Macroeconomic Theory and Policy
MacroeconomicTheory and Policy
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MacroeconomicTheory and Policy

(6 points) Based on the given Lagrangian, compute the representative consumer’s firstorder conditions with respect to consumption and with respect to labor. Clearly present the important steps and logic of your analysis.
Answer:Thefirstorder conditions (FOC) with regards to consumption and labor isexpressed as follows
^{1}_{c}−λ= 0
− An^{φ}+ λ(1 − t) w= 0

(7 points) Based on ONLY the firstorder condition with respect to labor computed in part a, qualitatively sketch two things in the diagram below. First, the general shape of the relationship between w and n (perfectly vertical, perfectly horizontal, upwardsloping, downwardsloping, or impossible to tell). Second, how changes in t affect the relationship (shift it outwards, shift it in inwards or impossible to determine). Briefly describe the economics of how you obtained your conclusions. (IMPORTANT NOTE: In this question, you are not to use the firstorder condition with respect to consumption nor any other conditions.)
Answer:The firstorder condition (FOC) with regards to labor can bereadjusted to (if one opts to put it in horizontal axis/ verticalaxis form).
w = 
1 
An^{φ} 

λ 
(1 −t) 

Assumingthat φ> 0, there is evidently an upward sloping relationship between nandw.While plotting this below, (as well as ignoring concavity/convexityconcerns which are overseen by the certain magnitude of φ)it offers an upward sloping relationship keeping t,A,in addition to λconstant (Chugh,2015).Such it denotes the labor supply function. Therefore, beginning fromsuch upwardsloping relationship, an increase in the tax rate t(assuming λ,A,and nconstant) forces the whole function to move inwards. Additionally,the latter effect happens owing to persons working minimal hours whenthe tax rate increase, all else equal owing to the decline in theiraftertax real wage (Chugh,2015).
Thereexists a perfectly horizontal labor supply function that comes up inthe diagram below. The reason is that nmerelyfails to appear in the FOC on labor. Secondly, since tfailsto appear, it forces the labor supply function to move shift down orup. Such labor supply is considered as perfectly elastic.
bor
bor
Problem4 continued

(4 points) Now based on both of the two firstorder conditions computed in part a, construct the consumptionleisure optimality condition. Clearly present the important steps and logic of your analysis.
Answer:Normally,this necessitates removing the Lagrange multiplier across the twoexpressions.Additionally,thefirstorder condition (FOC) on consumption cgiveslambda orλ=^{1}_{c}Inserting this into the FirstOrder Condition (FOC) on labor gives
An^{φ}= ^{(1−}_{c}^{t}^{)}^{w}.
Alternatively,with regards to algebraic rearrangement multiplying by c,the consumptionleisure optimality condition can be denoted as Ac=(1−t)w
_{An}^{φ}= −
(1^{–}t)w.
1/ c

(7 points) Based on both the consumptionleisure optimality condition obtained in part c and on the budget constraint, qualitatively sketches two things in the diagram below. First, the general shape of the relationship between w and n (perfectly vertical, perfectly horizontal, upwardsloping, downwardsloping, or impossible to tell). Second, how changes in t affect the relationship (shift it outwards, shift it in inwards, or impossible to determine). Briefly describe the economics of how you obtained your conclusions.
Answer:Thebudget constraint is expressed asc=(1−t)wn.Hence, substituting the equation into the consumptionleisureoptimality condition which was obtained from part C, we get theequation (1−t)wn⋅An^{φ}= (1−t).Additionally, the terms (1t)wonthe righthand and righthand sides clearly cancel what remains isthe n⋅An^{φ}=1 or combining the powers in An^{1+}^{φ}=1. Hence, when one plots such expression in the space above one gets
Theequation obviously is not reliant on the worthe real wage. The labor supply function is perfectly inelastic orperfectly vertical. A change in taxes fails to affect such perfectlyinelastic labor supply function.
Reference
Chugh,S. K. (2015). Modern macroeconomics. MIT Press.