14.661 F17 Lecture Note: Slutsky Equation

Slutsky for Hours (done in minutes) MIT 14.661 (FALL 2017) A Slutsky derivation Uncompensated and Compensated Labor Supply Utility is a function of consumption (x) and leisure (l), where h = T - l is hours worked. • Uncompensated (Marshallian) demands are a function of wages, prices, and unearned income {x(p, w, y), l(p, w, y)} = s.t. px = arg max U (x, l) w(T - l) + y This generates uncompensated labor supply: h(p, w, y) = T - l(p, w, y) • Compensated (Hicksian) labor supply is a function of wages, prices and utility {xc (p, w, ū), lc (p, w, ū), } = s.t. ū = arg min wl + px U (x, l) This is the dual problem, generating compensated labor supply: hc (p, w, ū) = T - lc (p, w, ū) The derivative of the compensated labor supply function is the substitution e↵ect. The excess expenditure function • A consumer spends this much to get to ū: E[p, w, ū] = pxc (p, w, ū) + wlc (p, w, ū) pxc (p, w, ū) + w(T - hc (p, w, ū)) 1 • I need this much cash to get to ū: E ⇤ [p, w, ū] = E[p, w, ū] - wT = pxc (p, w, ū) - whc (p, w, ū) Viewed as a function of prices wages, and my utility target, this is called the excess expenditure function. • The excess expenditure function has the following properties: 1. Shephard’s Lemma @E ⇤ [p, w, ū] @E[p, w, ū] = - T = lc (p, w, ū) - T = -hc (p, w, ū) @w @w This is the enevelope theorem in action. 2. Concavity @ 2 E ⇤ [p, w, ū] @ 2 E[p, w, ū] @lc (p, w, ū) <0 = = 2 2 @w @w @w How do we know this? The expenditure function is concave in prices because people reallocate away from the more expensive good when its price increases. In other words, spending (cost) goes up less than linearly in prices. From this, we conclude @hc (p, w, ū) >0 @w The substitution e↵ect on hours is positive. A useful identity • Compensated and uncompensated labor supply are related as follows hc (p, w, ū) = h(p, w, E ⇤ [p, w, ū]) (1) In other words, if I adjust your unearned income (compensate you) so as to keep you on ū while changing your wage, then I learn what happens when your wages change while you’re stuck on indi↵erence curve ū. Slutsky derived • Di↵erentiate both sides of (1) @hc (p, w, ū) @w @hc @w = =  @h(p, w, E[p, w, ū] - wT ) @h(p, w, E[p, w, ū] - wT ) @E[p, w, ū] + -T @w @y @w @h @h + [-hc (p, w, ū)] @w @y • Re-arrange to get the Slutsky equation for hours: @h @hc @h = + h @w @y @w Now, use it! 2 B Ashenfelter (1978) puts Slutsky to work Parameterizing policy: the Negative Income Tax • A stylized negative income tax (NIT) or similar welfare program provides a subsidy of G, reduced by amount t, for every dollar of earnings. Many such program emerged in the 1970s as an alternative to traditional welfare (we’ll come back to this) – Assuming (as is often the case) that only earnings are taxed, the program subsidy is S = G - twh when wh < G =B t and zero otherwise – B is the program breakeven • Ashenfelter (1978) asks: what does basic theory say about program e↵ects on labor supply? The paper then uses data from the Rural NIT to estimate key labor supply paremeters Theoretical NIT e↵ects • Define h(p, w, y) ¯ hc (p, w, u) = uncompensated .l.s = compensated .l.s – Uncompensated di↵erential dh = @h @h dw + dy @y @w Substituting with Slutsky  c @h @h @h dh = + h dw + dy @y @y @w • Rearranging and inserting program parameters dh = = = @h @hc dw + [hdw + dy] @y @w @h @hc [-twh + G] (-tw) + @y @w @h @hc S (-tw) + @y @w 3 (2) so dlnh = ⌘c (-t) + ⌘y S y (3) This is Ashenfelter (1978) “parameter scheme 2,” where S is replaced by S0 , an ex ante subsidy value based on pre-program earnings. Note that labor supply must fall. – “Parameter scheme 1” generates an estimation scheme for uncompensated elasticities by working directly with 2 to produce dlnh = ⌘u (-t) + ⌘y G y Estimates Rural NIT: 5 experimental treatments (G,T) plus controls in each of IA and NC. About 800 families exposed for 3 years; Average G=3,200 (50-100% pov level) and t=1/2 (.3-.7). Ashenfelter pools data for the three years and regresses change in log earnings on year e↵ects, tax rates, and guarantee or subsidies as proportion of unearned income. • See Tables 3-6. • Beware sinister misreporting and attrition! (Greenberg and Halsey, 1983; Ashenfelter and Plant, 1990) 4