Lecture 1: A Robinson Crusoe Economy

Transcription

1 Lecture 1: A Robinson Crusoe Economy Di Gong SBF UIBE & European Banking Center c Macro teaching group: Zhenjie Qian & Di Gong March 3, 2016 Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

2 The Robinson Crusoe economy Island economy A single agent Isolation, no trade Static problem: choose between consumption and leisure Limited resources Closed economy: North Korea Key question: how to split a unit of time into labor and leisure, and what factors determine that choice. Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

3 Intermezzo: Robinson Crusoe Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

4 Intermezzo: Cast Away Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

5 The Robinson Crusoe model: Technology Definition 1: Production function y = f(k, l) 1 Output y measured in units of consumption goods 2 Homogeneous input capital k and labor l 3 f(0, 0) = f(0, l) = f(k, 0) = 0: without capital or labor, nothing can be produced and output is zero. 4 f k ( ) > 0, f l ( ) > 0: any input is good for output Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

6 Technology continues Definition 2: The marginal product of factor (MPK, MPL) is the increment of output obtained by one unit increase in one factor while holding the rest factors fixed. MP K f(k, l) k = f k (k, l), MP L f(k, l) l = f l (k, l) e.g. Cobb-Douglas production function y = Ak 1 α l α MP K = (1 α)ak α l α = (1 α) y k MP L = αak 1 α l α 1 = α y l Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

7 Technology continues Assumption 1: The diminishing marginal returns. The decrease in the marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, while the amounts of all other factors of production stay constant. 2 f(k, l) 2 k < 0, 2 f(k, l) 2 l < 0 Assumption 2: Capital is fixed (k = k) y = f( k, l) = f(l) MPL is given by the slope of the production function. Diminishing marginal returns indicate that the slope of f(l) decreases as l increases while k is fixed. Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

8 Graph of production function and MPL Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

9 Preference Utility function Assume u(c, l), l [0, 1] 1 Utility from consumption c: MUC = u c = u(c,l) c > 0 2 Disutility from labor l: MUL = u l = u(c,l) l < 0 3 Leisure: 1 l Definition 3: An indifference curve Γ(ū) is the set of pairs of consumption and labor which deliver the same utility ū Γ(ū) = {c, l : c 0, l [0, 1], u(c, l) = ū} Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

10 A functional form Specify the utility function to a very specific functional form u(c, l) = ln(c) + ln(1 l) Set u = ū and solve for c as a function of l and constant ū ū = ln(c) + ln(1 l) c = eū 1 l For a constant utility ū, if we increase labor (decrease leisure), consumption will go up. Along any indifference curve, we have a negative (positive) relationship between consumption and leisure (labor). Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

11 Graph of the indifference curve Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

12 Preference continues We fully diferentiate ū = u(c, l) and obtain 0 = u c dc + u l dl Slope of the indiference curve dc dl = u l = MUL u c MUC Marginal rate of substitution of consumption for hours of work Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

13 Optimization problem Crusoe s problem subject to maxu(c, l) c,l c y (1) where y = f(l) Since MUC > 0, never optimal not to consume all of the production. Hence, (1) holds equality. Rewrite the problem as maxu(c, l) c,l subject to c = f(l) Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

14 Solution: Lagrange multiplier Construct Lagrangian function L(c, l, λ) = u(c, l) + λ[f(l) c] Set zero the derivatives of the Lagrangian function w.r.t. the choice variables and the Lagrangian multiplier λ 1 FOC c c L(c, l, λ) = u c(c, l ) + λ [ 1] = 0 (2) 2 FOC l 3 FOC λ l L(c, l, λ) = u l(c, l ) + λ [f (l )] = 0 (3) λ L(c, l, λ) = f(l ) c = 0 (4) Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

15 Solution: Lagrange multiplier Rearrange (2) and (3) Equivalently f (l) = u l u c MUC MP L = MUL Intuition: The increase in marginl utility of consumption induced by the incremental output by marginal product of labor should be able to compensate for the disutility from labor. Graphically MP L = MUL MUC 1 LHS: slope of production function 2 RHS: slope of indifference curve Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

16 Combine indifference curve and production function Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

17 An example Utility function u(c, l) = ln(c) + ln(1 l) Cobb-Douglas production function y = Al α Optimal solution (superscript indicates optimal values) l = α 1 + α, α c = A( 1 + α )α Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

18 The wealth effect Wealth effect: the change in the production function does not change the relative cost of leisure in terms of consumption. A positive wealth effect the consumer will be able to consume more while working the same hours. e.g. f 0 (l) = B 0 + Al α, f 1 (l) = B 1 + Al α If Robinson s output increases (B 1 > B 0 ), then 1 the production function shifts upward in a parallel manner 2 MPL (=Aαl α 1 ) does not change 3 but output per labor hour has increased, i.e., Crusoe can produce more output with the new technology by working the same hours. 4 consumption, leisure, labor l Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

19 Why consumption & labor l? Scenario 1: if c & l Higher c will lower MUC. higher l will lower MP L but raise disutility from labor MUL. Equation MUC MP L = MUL cannot hold. Contradiction! Scenario 2: if c & l stayed the same Higher c will lower MUC. Fixed l will not change MP L and MUL. Equation MUC MP L = MUL cannot hold. Contradiction! Scenario 3: if c This happens only if l. higher MUC and MP L, but lower MUL. Contradiction! Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

20 Graph: wealth effect Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

21 The substitution effect Substitution effect: increase in MPL without an upward shift in the production function Barro imagines that f(l) changes in such a way that it still passes through (c, l ) but MPL at the point increases At (c, l ), MUC and MUL are the same as before. High MPL indicates higher marginal reward to work. If Crusoe increases his work effort, the utility gain from the extra consumption goods will outweigh the utility loss from working harder. consumption, leisure, labor l, until MUC MP L = MUL is satisfied again. Crusoe substitutes extra consumption for less leisure. Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

22 Graph: substitution effect Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

23 Combining the wealth & substitution effects Increase in MPL & an upward shift in the production function at all values of l consumption effects on leisure & labor Positive wealth effect: Robinson feels wealthier and therefore tends to enjoy more leisure Substitution effect: He is incentivized to work harder as MPL is higher. Two effects offset one another, but unclear which one dominates. Note: zero effect on labor, because of the specific utility function and production function we have chosen. Mathematically dc da = ( α 1 + α )α > 0, dl da = 0 Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

24 Graph: Combining the wealth & substitution effects Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

25 Analysis of Comparative Statics Same preference u(c, l) = ln(c) + ln(1 l) Linear technology f(l) = B + Al with A > B > 0 (unlike the nonlinear production function f(l) = B + Al α in the previous example) Crusoe s problem subject to maxln(c) + ln(1 l) c,l B + Al = c Solution using the Lagrangian optimization c = A + B, l = A B 2 2A Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

26 Analysis of Comparative Statics: continues Pure wealth effect db > 0 dc db = 1 dl > 0, 2 db = 1 2A < 0 Intuition: both consumption and leisure increase (normal goods) Overall effects of wealth effect and substitution effect da > 0 dc da = 1 dl > 0, 2 da = B 2A 2 > 0 consumption, leisure as work becomes more productive Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27

27 Summary of the Robinson Crusoe model Crusoe: representative agent Crusoe s utility function how people on average trade-off between consumption and leisure Crusoe s resource constraint economy-wide production fuction aggregate over individuals to get the whole economy analyze how the economy responds to productivity shocks. Di Gong (UIBE & EBC) Intermediate Macro March 3, / 27