paying by the hour: are wages the cost of waiting?

Paying by the Hour: Are Wages the Cost of Waiting?

Achal BassambooKellogg School of Management, Northwestern University

Martin A. LariviereKellogg School of Management, Northwestern University

Simin LiKellogg School of Management, Northwestern University

August 31, 2019

Consumers dislike waiting. In modeling consumer behavior for services, this obvious fact is often captured

by assuming that consumers incur a cost per-unit-time spent waiting. A natural question that arises is:

where does this waiting cost come from? Is it merely foregone earnings or is it something different? In this

paper, we address these questions by assuming an agent faces a utility maximization problem that focuses on

time allocation subject to time and budget constraints. The cost of waiting is then the minimum monetary

compensation required to keep the agent utility level constant as the agents loses time due to waiting.

We show that the relationship between the cost of waiting and wages depends intricately on the agent’s

compensation structure. We prove that agent’s cost of waiting is equal to her hourly wage only when the

compensation structure is linear. The equivalence between hourly wages and waiting costs breaks down for

a more general compensation structures such as fixed shifts or even piece-wise linear wages. Furthermore,

we show that hourly wages can overestimate or underestimate the cost of waiting in the presence of such

non-linear compensation structure. These results are robust to several modifications of our basic model.

Key words : value of time, service operations, economics of queues

History :

1. Introduction

Why should service firms provide short waits? On the one hand, that seems a trivial question. Firms

offer short waits because customers do not like waiting. On the other hand, good service requires

limiting how heavily capacity is utilized and hence is expensive. Without an explicitly defined

benefit for reducing customer waits, any attempt to develop an optimal scheme for managing

a queue will reduce to maximizing the utilization of capacity. Consequently, dating back to at

least Naor (1969), it has been common in modeling service systems to assume that customers are

endowed with a per-unit-time cost of waiting. The cost of excess capacity can then be balanced

with the saving in customer delay costs.

But what determines customer delay costs? That is the research question that this paper seeks

to answer. Given that some link between consumer utility and delay is essential to many models in

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service management, it is important to have a firm understanding of what underpins how customers

value their time.

The most common assumption when researchers need to estimate consumer delay cost is to

assume that it equals to their hourly wages1. Intuitively, an hour spent in a queue is an hour that

one cannot be working and thus an hour of earnings is lost. Several papers take this approach.

One stream of literature discusses the allocation of services and goods through differentiating

customers’ costs in queuing. Nichols et al. (1971) assumes there is a tradeoff between the price

of a commodity and the waiting time to get it. The agents monetary budget is linear in its time,

with the slope being the wage rate. Similarly, Suen (1989) models an agent’s budget limit to be

linearly decreasing in waiting time at the rate that equals her wage. One major criteria Keller and

Laughhunn (1973) used to measure the adequacy of physician capacity is through patients’ waiting

costs, which are estimated by wages loss from being absent from their jobs. Another stream of work

exploits heterogeneous customer waiting costs to segment customers through pricing and a priority

service discipline. Shmanske (1993) focuses on the effectiveness of a set of price discriminating tolls.

Gavirneni and Kulkarni (2016) propose and evaluate the benefit of the fee-based premier service

options. Both papers use customers’ income as a proxy for the heterogeneous waiting costs.

Assuming that waiting costs equal wages ignores that consumers allocate their limited time across

multiple activities. Time in a queue may not necessarily be taken from income generating activities

(working). Beginning with Becker (1965), there is a literature on how consumers allocate their time.

In this framework, some activities may generate income (i.e., the agent is paid to work). Other

activities require time to enjoy (e.g., seeing a movie). Agents are then optimizing their choices

subject to a conventional budget constraint as well as a time constraint. DeSerpa (1971) shows

that an agent’s marginal value of time in such a setting is a rate of substitution between the agent’s

time and budget constraints. We show that it is generally not true that the rate of substitution

equals the hourly wage, which we will elaborate in Section 4 and Section 5.

