Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
✓ct
1 ht = (1 ⌧t)(1 ↵)(yt/ht) 1
THE LABOR WEDGE
This demonstrates how to construct the labor wedge for each country.
1 The Household
The household has a utility function
U =logCt+✓log(1 ht)
where C is consumption and h is hours worked. The household’s utility is therefore increasing in consumption and leisure. The household maximizes utility subject to a budget constraint:
Ct= (1 ⌧t)wtht
wherewtis the real wage and⌧ is the tax on labor.
TheÆrst order conditions for this problem yield an equilibrium condition:
✓ct
1 ht = (1 ⌧t)wt
Note that what this equilibrium condition states is that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indi erence curve (in absolute value). The after tax real wage is the slope of the budget line (in absolute value).
Graphically, we always show that the slope of the indi erence curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put di erently, the left-hand side is capturing the marginal beneÆt of leisure whereas the right-hand side of the equation is measuring the marginal cost of leisure. Thus, in equilibrium, the marginal beneÆt from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
2 Firm
TheÆrm wants to maximize proÆt:
P rof it=zth1t ↵ wtht
where z is some measure of productivity and ↵ is the labor share of aggregate income. Solving the proÆt-maximization problem yields:
(1 ↵)ztht↵=wt
Thus, the marginal product of labor is equal to the wage. Given the properties of the production function, we could write this as
(1 ↵)yt ht =wt
3 Equlibrium
Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
54
This equilibrium condition states that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indifference curve (in absolute value). The after-tax real wage is the slope of the budget line (in absolute value). Graphically, we always show that the slope of the indifference curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put differently, the left-hand side is capturing the marginal benefit of leisure whereas the right-hand side of the equation is
measuring the marginal cost of leisure. Thus, in equilibrium, the marginal benefit from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
Now we can look at this from the firm’s perspective. The firm wants to maximize profit,
where z is some measure of productivity and α is the labor share of aggregate income.
Solving the profit-maximization problem yields:
Thus, the marginal product of labor is equal to the wage. Given the properties of the
production function, we could write this as
Combining the labor supply curve and the labor demand curve, we can determine the THE LABOR WEDGE
This demonstrates how to construct the labor wedge for each country.
1 The Household
The household has a utility function
U =logCt+✓log(1 ht)
where C is consumption and h is hours worked. The household’s utility is therefore increasing in consumption and leisure. The household maximizes utility subject to a budget constraint:
Ct= (1 ⌧t)wtht wherewtis the real wage and⌧ is the tax on labor.
TheÆrst order conditions for this problem yield an equilibrium condition:
✓ct
1 ht = (1 ⌧t)wt
Note that what this equilibrium condition states is that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indi erence curve (in absolute value). The after tax real wage is the slope of the budget line (in absolute value).
Graphically, we always show that the slope of the indi erence curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put di erently, the left-hand side is capturing the marginal beneÆt of leisure whereas the right-hand side of the equation is measuring the marginal cost of leisure. Thus, in equilibrium, the marginal beneÆt from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
2 Firm
TheÆrm wants to maximize proÆt:
P rof it=zth1t ↵ wtht
where z is some measure of productivity and ↵ is the labor share of aggregate income. Solving the proÆt-maximization problem yields:
(1 ↵)ztht↵=wt
Thus, the marginal product of labor is equal to the wage. Given the properties of the production function, we could write this as
(1 ↵)yt
ht
=wt
3 Equlibrium
Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
✓ct
1 ht = (1 ⌧t)(1 ↵)(yt/ht) 1
THE LABOR WEDGE
This demonstrates how to construct the labor wedge for each country.
1 The Household
The household has a utility function
U =logCt+✓log(1 ht)
where C is consumption and h is hours worked. The household’s utility is therefore increasing in consumption and leisure. The household maximizes utility subject to a budget constraint:
Ct= (1 ⌧t)wtht where wt is the real wage and⌧ is the tax on labor.
TheÆrst order conditions for this problem yield an equilibrium condition:
✓ct
1 ht = (1 ⌧t)wt
Note that what this equilibrium condition states is that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indi erence curve (in absolute value). The after tax real wage is the slope of the budget line (in absolute value).
