Macroeconomics+

Labor+–+Leisure+Model+

ECON+321+

Prac:ce+Problems:+WDI+

1.+Explain+the+following:+When+looking+at+WDI,+

popula:on+weigh:ng+is+necessary,+but+not+sufficient.+

2.++

+a. +Claim+1:+“Income+inequali:es+within+countries+ +

+ +have+increased.”+: +TRUE+or+FALSE+

+b. +Claim+2:+“Income+inequali:es+across+countries+ +

+ +have+increased.”+: +TRUE+or+FALSE+

+c. +Conclusion:+“Global+income+inequali:es+have+also+

+ +increased.” +:+ +TRUE+or+FALSE+

+d. +Explain+how+your+answer+can+be+correct+(if+it+is).+

Business+Cycle+

Consump:on+vs+Leisure+

One+Period+Labor+–+Leisure+Model+

•  Robinson+Crusoe+Model+

– The+individual+is+the+household+and+the+firm.+

– No+government.+

– No+trade.+– No+savings.+– One+good,+which+is+perishable+(i.e.+coconut,+fish,+ect.)+

One+Period+Labor+–+Leisure+Model+

•  Produc:on+•  y+=+f(n)+

– where+n+=+labor+supplied.+•  c+=+y+

–  everything+produced+is+eaten+or+it+is+wasted.+

•  Assump:ons+

•  f’(n)++>+0+•  f’’(n)+<+0+

– where+f’(n)+is+the+marginal+product+of+labor.+

The+result+is+once+again+diminishing+returns.++We+get+the+

low+hanging+fruit+first.++We+keep+working+and+get+more+

coconuts,+but+we+have+to+climb+higher.+

Diminishing+Returns+

y"="f(n)"+

n+

y+

slope+is+the+marginal+

product+of+labor.+

"""""""n1+

Marginal+Product+of+Labor+

""MPL""""+

"""n""""+

Assume:+y+=+Anα,+where+0<α<1+

Then: +dy/dn+=++αAnαa1+>+0+

+ ++

+ +d2y/dn2+=+α(αa1)Anαa2++<+0+

Total+Factor+Produc:vity+

"y+=+A2nα+

n+

y+

y+=+A1nα+

n+

Where+A2+>+A1:+

At+the+same+level+of+labor,+the+

MPL+is+higher+with+a+higher+A.+

MPL+for+Different+TFP’s+

""MPL""""+

"""n""""+

+αA2nαa1+

+αA1nαa1+

Preferences+

•  Budget+Constraint: + +c+=+y+=+Anα+

•  Preferences+are+represented+by+the+U:lity+func:on:+U(c,l);++where+c+is+consump:on+and+l+is+leisure.+

U1+

U2+

Indifference+curves+represent+all+

possible+combina:ons+of+goods+

over+which+the+consumer+is+equally+

happy.++In+other+words….he+receives+

the+same+u:lity.+

U2++>+U1++with+more+of+both,+but+

points+along+the+same+indifference+

curve+leave+the+consumer+equally+

happy.+

c+

l+

U:lity+is+increasing+

this+way+

U:lity+

•  dU/dc+=+Uc+>+0+

•  dU/dl++=+Ul+>+0+

•  d2U/dc2+=+Uc<+0+

•  d2U/dl2++=+Ul+<+0+

Posi:ve+Marginal+U:lity+for+both+

goods.+

Diminishing+Marginal+U:lity+for+

both+goods.+

Convex+Indifference+Curves+

•  Convexity+of+the+indifference+curve:+1)  represents+diminishing+marginal+u:lity.+

2)  shows+preferences+for+variety++how+much+of+

one+good+do+you+have+to+give+up+for+another.+

•  The+marginal(rate(of(subs0tu0on((given+by+the+indifference+curves)+represents+the+

