recitation Notes 1
Konrad Menzel
September 18, 2006
1 Digression: Deriving the Slutsky Equation using Roy’s Iden­
tity
As in the lecture, pensated and pensated demand for leisure as l(p, w, y¯), and lc(p, w, u¯),
respectively. Analogously to the other derivation, we start from
l(p, w, y¯) = lc(p, w, v(p, w, y¯))
It turns out that it is more convenient to formulate this problem in terms of ”excess” demand for leisure
l(p, w, y¯) − T (. beyond the initial time endowment of T hours), so that we don’t have to worry about
changes in the value of the full e wT +y¯. In other words, we evaluate the consumer’s trading options
relative to an initial bundle (l0, x0) = (T, y¯) when we allow her to trade leisure for the consumption good
w
at a rate p , . the real wage. Excess demand for leisure is equal to the negative of hours worked and
must therefore be non-positive.
Taking derivatives with respect to y¯, we get
∂∂∂
l(p, w, y¯) = lc(p, w, v(p, w, y¯)) v(p, w, y¯) (∗)
∂y¯ ∂u¯ ∂y¯
Differentiation with respect to w gives us
∂∂ d
l(p, w, y¯) = l(p, w, y¯) − T = lc(p, w, v(p, w, y¯))
∂w ∂w dw
∂∂∂
= lc(p, w, v(p, w, y¯)) + lc(p, w, v(p, w, y¯)) v(p, w, y¯)
∂w ∂u¯ ∂w
Roy’s ID ∂∂∂
= lc(p, w, v(p, w, y¯)) − lc(p, w, v(p, w, y¯)) v(p, w, y¯) l(p, w, y¯) − T
∂w ∂u¯ ∂y¯
(∗) ∂∂
= lc(p, w, v(p, w, y¯)) − l(p, w,