Our model follows in the spirit of Becker (1965). We consider a world in which an agent’s utility

depends on the time and money she has available for leisure. She seeks to maximize her overall

utility subject to constraints on both her spending and time. Money is only earned by working. In

addition to working and leisure, time must be spent on overhead tasks (e.g., cleaning the house

or doing laundry). A queuing delay can thus be modeled as an increase in overhead. We allow

the agent to spend money to reduce time allocated to overhead tasks. That is, the agent can hire

someone to clean her house or pay for priority service.

This simplified setting allows us to study the impact of different compensation structures. Our

premise is that not all jobs paying a given average hourly wage are the same. For example, one

1 The argument that wages are the value of time also makes its way into the popular press (Pesca, 2018).

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job may offer the agent complete flexibility in terms of how long to work (akin to what the gig

economy promises). Another employment opportunity may pay the same hourly wage but require

the agent to commit to a shift of τ hours - no more, no less. Suppose that the agent would work

less than τ hours if she had complete flexibility. Then she will be time-constrained in a setting

where she must commit to τ hours for the same hourly wage. Conversely, if she would work more

than τ hours under flexibility, she will be earnings-constrained when locked into a shift. Altonji and

Paxson (1988) term the former as overemployment and the latter as underemployment. We show

that when the agent is time-constrained, hourly wages underestimate the cost of waiting while they

overestimate the cost of waiting when the agent is earnings-constrained.

Of course, issues of mis-measuring delay costs are uninteresting if pricing and admissions deci-

sions are not terribly sensitive to customer costs. Consequently, in Section 2, we first look at the

consequence of misjudging waiting costs when pricing entry to a queuing system based on the

classfical model of Naor (1969). Next, we present the basic model and rigorously define the marginal

cost of waiting. We also define an appropriate hourly wage for various compensation structures.

We present the results to the question we raised above under the two main types of compensation

structures in Section 4.1 and Section 4.2 respectively. In Section 5, we perform several robustness

checks on the results we develop under the basic model.

2. Mismeasurement of Delay Costs Leads to Sub-optimal SystemDesign

Why should operations researchers care about an accurate measure of consumers’ delay costs? We

study this question by revisiting the setting of Naor (1969) but assuming the system is managed

by a profit maximizer. Recall that Naor (1969) models a service system as an M/M/1 queue.

The arrival rate to the system (not the queue) and service rate are λ and µ respectively, and we

further define ρ= λ/µ. All customers value the service gross of waiting and out of pocket costs at

r while all have a per-unit-time cost of waiting of c. The queue is visible and arriving customers

consequently tradeoff the value of service, its posted price, and their expected wait given the

number of customers in line. Customers consequently have a cutoff value N and only join the queue

if the number of customers in the system is below N when they arrive. If the system is controlled

by a profit maximizer, the problem can be modeled as choosing which threshold n∗ to induce with

the corresponding price set to leave an arrival finding n∗ − 1 customers in the system indifferent

to joining. That is, upon seeing the queue length, not all consumers will join the queue, which in

turn leads to a corresponding blocking probability of any newly arrived customer, pn∗ .

This pricing scheme has an immediate implication: if the service provider underestimates the

cost of waiting by even a small amount, he will overprice the service. Customers will consequently

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use a cutoff value of n∗− 1 resulting in a discrete drop in throughput. This can be seen from the

left-hand panel of Figure 1. Let P(x|c) be the total expected revenue obtained when the service

provider set this price assuming consumers having waiting cost x> 0 when in fact they have waiting

cost c. Here we plot, the percentage of revenue loss, (P(w|c)−P(c|c))/P(c|c), that results from

the mis-measurement of delay costs against the percentage of mis-measurement, (w − c)/c. We

suppose the agent has a delay cost c but that the profit-maximizing firm approximates consumer

delay costs by the publicly known wage rate w. When the wage rate underestimates the delay cost,

that is (w− c)/c < 0, we see a sudden drop in revenue P(w|c), due to the drop in the threshold.

However, the revenue loss is mitigated as lower wage rates lead to greater misestimation and a

higher price. Intuitively, the consumer threshold is fixed for range of prices. Within this range,

revenue is maximized at the highest price. Indeed, it is this feature of the revenue-maximizing price

that results in revenue taking an immediate drop if delay costs are underestimated even slightly.

