Title: Econ: for comments: GE example with indivisible (dis)utility
Simple Robinson Crusoe (RC) economy.
There are 2 periods.
Price level p is given exogenously (e.g. through trade).
RC is the "capitalist," he earns competitive profits (=0) so he is indifferent between producing and not producing. He doesn't have to eat to live.
Friday is the laborer, he has a family and a (high) fixed disutility of labor: his disutililty as a function of hours worked H is C1(H) = f + c1 H for f > 0 and c1 > 0.
If the Friday family survive the first period then in the second period, Friday Jr. will be the laborer. He is much more efficient than his father. His disutility is C2(H) = 0 + c2 H where c2 < c1.
Friday's total utility is U = Y - C1 = wH - c1 H - f, where Y = income = consumption = wage x hours worked.
Suppose if Y < C1 then the Fridays cannot survive to the second period.
First-period neoclassical equilibrium is at dC1/dH = c1 = w; suppose c1 = 1 = w. At w = 1, suppose Friday works H = 1 hr. and earns Y = $1.
Further suppose f = $2 > $1.
Given these parameters, Friday works 1 hr. At the end of period 1, all Fridays die. In period 2 there is no production. Their total utility over two periods is then -infinity.
This is an inefficient outcome. Had Friday Jr. survived the first period, production would have folded by a factor of k > 1 in period 2 (i.e., Friday Jr. would have worked many hours more than his father). Total utility over two periods would then have been > 0.
Ends that can be tied together:
1. The minimum level of income necessary to survive period 1 can be tied to p.
2. The linear part of C(H) can be made into a backward-bending labor supply function, c(H), such that there are two equilibrium wage levels, w and W, where W > w. With a downward-sloping labor demand curve, W is an unstable equilibrium but w is stable. If equilibrium selection rule includes "stability," then W would be ruled out. This can further be tied to a low-wage equilibrium that ends up being dynamically inefficient because it does not generate sufficient income to survive the first period.