We use the following functional forms to simulate our results:
A Worker’s working time: tw(yw, αw) = (yw αw)2 A Worker’s learning cost: c(αw, i) = i − αw A Worker’s production knowledge: Z(yw) = y2w A Worker’s ability: αw = 0.5
Problem price: p(yw) = β · ywγ, β > 0, γ ≥ 0
A Manager’s working time: tm(ym, yw, αm) = (ym − yw αm )2 A Manager’s learning cost: c(αm, i) = i − αm
A Manager’s production knowledge: Z(ym) = ym2 A Manager’s ability: αm = 1
Problem distribution: yi = 1 − Zmin
Z , Zmin > 0, Z ≥ Zmin, i = w, m
In this paper, our predictions show results similar to the empirical find-ings above. That is, because of communication technology improvement, the low skill agents tend to ask their manager to solve problems rather than solving the problems by learning, which reduces their working time and increases their leisure. For the high skill agent (a manager), she needs to spend more time on solving higher proportion of the problems due to the technology improvement, so her working time increases and her leisure time decreases. These results explain why leisure inequality occurred between 1985 and 2005. At that time, wage inequality between high- and low-skilled agents increased dramatically. Wage inequality can also be illustrated by improvements in communication technology because the improvements lead to the patterns that the workers earn less and the manager earns more. Therefore, our simulation concludes that the com-munication technology improvement leads to the two patterns in leisure inequality and wage inequality between high skill agents and low skill
agents.
Figure 1: The proportion of the problem a manager solved ym versus the communication costs h.
ym = −0.571 h + 0.92 for h ∈ (0, 0.56]
yw = 0.0357 h + 0.2 for h ∈ (0, 0.56]
Figure 1 shows that improvements in communication technology, h, lead to an increase in the proportion of a problem which a manager can solve, ym = G(Z). As depicted, the improvements cause a manager to spend more time on solving a higher proportion of the problem, and thus her working time increases and leisure time decreases as shown in Figure 2.
Figure 2: The leisure of agent i versus the proportion of the problem.
lm = −0.14 ym + 0.589 for ym ∈ [0.6, 0.92]
lw = −1.7 yw + 1.18 for yw ∈ [0.2, 0.22]
Figure 3: Earning of agent i versus the proportion of the problem.
Em = 1.625 ym − 0.345 for ym ∈ [0.6, 0.92]
Ew = 4.1 yw − 0.612 for yw ∈ [0.2, 0.22]
In figure 2, the manager’s leisure moves from l0m to l00m. That means her leisure time decreases because of the technology improvement. Mean-while, her earning increases because she obtains more return on problem-solving after the improvement. The upward direction of her earning is shown in Figure 3. The point moves from Em0 to Em00 . The notable trend is the fraction of the problem a worker solved, yw = G(Z), see Figure 3.
Observe (from point Ew0 to point Ew00) in Figure 3, a worker tends to solve smaller fraction of the problem after the technology improvement.
That’s because the cost of asking a manager for directions is relatively lower than solving the problem by himself. This reduction implies a worker makes a less effort to work, and thus his leisure time increases as shown in Figure 2 (from point lw0 to lw00). All results are shown in Ta-ble 1 and the functional forms used in this simula-tion is presented as above. Therefore, these figures illustrate that the communication tech-nology improvement leads to increases in the manager’s earning and the worker’s leisure and decreases in the manager’s leisure and the worker’s earning. That is, the improvement in communication technology leads to wage inequality and leisure inequality between high skill agents and low skill agents.
Table 1.1 Agents’ earnings before and after technology improvement h = 0.56 h = 0.28 h = 0.01 other parameters
Em 0.63 0.89 1.1407 i = 1, αm = 1 Ew 0.29 0.248 0.209 αw = 0.5, i = 1
Table 1.2 Agents’ leisure before and after technology improvement h = 0.56 h = 0.28 h = 0.01 other parameters
lm 0.505 0.482 0.461 i = 1, αm = 1 lw 0.806 0.823 0.839 αw = 0.5, i = 1
To be clear, let us turn to the mathematics part. In the previous sec-tion, some functions are used to determine the equilibrium conditions.