In contrast, if the service provider overestimates the delay cost by a relatively small amount,

the consumer cutoff value remains n∗. Throughput is unchanged, but the service provider is not

charging the highest price that is consistent with the threshold n∗. In short, the service provider is

leaving money on the table. As the wage rate (and hence the degree of misestimation) increases,

the service provider decreases the price causing the revenue gap to expand. For a sufficiently high

perceived waiting cost, the service provider seeks to lower the throughput by raising his price to

reduce the consumer threshold to n∗−1. This causes the kink seen in Figure 1. A lower throughput

but a higher price results in the revenue gap growing at a lower rate.

To see how sensitive the revenue loss is to the percentage of cost mis-measurement, we plot the

largest revenue loss in ranges of mis-measurement (0, x), where x ∈ (0%,50%) and (x,0), where

x ∈ (−50%,0%). The figure reiterates that any mis-measurement in delay costs will lead to a

revenue loss. However, comparing overestimation and underestimation of the delay cost, the extent

of loss is asymmetric. The revenue gap is immediate and especially large when the delay cost is

underestimated by the wage rate. In the example presented here, the revenue loss can be up to 25%

when the customer’s wage rate underestimate her true delay cost. For the revenue gap is larger

when the system is more heavily loaded.

The numerical result supports our conjecture that mis-measurement in delay cost leads to a sub-

optimal design of the profit-maximizing service system, and the resulting revenue gap is especially

large for the over-employed, i.e. time-constrained, consumers when the service system is heavily

loaded. We further generalize the numerical study in the proposition below.

Proposition 1 (Revenue Gap for the Over-employed Consumers). Suppose the service

provider would induce a consumer threshold n∗ and a corresponding blocking probability pn∗ if

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Figure 1 The suboptimal system design under mis-measurement of delay costs. We fix the true delay cost

c= 35 and vary consumer’s average wage w in the range of wage rates, w ∈ (14,70). We set the service reward to

be r= 8.33 and the service rate µ= 30 per hour. In the left panel, we plot the percentage of revenue loss

(P(w)−P(c))/P(c) against the percentage of mis-measurement in the delay costs (w− c)/c. (w− c)/c < 0

indicates that the wage rates underestimate consumers’ actual delay costs. In the right panel, we plot the largest

possible revenue loss if wage is in a given distance away from the true delay cost. Formally, we use the percentage

of mis-measurement to represent the distance away from the true cost, that is x∈ (−50%,50%) on the horizontal

axis. The worst case loss, plotted in vertical axis, is then min(P(w)−P(c))/P(c) when

(w− c)/c∈ (min{0, x},max{0, x}).

-50% 0% 50% 100%

-25%

-20%

-15%

-10%

-5%

0%

=0.5

=0.7

=0.95

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

-25%

-20%

-15%

-10%

-5%

0%

=0.5

=0.7

=0.95

he knew the true delay cost. Consider the scenario where the consumer’s true delay cost c is

underestimated by her wage rate w by ε, i.e. w= c− ε, where ε > 0. The revenue loss of the service

provider under the mis-measurement scenario is given by,

L(pn∗ ;λ, c, r,µ)→ λ(r−n∗c/µ)(pn∗ −pn∗ρ

n∗

ρn∗ − pn∗) as ε→ 0. (1)

Note that, r−n∗c/µ would be the is the optimal price if the actual delay cost c is observed by

the service provider, which by design, is greater than 0. It is easy to show that pn − pnρn∗

ρn∗−pn

< 0.

Intuitively, the blocking probability increases when the actual delay cost is underestimated because

consumers use a lower threshold. Thus,

L(pn;λ, c, r,µ)< 0.

Corollary 1 (Non-zero Revenue Gap). The revenue gap incurred on over-employed (i.e.

time-constrained) consumers is always non-zero regardless of the magnitude of the mis-

measurement.

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The sensitivity of service providers’ revenue and more generally system design, to the measure-

ment of delay costs motivates us to establish a firm understanding of consumer delay costs under

various utility and compensation structures.

