The Labor - Leisure Choice 

 

The following was implemented in Maple by Marcus Davidsson (2009) davidsson_marcus@hotmail.com 

and is based upon Kejak & Duras (2007) Preparatory macroeconomics 

 

1) The Basic Labor-Leisure Model

1.1) The Derivation
 

We start by defining the following: 

 

The total number of hours available is denoted by  

The number of leisure hours is denoted by 

The number of working hours is denoted by  

The wage per hour is denoted by  

 

We now note that if a person works hours and the wage rate per hour is given by then the income is given by
 

 

We now assume that the income from working can be used to buy either consumption or leisure.


Our budget constraint can therefore be written as:
 

 

which says that the income from working must be equal to given by the cost of consumption where is the price of  consumption.
 

We now assume that the price of consumption is normalized to one so we get which means that:
 

 

which can be written as:
 

 

 

Which can be written in an equal to zero form as:
 

 

(1)

 

which says that income from working minus consumption must be equal to zero which means that the consumer always must run a

balanced budget ie he can not spend more money than he has earned from working. 

 

We now assume that the consumer get utility from consumption and leisure . We assume that the utility function is simply

given by a constant return to scale Cobb Douglas utility function given by
 

 

(2)

 

which can be plotted as





 

 

The consumer now want to find the optimal amount of consumption and leisure that will maximize his utility.


We therefore set up the Lagrange as follows:


 

(3)

 

The first order conditions are given by 

foc-1 

 

(4)

 

we solve for  

 

(5)

 

foc-2
 

 

(6)

 

we now add on both sides so we get: 

 

 

(7)

 

we divide both sides by
 

 

 

(8)

 

 

 

foc-3 

 

 

(9)

 

 

which is equal to our initial budget constraint 

 

 

 

 

 

Manipulation foc-1 and foc-2 

 

 

foc-1 is given by 

 

 

 

(10)

 

foc-2 is given by 

 

 

 

(11)

 

 

We now set foc-1 equal to foc-2 

 

 

 

(12)

 

 

we solve for  

 

 

(13)

 

 

 

if we plug in the expression for c into our budget constraint we get: 

 

 

 

(14)

 

We solve for  

 

 

 

(15)

 

 

 

 

 

Which is the final expression for the optimal amount of leisure. 

 

 

 

Since L was still present on the right hand side in our consumption equation we have to remove L by substituting in the expression for L 

 

 

 

(16)

 

 

 

 

Which is the final expression for the optimal amount of consumption 

 

 

 

 

 

 

 

1.2) Checking and Plotting the Model 

 

 

 

We start by checking that we indeed have solved for the optimal values of leisure and consumption 

 

 

 

 

 

 

 

	(17)

 

 

 

We now note that that our budget constraint is given by 

 

 

(18)

 

 

 

We solve for since we assume that consumption is located on the y-axis 

 

 

(19)

 

 

 

 

We can now plot our budget constraint and the indifference curves 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2) The Basic Labor-Leisure Model with Labor Taxes
 

2.1) The Derivation

 

We again start by defining the following: 

 

The total number of hours available is denoted by  

The number of leisure hours is denoted by 

The number of working hours is denoted by  

The wage per hour is denoted by  

The tax rate is denoted by  

We now note that if a person works hours and the wage rate per hour is given by and the tax rate is given by

then the after tax income is given by:

 

 

We now assume that the after tax income from working can be used to buy either consumption or leisure.


Our budget constraint can therefore be written as:

 

 

which says that the after tax income from working must be equal to given by the cost of consumption where is the price of  consumption.
 

We now assume that the price of consumption is normalized to one so we get which means that:
 

 

which can be written as:
 

 

 

Which can be written in an equal to zero form as:
 

 

(20)

 

which says that the after tax income from working minus consumption must be equal to zero which means that the

consumer always must run a balanced budget ie he can not spend more money than his after tax earnings from working. 

 

We now assume that the consumer get utility from consumption and leisure . We assume that the utility function is simply

given by a constant return to scale Cobb Douglas utility function given by 

 

 

(21)

 

which can be plotted as





 

 

The consumer now want to find the optimal amount of consumption and leisure that will maximize his utility.


We therefore set up the Lagrange as follows:


 

(22)

 

The first order conditions are given by 

foc-1 

 

(23)

 

we solve for  

 

(24)

 

foc-2
 

 

(25)

 

we now add on both sides so we get: 

 

 

(26)

 

we divide both sides by
 

 

 

(27)

 

 

 

foc-3 

 

 

(28)

 

 

which is equal to our initial budget constraint 

 

 

 

 

Manipulation foc-1 and foc-2 

 

 

foc-1 is given by 

 

 

 

(29)

 

foc-2 is given by 

 

 

 

(30)

 

 

We now set foc-1 equal to foc-2 

 

 

 

(31)

 

 

we solve for  

 

 

(32)

 

 

if we plug in the expression for C into our budget constraint we get: 

 

 

 

(33)

 

We solve for  

 

 

 

(34)

 

 

 

Which is the final expression for the optimal amount of leisure. 

 

 

 

 

Since L was still present on the right hand side in our consumption equation we have to remove L by substituting in the expression for L 

 

 

 

(35)

 

 

 

 

Which is the final expression for the optimal amount of consumption 

 

 

 

 

 

 

 

 

2.2) Checking and Plotting the Model 

 

 

 

 

We start by checking that we indeed have solved for the optimal values of leisure and consumption 

 

 

 

 

 

 

 

	(36)

 

 

 

We now note that that our budget constraint is given by 

 

 

(37)

 

 

 

We solve for since we assume that consumption is located on the y-axis 

 

 

(38)

 

 

 

We can now plot our budget constraint and the indifference curves with an Labor tax
 

 

 

 

 

 

 

 

 

 

 

We can see that the optimal amount of consumption is decreasing when a labor tax is introduced but a labor tax does not affect the

optimal amount of leisure hours or working hours for that matter since  

 

 

 

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