<p>Note first that if an individual generally reacts to increases in the wage by a decrease in labor supply, then the individual must have maximal labor supply at a wage of 0. I will supply such an example but the example can be adjusted to a more realistic setting.</p>

<p>Let's start with the consumer problem. Suppose there is no capital income, such that $c=w\cdot l$ where $w$ is the real wage. If we think of a reason why labor supply is decreasing in income, then one idea is that the higher the consumption, the larger the disutility from labor. (This is actually not that far fetched, think of how lovely leisure can be with a yacht.)</p>

<p>So let's take a standard quasilinear utility function and adjust it to $u=\ln c+(1-l\cdot c)$. Inserting the budget constraint yields: $u=\ln (w \cdot l) + (1-w\cdot l^2)$. As the function is concave in $l$, first order conditions are necessary and sufficient for a maximum:
$$1/l - 2wl = 0$$
from which follows
$l=(1/(2w))^{.5}$ and labor supply is decreasing in the wage. We could generalize this function by adding an offset to $c$ and $l$ in order to create the income effect dominance only after a certain wage rate.</p>

CC BY-SA 3.0
2015-09-29T17:11:49.313Z
(+1). Indeed, separable utility in consumption and leisure, is totally unrealistic. The fact that "leisure" does not mean "not working and staying still" but engaging in enjoyable activities (and this always implies consumption even if only of calories and a bit of wear-off of durable goods), is a good starting point to rethink the modelling of utility from consumption and leisure.
Alecos Papadopoulos
CC BY-SA 3.0
2015-09-28T16:27:40.103Z
2015-09-28T16:27:40.103Z
HRSE
<p>I think it was in one of Joan Robinson's books where a not just backward-bending but "mirrored S"-shaped labour supply was presented, referring to agricultural societies:</p>

<p><a href="https://i.sstatic.net/Brwnr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Brwnr.png" alt="enter image description here"></a></p>

<p>The rationale of course is that if your productivity is very low, you have to work very long and exhausting hours to even survive. As your productivity raises, what you want is not to work even more for the extra income, but thankfully to get some rest.<br>
The empirical regularity is that most of workers in the world are away from the upper "backward" movement of labor supply - but in poor countries, they may very well be in the lower backward part. And I think this resonates with the information presented in @JohnL. answer. So chances are, you will find such preferences modeled in studies that focus on poor/developing economies.</p>

CC BY-SA 3.0
2015-08-27T21:09:44.123Z
2015-08-27T21:09:44.123Z
Alecos Papadopoulos
<p>In open macroeconomic models some cases of logarithmic preferences cause the wealth effect to dominate the intertemporal substitution effect.</p>

<p>If I recall correctly, at least Vegh's book "Open Economy Macroeconomics in Developing Countries" (2013, MIT Press) presents this case.</p>

CC BY-SA 3.0
2015-08-25T14:10:36.090Z
2015-08-25T14:10:36.090Z
John L.
<ul>
<li>King-Plosser-Rebelo preferences satisfy balanced growth requirements, we have that income and substitution effects of labor cancel. Labor does not respond to a change in the wage level.</li>
<li>Greenwood-Hercowitz-Huffman preferences shut off the wealth channel: There is only a substitution effect left, hence labor co-moves with wages.</li>
<li>Jaimovich-Rebelo create the nested case with a wealth effect within the two aforementioned.</li>
</ul>

<p>Is there any commonly used preference specification for which the wealth effect <strong>dominates</strong> the substitution effect, and hence labor supply reduces as a response to an increase in wages?</p>

CC BY-SA 3.0
2015-09-28T17:52:32.937Z
Just realized that I need a clarification, to be on the safe side: are you looking for a comparative static result in a given time period, i.e. something like $dL/dw <0$, ($L$ is labor) or for a result in rates of change through time, i.e. something like $\dot w > 0 \implies \dot L <0$?
Alecos Papadopoulos
CC BY-SA 3.0
2015-09-28T19:10:58.703Z
@AlecosPapadopoulos I'm looking for comparative statics.
FooBar
CC BY-SA 3.0
2015-09-30T01:33:44.820Z
Please specify whether you are looking for a dominating wealth effect locally or for all budget constraints and what these budget constraints include (e.g. capital income).
HRSE
CC BY-SA 3.0
2015-09-30T05:23:50.570Z
@HREcon I'm open to local and global domination. Regarding budget constraints, I prefer the case with labor income only.
FooBar
<utility><preferences>
Preferences where wealth effect dominates
FooBar
# Preferences where wealth effect dominates