3. The Model3.1. Utility Maximizing Agent’s Endogenous Cost of Waiting

Consider an agent who gains utility of U(tl, cl) from leisure time tl and leisure consumption cl.

We assume U(tl, cl) to be continuous, non-decreasing and concave in both arguments. In addition,

U(0, cl) and U(tl,0) are assumed to be non-negative for all cl ≥ 0 and tl ≥ 0. We further assume

that the cross partial derivatives of the utility function are positive. This implies that the agent

values more leisure time when she has more money available to spend on leisure. She also values

leisure consumption more if she has more time to spend in leisure. Occasionally, for analytical

tractability and to draw insights, we impose separability on the utility function, either additive or

multiplicative. That is, we assume U(tl, cl) = h(cl) + g(tl) or U(tl, cl) = h(cl)g(tl), where h(·), g(·)

are non-negative, non-decreasing and concave. Examples of such utility functions include linear

utility function U(tl, cl) = atl + bcl, where a, b > 0 and modified Cobb-Douglas utility function

U(tl, cl) = (tl + 1)a(cl + 1)b, where a, b∈ (0,1).

Following classical theory, the utility maximizing agent is subject to two resource constraints,

time and budget – see Becker (1965). Suppose the agent has T units of time per day and she

spends time in three types of activities: leisure tl, work tw and overhead activities O. Examples of

overhead activities include mowing the lawn, doing laundry and other housework. Instead of doing

all overhead activities by herself, the agent might hire a professional to do or lease equipment to

speed the work. As a result, the hours of overhead activities, the agent has to accomplish is a

function of the agent’s expenditure cO. We assume O(cO) decreases in cO.

Lastly, the agent earns money from working and the wage is then spent on both leisure consump-

tion cl and overhead activities cO. For simplicity, we first assume the agent has no savings. Denote

agent’s earnings from working as Π(tw), where Π(·) is the compensation function that depends on

how many time units the agent works. We assume agent’s income comes entirely from working2.

We refer to the decision time horizon to be a day for convenience. On a typical day, the agent

decides on how to allocate time (t∗l , t∗w) and budget (c∗l , c

∗O) to get an optimal utility u∗. Formally,

the agent solves the following problem:

(P ) u∗ = max(tl,cl,tw,cO)

U(tl, cl)

2 A model that includes savings will be discussed in the Appendix B. Our main insights continue to hold. Also, wehave Π(0) = 0 and Π′(t)≥ 0.

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s.t. tw + tl +O(cO)≤ T,

cl + cO ≤Π(tw),

tl, cl, tw, cO ≥ 0.

We denote u∗ as the optimal utility to problem (P ) with original time budget T . It is easy to see

that the optimal allocation of time and budget will change if an agent “loses time”. We therefore

want to formalize agent’s cost of waiting based on the above decision problem. Suppose the agent

waits in a queue, on a given day for δ time units. During the time in queue, no leisure or work can

be done. It is equivalent to say the agent’s overhead increases from O(cO) to O(cO) =O(cO) + δ. A

natural question will then be what is the minimum monetary compensation the agent must receive

so that her optimal utility is unaltered. Given that the agent’s available time decreases by δ units

(i.e. the revised time constraint in (P ′)), by how much must the earnings be augmented (i.e. added

to the budget constraint in (P ′)) to ensure she still achieves a utility of u∗ (i.e. utility constraint

in (P ′)).

(P ′) min(tl,cl,tw,cO,ε)

ε

s.t. tl + tw + (O(cO) + δ)≤ T,

cl + cO ≤Π(tw) + εδ,

U(tl, cl)≥ u∗,

tl, cl, tw, cO, ε≥ 0.

We claim that ε∗ as δ→ 0 is the marginal rate of substitution of time for money measured by

the ratio of shadow prices of time and budget constraints, this can be interpreted as the agent’s

value of time as found in DeSerpa (1971). Equivalently, this is the cost of waiting. We establish

the equivalence formally in the next proposition.

Proposition 2. Denote λ∗1, λ∗2 to be the shadow prices of time and budget constraints respec-

tively in problem (P ). The shadow price ratio is then R , λ∗1/λ∗2. Suppose ε∗ is the solution to

problem (P ′) as δ→ 0. We have R = ε∗. That is, agent’s marginal cost of waiting equals to her

shadow price ratio.