By using the functional forms as page 16, we do comparative status to understand how the technology change affects the equilibrium quantities.
From equation (14) in Section II, the manager’s wage per unit of work-ing time is given by
(25) : wm = 2(1 − yw) h + 2(ym− yw).
If the communication technology improves, h ↓, her wage increases.
That is, the partial derivative of equation (18) with respect to h increases as
(26) : ∂wm
∂h = − 2(1 − yw)
(h + 2(ym − yw))2 < 0.
Proposition 1 The improvement in communication technology leads to an increase in the manager’s wage.
As shown in Figure 1, due to improvements in communication tech-nology, the equilibrium quantities of ym increases and yw decreases. The following propositions is attributed to this effect:
Proposition 2 The improvement in communication technology leads to the opposite effect on production knowledge between high- and low-skilled agents: an increase in high-skilled agents’ production knowledge and an decrease in low-skilled agents’ production knowledge.
From equation (18), we have
From proposition 1, we know that the wage of the manager increases.
The increase implies the marginal rate of substitution between her con-sumption and leisure rises. That is, due to the technology improvement, the manager’s consumption increases from c0m to c00m and her leisure time declines from lm0 to lm00 . Hence, we can obtain the following proposition:
Proposition 3 The improvement in communication technology leads to a decline in leisure for the manager.
From equation (7) in Section II, the worker’s wage per unit of working time is given by
(28) : ww = 2 − 2yw(1 + (i − 0.5))
4yw .
The partial derivative of equation (29) with respect to i can be obtained by
(29) : ∂ww
∂i = −1/2 < 0.
This derivative means a decrease in ww is caused by an increase in the cost of acquiring production knowledge, i. Thus, for a worker, when the cost of asking a manager for help is relatively lower than learning and solving by himself, he tends to solve lower proportion of the problem, which decreases his available income on consumption. The prediction is the same as the trend in figure 3. The prediction of equation (30) leads to the following proposition:
Proposition 4 The wage of a worker decreases due to an increase in the cost of acquiring production knowledge.
According to equation (5), we have
(30) : lw = 1 − (yw αw)2.
As the manager’s part we discussed, if there is a reduction in the com-munication cost h, a worker tends to solve smaller fraction of the problem, yw ↓. This decline leads to an increase in his leisure. The prediction is also the same as we showed in the previous figure. The prediction is stated with the following proposition as well:
Proposition 5 The improvement in communication technology leads to an increase in leisure for a worker.
In conclusion, our simulation is consistent with empirical findings. Leisure inequality and wage inequality between high- and low-skilled agents are caused by the improvement in communication technology.
5 Conclusion
Over decades, wage inequality and leisure inequality have dramatically increased across agents with different levels of skills. In this paper, we have explored how technology improvements affect both inequalities. [1]
states that wage inequality between high- and low-skilled agents is caused by the improvement in communication technology. Following their the-ory, our paper considers labor-leisure choices among agents. We under-stood how technology change influences leisure inequality among agents.
Our results show that improvements in communication technology lead to leisure inequality between high- and low-skilled agents besides wage inequality. Because of reductions in communication costs of solving prob-lems, the two types of agents have different decisions on their working time and leisure. That is, the low-skilled agents choose leisure more than work on their job. Oppositely, under the technology change, the high-skilled agents work more than before. Our findings are consistent with the empirical data presented in [5,6]. Of course, our work has limitations:
our theory does not include leisure goods. The goods used in leisure have been discussed in recent papers such as [7, 9]. Futhermore, our paper does not consider bargaining power between buyer and seller, and match-ing problem among the high-skilled and the low-skilled agent. It would be interesting to consider these extensions.
References
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