Consider a household that is born with no assets and lives for two periods (call them...
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Consider a household that is born with no assets and lives for two periods (call them "young" and "old"). The household is endowed with 1 unit of time in each period of life, which it can spend as labor or enjoyed as leisure. and Wor Taking wages (when young and old), Wy and net interest rate on savings, r, as given, the household chooses consumption in each period, cy and co, and fraction of time worked in each period, hy and h, and savings s in order to maximize its utility subject to the budget constraints when young and old: In cy+nln (1- hy) + 3 [ln co + n ln (1- ho)] max (cy,Co,hy,ho,s) subject to [bcy] cy + s = wyhy [bco]: Co Cy, Co, hy, ho 20 Note that 1-hy (and 1-ho) represent leisure when young - that's what enters utility. (0,1) is the discount factor. We also allow for s to be negative or positive. Negative savings would represent borrowing. Solving for s from [BC] and substituting into [BCy] yields the lifetime budget constraint: Cy + - Co 1+r Hence we reduce our optimization problem to: Co 1+r woho+ (1+r)s Cy + max (cv,co,hv,ho) subject to max Cy,Co,hy,ho,A =Wyhy + woho (1+r) In cy + n ln (1- hy) + B [Inco + nln (1-ho)] Because of log utility, we do not worry about the agent choosing nonpositive consumption or nonpositive labor supply, so we can ignore the non-negativity constraints. We denote the Lagrangian multiplier by X and write down our Lagrangian: Wyhy + L = lncy + n ln (1- hy) + B [Inco + n ln (1-ho)] - x [c + [Cop - why - (1+r)] Cy 1+r We take first order conditions with respect to Cy, Co, hy, ho, and set each to 0. We then combine the resulting equations so as to substitute out for A. We end up with four equations in four unknowns: [Intertemp. Tradeoff] : [Intratemp. Tradeoff y ] : Co 1+r [Intratemp. Tradeoff o ] : [BC] : cy +: Co Cy = woho (1+r)* 1 Wyhy + = B(1+r) Cy Co (1 - hy) Wy n (1-ho) wo woho (1+r) To solve this model numerically (for given parameter values p, r,n, wy and wo), it suffices to solve the above system of four non-linear equations for the four unknowns Cy, Co, hy, ho. Thus, consider the following table: Parameter B r Wy Wo Table 1: Description utility weight on leisure discount Factor net interest rate on savings wage when young wage when old Value 0.5 0.8 0.1 1.00 0.5 (a) What are the household's equilibrium levels of consumption and labor supply for each stage in its life cycle (ie: young v.s. old)? Write a script to solve this model using Matlab's fsolve function. (b) Do your results make economic sense? If so, what does the model predict about labor supply and consumption at different parts of a household's life cycle? (c) BONUS: Could you solve this system using Matlab's fmincon function? If solve, write an alternative script to achieve that. Are your results from part (a) confirmed? Consider a household that is born with no assets and lives for two periods (call them "young" and "old"). The household is endowed with 1 unit of time in each period of life, which it can spend as labor or enjoyed as leisure. and Wor Taking wages (when young and old), Wy and net interest rate on savings, r, as given, the household chooses consumption in each period, cy and co, and fraction of time worked in each period, hy and h, and savings s in order to maximize its utility subject to the budget constraints when young and old: In cy+nln (1- hy) + 3 [ln co + n ln (1- ho)] max (cy,Co,hy,ho,s) subject to [bcy] cy + s = wyhy [bco]: Co Cy, Co, hy, ho 20 Note that 1-hy (and 1-ho) represent leisure when young - that's what enters utility. (0,1) is the discount factor. We also allow for s to be negative or positive. Negative savings would represent borrowing. Solving for s from [BC] and substituting into [BCy] yields the lifetime budget constraint: Cy + - Co 1+r Hence we reduce our optimization problem to: Co 1+r woho+ (1+r)s Cy + max (cv,co,hv,ho) subject to max Cy,Co,hy,ho,A =Wyhy + woho (1+r) In cy + n ln (1- hy) + B [Inco + nln (1-ho)] Because of log utility, we do not worry about the agent choosing nonpositive consumption or nonpositive labor supply, so we can ignore the non-negativity constraints. We denote the Lagrangian multiplier by X and write down our Lagrangian: Wyhy + L = lncy + n ln (1- hy) + B [Inco + n ln (1-ho)] - x [c + [Cop - why - (1+r)] Cy 1+r We take first order conditions with respect to Cy, Co, hy, ho, and set each to 0. We then combine the resulting equations so as to substitute out for A. We end up with four equations in four unknowns: [Intertemp. Tradeoff] : [Intratemp. Tradeoff y ] : Co 1+r [Intratemp. Tradeoff o ] : [BC] : cy +: Co Cy = woho (1+r)* 1 Wyhy + = B(1+r) Cy Co (1 - hy) Wy n (1-ho) wo woho (1+r) To solve this model numerically (for given parameter values p, r,n, wy and wo), it suffices to solve the above system of four non-linear equations for the four unknowns Cy, Co, hy, ho. Thus, consider the following table: Parameter B r Wy Wo Table 1: Description utility weight on leisure discount Factor net interest rate on savings wage when young wage when old Value 0.5 0.8 0.1 1.00 0.5 (a) What are the household's equilibrium levels of consumption and labor supply for each stage in its life cycle (ie: young v.s. old)? Write a script to solve this model using Matlab's fsolve function. (b) Do your results make economic sense? If so, what does the model predict about labor supply and consumption at different parts of a household's life cycle? (c) BONUS: Could you solve this system using Matlab's fmincon function? If solve, write an alternative script to achieve that. Are your results from part (a) confirmed?
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ANSWER a To solve the model using Matlabs fsolve function we need to define the equations and the va... View the full answer
Related Book For
Fundamentals Of Taxation 2015
ISBN: 9781259293092
8th Edition
Authors: Ana Cruz, Michael Deschamps, Frederick Niswander, Debra Prendergast, Dan Schisler, Jinhee Trone
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