In contrast to the common wisdom that equates the marginal cost of time with hourly wage, the

above proposition suggests that the marginal cost of waiting is in fact an endogenously determined

rate of substitution. It is unclear whether this endogenous cost can always be fairly estimated as

simply agent’s marginal wage. To address this question, we need to first define the “marginal

wage” for various compensation structures. There are various compensation structures that can be

found in practice. When the marginal wage is not defined under certain structures, we define an

appropriate definition for the same.

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3.2. Agent’s Effectively Hourly Wage

There are two classes of compensation structures we would like to consider in this paper. First, con-

sider an agent working in the gig economy. She has a smooth compensation structure. Particularly,

she might have a linear compensation function. For instance, a freelance worker taking a job on

Taskrabbit is paid hourly at a fixed rate. However, many compensation structure are not smooth.

If a ride-sharing driver encounters a surge price, her compensation structure will have a kink due

to the different hourly earnings. Under both cases, the compensation structure is continuous, but

the former is smooth, the latter non-smooth. The marginal wage is not defined everywhere when

it has kinks. We thus interpret the left derivative of the compensation function with kinks as the

marginal wage. Second, in contrast to the agents working in the gig economy, most traditional jobs

cannot allow workers to choose freely the number of hours to work on a day-to-day basis. The

agent has to work for, for example at least eight hours per day to get paid. In addition, since the

job requires coordination with other workers, the agent cannot unilaterally work for nine hours and

get extra pay. The shape of such a compensation structure is discontinuous with a jump while the

left derivative of the wage is zero almost everywhere. In this case, it is natural to think of agent’s

average hourly wage, i.e. total earnings divide by the mandatory shift length. We summarize the

above discussions on marginal wage as follows. Instead of marginal wage, we use the term effective

hourly wage and denote it as w.

Definition 1 (Effective Hourly Wage w).

1. (Gig Economy Smooth Compensation Function) Given Π(tw)∈ C1, i.e. the compensation func-

tion is continuously differentiable, we define the agent’s effective hourly wage as the first

derivative of the compensation function evaluating at the optimal working time, that is, w=∂Π(tw)

∂tw|t∗w .

2. (Gig Economy Non-smooth Compensation Function) Given Π(tw) continuous but non-smooth,

and denote the left derivative as∂−Π(tw)

∂tw|t∗w . We define the agent’s effective hourly wage as the

left derivative of the compensation function evaluating at the optimal working time, that is,

w= limh→0−Π(t∗w+h)−Π(t∗w)

h.

3. (Traditional Economy) Given Π(tw) as a step function with a jump discontinuity at t0, we

then define the agent’s effective hourly wage as the average wage rate, that is, w= Π(t∗w)

t∗w.

We have now established the definition of effective hourly wage w and a measure of the marginal

cost of waiting, the shadow price ratio R. The question we are interested in, essentially becomes,

under the various compensation structures Π(tw) presented above, is marginal cost of waiting equal

effectively hourly wage, i.e. does R equal w? Furthermore notice that, agent’s time constraint does

not only depend on the working time but also the time spent in overhead activities and a possibly

non-negligible commute time to work (see Becker 1965). In the basic model we presented in section

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3, we assume a continuous overhead structure and a negligible commute time to work. It is then

also interesting to see whether the results from the basic model are robust to different structures

of the overhead activities and commute time. To start with, let us first establish the results for the

basic model.

4. Wage as a Cost of Waiting Estimator4.1. Smooth Compensation Function

First, consider a smooth compensation function. We have the following result.

Proposition 3. Suppose Π(tw)∈ C1, i.e. continuously differentiable, then

R=∂Π(tw)

∂tw|t∗w = w

whenever t∗w > 0.3

The equality indicates that, regardless of the number of hours the agent decide to work, her

cost of waiting is equal to the amount she is paid per marginal unit of time. A special case of a

smooth compensation function is the linear wage, Π(tw) =wtw. Using Definition 1, the effectively

hourly wage w is equal to the hourly wage, denoted as w, in the example. Following from the above

proposition, we have R=w. This suggests the common wisdom that hourly wage is a fair estimator

of agent’s cost of waiting holds when the agent has a linear compensation function.

