HW_6_Sketch_Solutions_-_Fall_2022

ECO 206Y1Y: Microeconomic Theory Prof. Freitas, University of Toronto Eco 206 HW6 , Sketch Solutions These are sketch solutions only. If you still have questions, please stop by office hours. Q1 Consider a representative worker’s preferences over leisure and the composite good given by u ( l, c ) = αln ( l ) + c . Their leisure endowment is L . Denote their non-labour income by M and the wage rate as w . (a) Derive their labour supply function for M = 0 (b) Derive their labour supply function for M = 1000 . (c) For the labour supply function in part (b), how does it change when M increases? Movement of the curve or along the curve? Hints: (1) When M increases, is it an income effect or a substitution effect or both? (2) Think about how the kink point of the budget constraint changes with M . Solution: Solution: For both (a) and (b) I’m going to derive the same function and then just set M = 0 and M > 0 to see the differences. As this is Quasilinear I can use Lagrangians but I need to watch out for corner solutions or solutions at the kink point for M > 0 . L = αln ( l ) + c + λ [ M + wL - c - wl ] FOC α l = λw 1 = λ M + wL = c + wl α l = w ⇒ l = α w M + wL = c + w α w ⇒ c = M + wL - α (a) The tangency condition gives us h = L - l = L - α w . Notice this holds only if 0 ≤ h ≤ L and c ≥ 0 . Therefore, the full labour supply curve is h = L - α w w ≥ α L 0 w < α L (b) The same reasoning gives us the labour supply for M > 0 : h = L - α w w ≥ α L 0 w < α L Why doesn’t M matter? We have two constraints: 0 ≤ h ≤ L and c ≥ 0 . The following conditions are equivalent to c ≥ 0 : M = 0 : c ≥ 0 ⇒ wL - α ≥ 0 ⇒ w ≥ α L M > 0 : c ≥ 0 ⇒ M + wL - α ≥ 0 ⇒ w ≥ α L - M L For this particular problem, 0 ≤ h ≤ L = ⇒ c ≥ 0 , which is why M does not show up in the optimal choice of h . (c) As M increases, the labour supply curve doesn’t change. We can see this algebraically, given our labour supply functions derived above. Note that an increase in M is just an income effect. This are quasilinear preferences so there is no income effect on l = L - h here. Q2 Consider a worker in the restaurant industry. The article linked in the module discusses the labour market in 2022. In class we interpreted it as a worker not changing her hours worked when given a higher wage. Here we will consider another interpretation- for the same wage, she chooses to work fewer hours. For your choice of preferences... (1) Explain why a worker’s labour supply would change this way? 202209 1 of 2 HW6