Graphically, we always show that the slope of the indi erence curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put di erently, the left-hand side is capturing the marginal beneÆt of leisure whereas the right-hand side of the equation is measuring the marginal cost of leisure. Thus, in equilibrium, the marginal beneÆt from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
2 Firm
TheÆrm wants to maximize proÆt:
P rof it=zth1t ↵ wtht
where z is some measure of productivity and ↵ is the labor share of aggregate income. Solving the proÆt-maximization problem yields:
(1 ↵)ztht↵=wt
Thus, the marginal product of labor is equal to the wage. Given the properties of the production function, we could write this as
(1 ↵)yt ht
=wt
3 Equlibrium
Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
✓ct
1 ht = (1 ⌧t)(1 ↵)(yt/ht) 1
THE LABOR WEDGE
This demonstrates how to construct the labor wedge for each country.
1 The Household
The household has a utility function
U =logCt+✓log(1 ht)
where C is consumption and h is hours worked. The household’s utility is therefore increasing in consumption and leisure. The household maximizes utility subject to a budget constraint:
Ct= (1 ⌧t)wtht
wherewtis the real wage and⌧ is the tax on labor.
TheÆrst order conditions for this problem yield an equilibrium condition:
✓ct
1 ht = (1 ⌧t)wt
Note that what this equilibrium condition states is that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indi erence curve (in absolute value). The after tax real wage is the slope of the budget line (in absolute value).
Graphically, we always show that the slope of the indi erence curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put di erently, the left-hand side is capturing the marginal beneÆt of leisure whereas the right-hand side of the equation is measuring the marginal cost of leisure. Thus, in equilibrium, the marginal beneÆt from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
2 Firm
TheÆrm wants to maximize proÆt:
P rof it=zth1t ↵ wtht
where z is some measure of productivity and ↵ is the labor share of aggregate income. Solving the proÆt-maximization problem yields:
(1 ↵)ztht↵ =wt
Thus, the marginal product of labor is equal to the wage. Given the properties of the production function, we could write this as
(1 ↵)yt
ht =wt
3 Equlibrium
Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
✓ct
1 ht = (1 ⌧t)(1 ↵)(yt/ht) THE LABOR WEDGE
This demonstrates how to construct the labor wedge for each country.
1 The Household
The household has a utility function
U =logCt+✓log(1 ht)
where C is consumption and h is hours worked. The household’s utility is therefore increasing in consumption and leisure. The household maximizes utility subject to a budget constraint:
Ct= (1 ⌧t)wtht where wt is the real wage and⌧ is the tax on labor.
TheÆrst order conditions for this problem yield an equilibrium condition:
✓ct 1 ht
= (1 ⌧t)wt
Note that what this equilibrium condition states is that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indi erence curve (in absolute value). The after tax real wage is the slope of the budget line (in absolute value).
Graphically, we always show that the slope of the indi erence curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put di erently, the left-hand side is capturing the marginal beneÆt of leisure whereas the right-hand side of the equation is measuring the marginal cost of leisure. Thus, in equilibrium, the marginal beneÆt from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
2 Firm
TheÆrm wants to maximize proÆt:
P rof it=zth1t ↵ wtht
where z is some measure of productivity and ↵ is the labor share of aggregate income. Solving the proÆt-maximization problem yields:
(1 ↵)ztht↵ =wt
Thus, the marginal product of labor is equal to the wage. Given the properties of the production function, we could write this as
(1 ↵)yt
ht =wt
3 Equlibrium
Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
✓ct
= (1 ⌧ )(1 ↵)(y/h)
55 equilibrium in the labor market:
Before discussing the labor wedge, one can look at what this says intuitively. First, consider the equilibrium in the absence of the tax on labor:
Written in this form, the left-hand side represents the marginal rate of substitution between consumption and leisure. The right-hand side represents the marginal product of labor. Since an equilibrium must be consistent with utility maximization and profit maximization, in the absence of taxes the marginal rate of substitution must be equal to the marginal product of labor:
Taxes generate a wedge between these values, for example like the wedge that taxes generate between the prices that buyers pay and that sellers receive. In the context of our framework, with taxes, it is true that:
Thus, it is no longer the case that the marginal product of labor is equal to the marginal rate of substitution. In addition, the larger the tax rate, the larger the
difference between MRS and MPL (i.e. the more leisure and the less hours worked in equilibrium).
