Consider the following one-period model. Assume that the consumption good is produced by a linear technology: Y = zND where Y is the output of the con- sumption good, z is the exogenous total factor productivity, ND is the labour hours. Government has to finance its expenditures, G, using a lump-sum tax, T, on the rep- resentative consumer. There is no other tax in the economy. The firm is owned by the representative consumer who is endowed with h hours of time she can allocate between work, NS and leisure, l. Preferences of the representative consumer are: U (c, l) = α ln c + (1 − α) ln l (1) where 0 < α < 1 is a parameter. Answers for part b below: A consumer's choice of optimizing its consumption and labor hours (h - l ) is given by the point where , MRS(c , l) = wage rate Now , MRS (c, l) = MU(c) /MU(l) MU(c) = dU/d c = a/c MU(l) = dU /dl = (1-a )/l MRS = a (l)1−a (c)a (l)1-a (c) Putting this in optimal condition we have : a (l)(1−a) (c)a (l)(1-a) (c) = w --- (i) l = w *c (1- a)/a We put this value into budget constraint : c* = w ( h - w *c (1- a)/a ) - T (Optimal Consumption ) From equation (i) we get : c = a (l) / (1-a) w Putting this value into budget constraint : a (l) / (1-a) w = w (h -l ) - T l* = wh − T (a(1−a)w+w )wh - T (a(1-a)w+w ) (Optima Leisure ) We know , l + N = h N* = h - l* N* = h - wh − T (a(1−a)w+w )wh - T (a(1-a)w+w ) (Optimal Labor / Employment ) Putting this value into production function we get : Y* = zN* Y* = z (h - wh − T (a(1−a)w+w )wh - T (a(1-a)w+w ) ) (Optimal Output level ) Answers in part (b) ^ Solve Below Suppose that the government spending, G, increases. Using your answers in (1b), determine and explain how endogenous quantities and prices behave in this economy. Can changes in G be responsible for business cycles we observe? Explain. (Note that consumption, employment, and wages are all pro-cyclical in data.) (e) Suppose that the total factor productivity, z, increases. Using your answers in (1b), determine and explain how endogenous quantities and prices behave in this economy. Can changes in z be responsible for business cycles we observe? Explain. (Note that consumption, employment, and wages are all pro-cyclical in data.) (f) Is there a Laffer Curve for this economy? Explain.
Consider the following one-period model. Assume that the consumption good is produced by a linear technology: Y = zND where Y is the output of the con- sumption good, z is the exogenous total factor productivity, ND is the labour hours. Government has to finance its expenditures, G, using a lump-sum tax, T, on the rep- resentative consumer. There is no other tax in the economy. The firm is owned by the representative consumer who is endowed with h hours of time she can allocate between work, NS and leisure, l. Preferences of the representative consumer are:
U (c, l) = α ln c + (1 − α) ln l (1) where 0 < α < 1 is a parameter.
Answers for part b below:
A consumer's choice of optimizing its consumption and labor hours (h - l ) is given by the point where , MRS(c , l) = wage rate
Now ,
MRS (c, l) = MU(c) /MU(l)
MU(c) = dU/d c = a/c
MU(l) = dU /dl = (1-a )/l
MRS = a (l)1−a (c)a (l)1-a (c)
Putting this in optimal condition we have :
a (l)(1−a) (c)a (l)(1-a) (c) = w --- (i)
l = w *c (1- a)/a
We put this value into budget constraint :
c* = w ( h - w *c (1- a)/a ) - T (Optimal Consumption )
From equation (i) we get :
c = a (l) / (1-a) w
Putting this value into budget constraint :
a (l) / (1-a) w = w (h -l ) - T
l* = wh − T (a(1−a)w+w )wh - T (a(1-a)w+w ) (Optima Leisure )
We know , l + N = h
N* = h - l*
N* = h - wh − T (a(1−a)w+w )wh - T (a(1-a)w+w ) (Optimal Labor / Employment )
Putting this value into production function we get :
Y* = zN*
Y* = z (h - wh − T (a(1−a)w+w )wh - T (a(1-a)w+w ) ) (Optimal Output level )
Answers in part (b) ^
Solve Below
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Suppose that the government spending, G, increases. Using your answers in (1b), determine and explain how endogenous quantities and prices behave in this economy. Can changes in G be responsible for business cycles we observe? Explain. (Note that consumption, employment, and wages are all pro-cyclical in data.)
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(e) Suppose that the total factor productivity, z, increases. Using your answers in (1b), determine and explain how endogenous quantities and prices behave in this economy. Can changes in z be responsible for business cycles we observe? Explain. (Note that consumption, employment, and wages are all pro-cyclical in data.)
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(f) Is there a Laffer Curve for this economy? Explain.
Hi there , as you have posted multiple sub parts so as per our guidelines we will only solve first three . Kindly repost the remaining parts to get them solved .
Production function : Y = zNd
Utility function : U = a ln c + (1-a) ln (l)
Where , c = consumption , l = leisure
Total number of hours available = h
Time Constraint : N + l = h (where , N = Labor hours )
Budget Contraint : Consumption = Income - Taxes => c = wNs - T
=> c = w (h - l) - T
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