(Asked by: username_0 on 2015-08-25)

* King-Plosser-Rebelo preferences satisfy balanced growth requirements, we have that income and substitution effects of labor cancel. Labor does not respond to a change in the wage level.
* Greenwood-Hercowitz-Huffman preferences shut off the wealth channel: There is only a substitution effect left, hence labor co-moves with wages.
* Jaimovich-Rebelo create the nested case with a wealth effect within the two aforementioned.

Is there any commonly used preference specification for which the wealth effect **dominates** the substitution effect, and hence labor supply reduces as a response to an increase in wages?

**Tags:** utility, preferences

> **Comments:**
> **username_1**: Just realized that I need a clarification, to be on the safe side: are you looking for a comparative static result in a given time period, i.e. something like $dL/dw <0$, ($L$ is labor) or for a result in rates of change through time, i.e. something like $\dot w > 0 \implies \dot L <0$?  
> **username_0**: @username_1 I'm looking for comparative statics.  
> **username_2**: Please specify whether you are looking for a dominating wealth effect locally or for all budget constraints and what these budget constraints include (e.g. capital income).  
> **username_0**: @HREcon I'm open to local and global domination. Regarding budget constraints, I prefer the case with labor income only.  

---

## 3 answers

### Answer by username_2 on 2015-09-28. Score: 3

Note first that if an individual generally reacts to increases in the wage by a decrease in labor supply, then the individual must have maximal labor supply at a wage of 0. I will supply such an example but the example can be adjusted to a more realistic setting.

Let's start with the consumer problem. Suppose there is no capital income, such that $c=w\cdot l$ where $w$ is the real wage. If we think of a reason why labor supply is decreasing in income, then one idea is that the higher the consumption, the larger the disutility from labor. (This is actually not that far fetched, think of how lovely leisure can be with a yacht.)

So let's take a standard quasilinear utility function and adjust it to $u=\ln c+(1-l\cdot c)$. Inserting the budget constraint yields: $u=\ln (w \cdot l) + (1-w\cdot l^2)$. As the function is concave in $l$, first order conditions are necessary and sufficient for a maximum:
$$1/l - 2wl = 0$$
from which follows
$l=(1/(2w))^{.5}$ and labor supply is decreasing in the wage. We could generalize this function by adding an offset to $c$ and $l$ in order to create the income effect dominance only after a certain wage rate.

> **Comments:**
> **username_1**: (+1). Indeed, separable utility in consumption and leisure, is totally unrealistic. The fact that "leisure" does not mean "not working and staying still" but engaging in enjoyable activities (and this always implies consumption even if only of calories and a bit of wear-off of durable goods), is a good starting point to rethink the modelling of utility from consumption and leisure.  

---

### Answer by username_1 on 2015-08-27. Score: 2

I think it was in one of Joan Robinson's books where a not just backward-bending but "mirrored S"-shaped labour supply was presented, referring to agricultural societies:

[![enter image description here](https://i.sstatic.net/Brwnr.png)](https://i.sstatic.net/Brwnr.png)

The rationale of course is that if your productivity is very low, you have to work very long and exhausting hours to even survive. As your productivity raises, what you want is not to work even more for the extra income, but thankfully to get some rest.  
The empirical regularity is that most of workers in the world are away from the upper "backward" movement of labor supply - but in poor countries, they may very well be in the lower backward part. And I think this resonates with the information presented in @JohnL. answer. So chances are, you will find such preferences modeled in studies that focus on poor/developing economies.


---

### Answer by username_3 on 2015-08-25. Score: 2

In open macroeconomic models some cases of logarithmic preferences cause the wealth effect to dominate the intertemporal substitution effect.

If I recall correctly, at least Vegh's book "Open Economy Macroeconomics in Developing Countries" (2013, MIT Press) presents this case.


---