4.2. General Compensation Function

In this section, we move away from the nice and smooth case, and look at two families of the

non-smooth compensation functions: First, non-smooth but continuous compensation structures

and second, the discontinuous compensation structures.

4.2.1. Continuous and Piecewise Smooth Compensation Function Referring back to

the smooth wages in Section 4.1, although gig economy usually has a smooth or even linear com-

pensation function, it is subject to adjustments in reality. Surge price is one good example. Suppose

a driver starts working at the start of the morning peak. She gets a surge price of two times the

usual rate. After two hours of peak time, the rate return to the normal level. Suppose further,

the driver earns at a fixed rate within each hour. The compensation function is then a piecewise

linear function, and furthermore, the second slope is smaller than the first slope. Denote the point

where the compensation function is not smooth, i.e., the point of the kink, as t1. The agent’s

compensation function can be written as Π(tw) = w1twI{t≤t1} + (w2(tw − t1) +w1t1)I{t>t1}, where

0<w2 <w1. This piecewise linear compensation structure falls under a more general group of con-

tinuous and piecewise smooth compensation functions, which is made up from a finite collection

3 When t∗w = 0, the agent does not participate in the workplace.

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of the strictly increasing C1 functions. In this case, one-sided derivatives exist on the boundary

points. Denote each piece of the compensation function to be Πi(·) for i ∈ {1,2, . . . , n+ 1}. The

compensation function, Π(tw) is then differentiable almost everywhere except at a finite set of

kinks, t0 = 0< t1 < t2 < · · ·< tn <T . We would then have,

Π(tw) =k−1∑i=1

Πi(ti− ti−1) + Πk(tw− tk−1), for tw ∈ [tk−1, tk), for all 1≤ k≤ n (2)

There are two scenarios an agent might face. The agent’s optimal working time t∗w is either in

the interior of one of the segements or equal to one of the kinks. For the first case, the agent has

t∗w 6= tk for all k, and Proposition 2 continues to apply. This also covers scenarios in which the agent

receives overtime pay. When the marginal wage offered to work for the next shift is higher than

the previous one, it is never optimal to work at tk hours, for all k. When t∗w = tk for some k, the

compensation function is not differentiable at t∗w = tk. In fact, for the latter case when t∗w = tk for

some k, the equality established in Proposition 3 breaks down.

Proposition 4. Suppose Π(tw) is defined as in (2). If t∗w = tk for some k, then

w≤R≤ w

where by part 2 of Definition 1, w=∂−Π(tw)

∂tw|t∗w . Also, we denote the right derivative of the compen-

sation function at the kink to be w= limh→0+Π(tw+h)−Π(tw)

h=

∂+Π(tw)

∂tw|t∗w .

However, it is not straightforward to see that there indeed exist compensation functions under

which the agent will allocate exactly tk time for working. We establish sufficient conditions for a

piecewise linear compensation function so that the agent will work for tk units of time in Appendix

A. The inequality holds, whenever the agent is offered more than they require to work for the first

tk hours. However, once they reach tk hours of work, the next offer is insufficient to induce them

to work longer.

Example To illustrate the inequalities using an example, consider an agent earns an hourly

wage of $12 for the first for 8 hours. The later work only only secures $8 hourly. Under a modified

Cobb-Douglas utility function (cl + 1)b(tl + 1)a, where a = 0.8, b = 0.45, her cost of waiting at

t∗w = 8 in this case falls in the range of the two offers w = $8 and w = $12, in particular, R≈ $10.

Intuitively, the first slope needs to be higher than agent’s cost of time so that she is willing to work

for at least eight hours, however the next slope must be lower than her cost of time so that it is

sub-optimal for her to work over the kink.

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4.2.2. Shift Pay Compensation Functions In contrast to the gig economy in which the

freelancers can make granular, hour-by-hour decisions about whether they want to work or play,

office or factory workers generally have to commit to a threshold number of hours to get paid

- see Fisman and Luca (2017). Additionally, they cannot unilaterally choose to work additional

hours for additional pay. Their compensation structure is thus a step function. Denote t1 to be the

discontinuity point. The compensation function can be formulated as follows.