THE LABOR WEDGE
This demonstrates how to construct the labor wedge for each country.
1 The Household
The household has a utility function
U =logCt+✓log(1 ht)
where C is consumption and h is hours worked. The household’s utility is therefore increasing in consumption and leisure. The household maximizes utility subject to a budget constraint:
Ct= (1 ⌧t)wtht where wt is the real wage and⌧ is the tax on labor.
TheÆrst order conditions for this problem yield an equilibrium condition:
✓ct
1 ht = (1 ⌧t)wt
Note that what this equilibrium condition states is that the marginal rate of substitution is equal to the after-tax wage earned by the worker. The marginal rate of substitution is the slope of the indi erence curve (in absolute value). The after tax real wage is the slope of the budget line (in absolute value).
Graphically, we always show that the slope of the indi erence curve is equal to the slope of the budget line. This equilibrium condition is the mathematical analog to the graph. Put di erently, the left-hand side is capturing the marginal beneÆt of leisure whereas the right-hand side of the equation is measuring the marginal cost of leisure. Thus, in equilibrium, the marginal beneÆt from leisure must be equal to the marginal cost of leisure. Otherwise, we aren’t maximizing utility. Finally, we can think of this equilibrium condition as the labor supply curve.
2 Firm
TheÆrm wants to maximize proÆt:
P rof it=zth1t ↵ wtht
where z is some measure of productivity and ↵ is the labor share of aggregate income. Solving the proÆt-maximization problem yields:
(1 ↵)ztht↵=wt
Thus, the marginal product of labor is equal to the wage. Given the properties of the production function, we could write this as
(1 ↵)yt ht =wt
3 Equlibrium
Combining the labor supply curve and the labor demand curve, we can determine the equilibrium in the labor market:
✓ct
1 ht = (1 ⌧t)(1 ↵)(yt/ht) 1
Before discussing the labor wedge, let’s consider what this says intuitively. First, consider the equilib- rium in the absence of the tax on labor:
✓ct
1 ht = (1 ↵)(yt/ht)
Written in this form, the left-hand side represents the marginal rate of substitution between consumption and leisure. The right-hand side represents the marginal product of labor. Since an equilibrium must be consistent with utility maximization and proÆt maximization, in the absence of taxes the marginal rate of substitution must be equal to the marginal product of labor:
M RS=M P L
or M RS
M P L = 1
Taxes generate a wedge between these values. (To understand the concept of a labor wedge, think back to principles of microeconomics and the cost of taxation. Taxes generate a wedge between the price that buyers pay and that sellers receive.) In the context of our framework, with taxes, it is true that:
M RS
M P L = 1 ⌧ (1)
Thus, it is no longer the case that the marginal product of labor is equal to the marginal rate of substitution. In addition, the larger the tax rate, the larger the di erence between MRS and MPL (i.e. the more leisure and the less hours worked in equilibrium).
However, if we generalize equation (1), we can think of ⌧ as not simply the tax rate, but as simply anything that might drive a wedge between the MRS and the MPL. THIS is what we mean by the labor wedge. The size of the labor wedge might therefore depend on factors other than tax rates. It might, for example, depend on the degree of regulation in the labor market. As a result, we would like to estimate the labor wedge for each country by using our equilibrium condition. Re-arranging our equilibrium condition above, we have:
⌧ = 1 ✓ 1 ↵
ct yt
ht 1 ht
We can then calculate⌧ by making the following assumptions about the parameters:
✓= 0.7 1 ↵=.67
In addition, since we have normalized the amount of time that a worker can devote to work and leisure to sum to unity, we can normalize hours worked by making the total amount of time available equal to the total number of potential hours. For example, let’s assume that workers can work 5 days per week and that they must get 8 hours of sleep per day. Thus, the total number of hours that someone could work in a year is5⇤16⇤52 = 4,160. Thus, to calculateh take the total number of hours worked per person per year and divide this number by 4,160. One can then use this normalized measure of hours, the parameter values above, and the consumption-to-GDP ratio to calculate⌧.
