Macroeconomic Theory and Policy

MacroeconomicTheory and Policy

MacroeconomicTheory and Policy

  1. (6 points) Based on the given Lagrangian, compute the representative consumer’s first-order conditions with respect to consumption and with respect to labor. Clearly present the important steps and logic of your analysis.

Answer:Thefirst-order conditions (FOC) with regards to consumption and labor isexpressed as follows

1cλ= 0

Anφ+ λ(1 − t) w= 0

  1. (7 points) Based on ONLY the first-order 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, upward-sloping, downward-sloping, 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 first-order condition with respect to consumption nor any other conditions.)

Answer:The first-order condition (FOC) with regards to labor can bereadjusted to (if one opts to put it in horizontal axis/ verticalaxis form).

w =

1

Anφ

λ

(1 −t)

Assumingthat φ&gt 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 upward-sloping 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 theirafter-tax 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

  1. (4 points) Now based on both of the two first-order conditions computed in part a, construct the consumption-leisure optimality condition. Clearly present the important steps and logic of your analysis.

Answer:Normally,this necessitates removing the Lagrange multiplier across the twoexpressions.Additionally,thefirst-order condition (FOC) on consumption cgiveslambda orλ=1cInserting this into the First-Order Condition (FOC) on labor gives

Anφ= (1−ct)w.

Alternatively,with regards to algebraic rearrangement multiplying by c,the consumption-leisure optimality condition can be denoted as Ac=(1−t)w

Anφ= −

(1t)w.

1/ c

  1. (7 points) Based on both the consumption-leisure 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, upward-sloping, downward-sloping, 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 consumption-leisureoptimality condition which was obtained from part C, we get theequation (1−t)wnAnφ= (1−t).Additionally, the terms (1-t)wonthe right-hand and right-hand sides clearly cancel what remains isthe nAnφ=1 or combining the powers in An1+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).&nbspModern macroeconomics. MIT Press.