Π(tw) = Π1I{tw≥t1} (3)

where Π1 is a fixed constant and t1 is the threshold. We call such compensation structure as a

shift pay compensation function - the agent can only choose from working for t1 hours and to earn

Π1 units of money or do not work at all and earn nothing. Intuitively, owing to the discontinuity,

agents who are now participating in the workplace might not always want to work at exactly t1

hours. They might either be coerced into work more than what they would ideally want or they

are willing to work for longer hours but have no additional opportunities. Studies based on PSID

(Panel Study of Income Dynamics) found there are workers who reported their work hours have

been constrained. In particular, 41% wanted to work more time while 14% wanted to work less

time (see Dickens and Lundberg 1985). Recall in contrast under a linear compensation function,

the agent can choose freely the number of hours they are willing to spend in working. This suggests

that a wage rate from a linear compensation function as a useful reference wage level when we

establish results for the shift pay structure. We thus start with defining the equivalent linear wage

as follows.

Definition 2 (Equivalent linear wage at t, we(t)). Given problem (P ) with shift pay

compensation structure and a constant time t. The equivalent linear wage is the minimum wage

rate w that induces the agent work for at least t units of time under the linear compensation

function Π(tw) =wtw, holding all other parameters in (P ) the same. That is,

we(t) = inf{w|t∗w(w)≥ t}

We formalize the intuition mentioned above in the following proposition.

Proposition 5. Suppose the agent has a shift pay compensation function of form (3). We

further impose separability on the utility function and convexity on the overhead structure. Denote

the equivalent linear wage at time t1, to be we(t1). Consider only the wage levels that make agents

participate in the workplace. We have,

R< w, for all w > we(t1),

R≥ w, for all w≤we(t1).(4)

In addition, under the shift pay structure, trivially we have t∗w = t1. Therefore, by part 3 of Definition

1, w= Π1/t1.

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Figure 2 The figure illustrate the inequalities between the agent’s cost of waiting R and her effectively hourly

wage w. The agent has a modified Cobb-Douglas utility function U(tl, cl) = (tl + 1)a(cl + 1)b and receive pay under

a shift pay compensation structure. The dotted line shows her pay is effectively $13.25 per hour. Her minimum

commitment to get paid is 8 hours per day. We fixed elasticity of leisure consumption b= 0.45, varied elasticities

of leisure time a over a range of [0.2,4], we see the inequality changes from R< w to R≥ w.

Example We illustrate the above inequalities using a numerical example in Figure 2. The agent’s

effective hourly wage is set to be $13.25, which is the highest minimum wage by state observed

in the year of 2018 in US (Institute (2018)). The agent needs to commit to at least 8 hours per

day to get paid, and thus her wage will be $106 per days. He has a modified Cobb-Douglas utility

U(tl, cl) = (tl + 1)a(cl + 1)b, with elasticity of leisure consumption set to b = 0.45. Varying the

elasticity of leisure time over a ∈ [0.2,4], we can see the inequality between the waiting costs R

and hourly wage w changing from R< w to R≥ w. This indicates that depending on agents’ value

in leisure time, her waiting cost is likely not well approximated by her hourly wage. In fact, the

hourly wage by itself tells almost nothing about agent’s cost of waiting.

Note that when Π(tw) =we(t1)t1I{tw≥t1}, we have t∗w = t1. That is, the minimum wage level for

agents to participate in workplace under the shift pay compensation structure is smaller than or

equal to we(t1). In addition, the result still holds when we(t1) is infinite. This might happen when

the agent values leisure time much more than the leisure consumption. In such cases, R≥ w for all

w.

:13

5. Robustness Checks: Variations of the Basic Model

We have so far discussed a smooth overhead structure and a negligible commute time to work. In

this section, we extend the basic model by relaxing these assumptions. Our key question of interest

is the relationship between R and w, and the impact of different overhead and commute structures.

We assume first the compensation function is smooth.