2
Before discussing the labor wedge, let’s consider what this says intuitively. First, consider the equilib- rium in the absence of the tax on labor:
✓ct
1 ht = (1 ↵)(yt/ht)
Written in this form, the left-hand side represents the marginal rate of substitution between consumption and leisure. The right-hand side represents the marginal product of labor. Since an equilibrium must be consistent with utility maximization and proÆt maximization, in the absence of taxes the marginal rate of substitution must be equal to the marginal product of labor:
M RS =M P L
or M RS
M P L = 1
Taxes generate a wedge between these values. (To understand the concept of a labor wedge, think back to principles of microeconomics and the cost of taxation. Taxes generate a wedge between the price that buyers pay and that sellers receive.) In the context of our framework, with taxes, it is true that:
M RS
M P L = 1 ⌧ (1)
Thus, it is no longer the case that the marginal product of labor is equal to the marginal rate of substitution. In addition, the larger the tax rate, the larger the di erence between MRS and MPL (i.e. the more leisure and the less hours worked in equilibrium).
However, if we generalize equation (1), we can think of ⌧ as not simply the tax rate, but as simply anything that might drive a wedge between the MRS and the MPL. THIS is what we mean by the labor wedge. The size of the labor wedge might therefore depend on factors other than tax rates. It might, for example, depend on the degree of regulation in the labor market. As a result, we would like to estimate the labor wedge for each country by using our equilibrium condition. Re-arranging our equilibrium condition above, we have:
⌧ = 1 ✓ 1 ↵
ct yt
ht 1 ht
We can then calculate⌧ by making the following assumptions about the parameters:
✓= 0.7 1 ↵=.67
In addition, since we have normalized the amount of time that a worker can devote to work and leisure to sum to unity, we can normalize hours worked by making the total amount of time available equal to the total number of potential hours. For example, let’s assume that workers can work 5 days per week and that they must get 8 hours of sleep per day. Thus, the total number of hours that someone could work in a year is5⇤16⇤52 = 4,160. Thus, to calculateh take the total number of hours worked per person per year and divide this number by 4,160. One can then use this normalized measure of hours, the parameter values above, and the consumption-to-GDP ratio to calculate⌧.
2
Before discussing the labor wedge, let’s consider what this says intuitively. First, consider the equilib- rium in the absence of the tax on labor:
✓ct
1 ht = (1 ↵)(yt/ht)
Written in this form, the left-hand side represents the marginal rate of substitution between consumption and leisure. The right-hand side represents the marginal product of labor. Since an equilibrium must be consistent with utility maximization and proÆt maximization, in the absence of taxes the marginal rate of substitution must be equal to the marginal product of labor:
M RS =M P L
or M RS
M P L = 1
Taxes generate a wedge between these values. (To understand the concept of a labor wedge, think back to principles of microeconomics and the cost of taxation. Taxes generate a wedge between the price that buyers pay and that sellers receive.) In the context of our framework, with taxes, it is true that:
M RS
M P L = 1 ⌧ (1)
Thus, it is no longer the case that the marginal product of labor is equal to the marginal rate of substitution. In addition, the larger the tax rate, the larger the di erence between MRS and MPL (i.e. the more leisure and the less hours worked in equilibrium).
However, if we generalize equation (1), we can think of⌧ as not simply the tax rate, but as simply anything that might drive a wedge between the MRS and the MPL. THIS is what we mean by the labor wedge. The size of the labor wedge might therefore depend on factors other than tax rates. It might, for example, depend on the degree of regulation in the labor market. As a result, we would like to estimate the labor wedge for each country by using our equilibrium condition. Re-arranging our equilibrium condition above, we have:
⌧ = 1 ✓ 1 ↵
ct yt
ht 1 ht
We can then calculate⌧ by making the following assumptions about the parameters:
✓= 0.7 1 ↵=.67
In addition, since we have normalized the amount of time that a worker can devote to work and leisure to sum to unity, we can normalize hours worked by making the total amount of time available equal to the total number of potential hours. For example, let’s assume that workers can work 5 days per week and that they must get 8 hours of sleep per day. Thus, the total number of hours that someone could work in a year is5⇤16⇤52 = 4,160. Thus, to calculate h take the total number of hours worked per person per year and divide this number by 4,160. One can then use this normalized measure of hours, the parameter values above, and the consumption-to-GDP ratio to calculate⌧.