willingness+of+a+consumer+to+give+up+one+good+

for+another+holding+u:lity+constant: ++

• MRS$=$&$(dU/dl)/(dU/dc)$

Marginal+Rate+of+Subs:tu:on+

c+

l+

B+

A+

At+point+A+you+only+need+a+small+

increase+in+consump:on+to+give+

up+some+leisure.++

At+point+B+you+don’t+have+a+lot+of+

leisure+so+you+need+a+lot+of+

consump:on+to+give+up+some+

leisure.+

Example+

•  U(c,l)+=+cl+•  Maxc,l+++++++U(c,l)+=+cl+

–  subject+to+the+budget+constraints.+1.  c+=+y+=+An+

2.  l+n+=+H; + +where+H+is+the+:me+endowment+and+n+is+

+ + +labor+supplied.+

a)  Subs:tute+the+constraints+into+the+objec:ve+func:on.+

b)  Represent+both+consump:on+and+leisure+in+terms+of+n.+

1.  c+=+An +; +l+=+H+a+n+

Example+•  Maxn+++++++U+=+(An)(Han)+

»  where+(an)+represents+the+disu:lity+of+labor+in+the+labor/leisure+

tradeoff.+

•  First+Order+Condi:on+dU/dn+=+(A)(Han)+–+An+=+0+

+where: ++(A)(Han)+is+the+marginal+u:lity+of+consump:on.++

++++++and: +(–+An)+is+the+marginal+disu:lity+of+work.+ ++

Ah+=+2An+

n*+=+H/2+;+which+is+the+labor+supply+in+terms+of+all+exogenous+variables.++

l+ +=+Han+=+HaH/2+=+(2HaH)/2+

l* +=+H/2+;+which+is+leisure+demand+in+terms+of+all+exogenous+variables.++

c* +=+AH/2+;+which+is+consump:on+demand+in+terms+of+all+exogenous+

variables.++

Budget+Constraint+

Not+Possible+

Possible+but+

not+efficient+

c+

l+H+++++l*+

c*+

++++++n*+

l+=+H+a+n+

c+=+An+=+A+(H+a+l)+

slope+=+aA+

Δ=aA+

Marginal+Rate+of+Subs:tu:on+

c+

l+

+++c*+

+++++++l*+ H+

+n*+

U(c,l)+=+cl+

dU/dc+=+l+=+H/2+

du/dl+=+c+=+AH/2+

MRS+=+a+(du/dl)/(du/dc)+

+=+a[A(H/2)]/[H/2]+

+=+aA+

ASIDE+

•  U:lity+is+ordinal,+which+means+the+actual+

number+value+doesn’t+mean+anything,+only+

the+order+which+it+falls+in+for+like+comparisons.++

– You+can’t+compare+two+different+consumers+U:lity,+

but+you+can+compare+amounts+for+the+same+

person.+

•  Produc:on+is+cardinal,+which+means+the+values+

mamer.+

•  $200+dollars+is+bemer+than+$100.+

Consump:on+and+Labor+:+Mirror+Image+

c+

nH+++++++n*+

c*+

++++++l*+

n+=+H+a+l+

c=+A+(H+–+l)+=+An(

U:lity+is+increasing+

this+way+

Consump:on+and+Labor+

c+

nH+++++++n*+

c*+

++++++l*+

Δ+=+w+=+marginal+

product+of+labor.+

U(c,l)+

c+=+wn+x+;+

+w+=+the+wage+rate+

+x+=endowment+of+wealth+

Consump:on+and+Labor+

c+

n+++++++++n1+

c1+

++++++l*+

Suppose+X+increases:+How+does+it+affect+labor+supply+and+consump:on?+

+ ++consump:on+increases+$

+ ++labor+supply+decreases++

+++++++n2+

c2+

c+=+wn+x2++

c+=+wn+x1++

x1+

x2+

parallel+

shiq+

Consump:on+and+Labor+

•  Consump:on+and+leisure+are+normal(goods.+

–  The+more+income+you+have,+the+more+you+want.+

–  Since+leisure+is+a+normal+good,+a+rise+in+income+should+

cause+labor+supply+to+decrease.+

•  Inferior+goods+are+just+the+opposite+of+normal+goods.+

–  i.e.+public+transporta:on.+•  Labor(Supply,(Income((Wealth)(Effect:((+ +An+increase+in+the+income+or+wealth+means++the+consumer+

demands+more+leisure+because+it+is+a+normal+good+and+labor+

supply+falls.+

Consump:on+and+Labor+

c+

n+++++++++n*+

c*+

Suppose+w+increases:+How+does+it+affect+labor+supply+and+consump:on?+

+ ++consump:on+increases++++ +$+ ++labor+supply+is+Ambiguous ?+

c+=+w2n+x++

c+=+w1n+x++

Consump:on+and+Labor(

•  Labor(Supply,(Subs0tu0on(effect:(–  A+change+in+w+means+a+change+in+the+price+of+leisure,+or+the+opportunity(cost(of+leisure+has+risen.++Changes+in+rela:ve+prices+will+induce+subs:tu:on+to+the+rela:vely+cheaper+good.+

–  The+subs:tu:on+effect+of+an+increase+in+the+wage+rate+makes+leisure+more+expensive+rela:ve+to+consump:on,+which+leads+to+an+increase+in+the+supply+of+labor+and+a+decrease+in+the+demand+for+leisure.+

–  However,+and+increase+in+w(will+also+have+a+wealth+effect+decreasing+the+labor+supply.++This+leaves+the+overall+result+ambiguous.++