5.1. Wage is a Fair Estimator Under Discontinuous Overhead Structures

Overhead activities can have various structures. If a house cleaning specialist is hired and she is

paid by hours of working, O(cO) has a linear structure. Alternatively, suppose an agent pay to join

a priority queue in call centers, banks or theme parks for instance. Her overhead time then drops

by a fixed amount after a one time payment. O(cO) in this case is a step function with one jump

discontinuity point. Therefore again, we need to consider three cases, O(cO) ∈ C1, O(cO) ∈ C0 and

O(cO) discontinuous. Suppose Π(tw)∈ C1, in either of the three cases, we have the following result.

Proposition 6. Suppose Π(tw) ∈ C1, under either cases where O(cO) ∈ C1,C0 or discontinuous

with jump discountinuities, the equality is preserved.

R= w

By definition w= ∂Π(tw)

∂tw|t∗w .

The proposition claims that, no matter how much and in what way the agent pays to reduce her

overhead, it should not affect her cost of waiting. The cost of joining the priority queue does not

affect the agent’s cost of waiting. In fact, without the smoothness condition of the compensation

function, our inequality results, in particular Proposition 4 and Proposition 5 are also robust to

discontinuous overhead structures as well.

5.2. Transportation as an Endogenous Cost Determinant

Long commutes are a part of life for many. In 2015, an average US worker spent 52 minutes

commuting each day. More than 2.6% of the population are mega commuters who spend 3 hours

per day on travel (Ingraham 2016). According to the national census, the average commute time is

consistently growing - see Ingraham (2016). This suggests that transportation time is an essential

element in agents’ utility maximization problem. However, how transportation time enters into the

model varies. In this section, we will discuss three different models of transportation time, and

check if the preserved equality under smooth compensation structures established above still hold

with non-negligible commutes.

14

5.2.1. Fixed Commute Time The most common case is that the agent uses the same means

of transportation and route to commute to the same location on a daily basis. The two-way

commute time per day is fixed, say tc units of time. Suppose working from home is not an option

for the agent. She then has only two options: commute to work and get paid or stay at home and

get nothing. The modified problem can be formulated as follows.

(Pft) max(tl,cl,tw,cO)

U(tl, cl)

s.t. tl + tw +O(cO) + tcItw>0 ≤ T

cl + cO ≤Π(tw)

tl, cl, tw, cO ≥ 0

Here ft stands for fixed commute time. When the agent is in the workplace, i.e. t∗w > 0, her total

available time is essentially reduced by the time she spent on commute, i.e. T − tc. If we relabel

the total available time, we should expect all results stated in Section 4.1 and 4.2 hold under the

(Pft) as well. Therefore, we state the following results.

Proposition 7. Denote the shadow price ratio to problem (Pft) as Rft. Denote the shadow

price ratio of the basic model to be R. Consider wage levels that make the agent participate in the

workplace {w|t∗w > 0},1. Suppose Π(tw)∈ C1, R=Rft = w.

2. Suppose Π(tw)∈ C0, R=Rft ≤ w.

3. Suppose Π(tw) is of the form 3, which is discontinuous, denote we,ft(t1) as the equivalent linear

wage under model (Pft).

Rft > w for all w<we,ft(t1),

Rft ≤ w for all w≥we,ft(t1),

The results directly follow from the proofs of Proposition 3, 4 and 5 in appendix. The key observa-

tion is that, the proofs are all independent of the magnitude of total time budget T . By relabeling

T − tc, proof of the above proposition is along the same line as for the basic model.

5.2.2. Relocation During the Day An alternative form of commuting arises when an agent

must relocate during a work period and thus has to accept a stretch of time with no pay in order to

extend her hours of working. Consider, for example, a ride-share driver who drops off a customer

in a neighborhood with low demand origination. To continue working and earning the driver would

need to move to a different neighborhood with more customers looking to start trips. Alternatively,

she could truncate her workday. If she opts to change neighborhoods, she faces a flat portion of

the compensation function, which she must weight against potential future earnings.

:15

Proposition 8. Suppose the agent has an option to relocate after t1 time units of working and

she could reach the new location by time t2. There is no earnings during the relocation period.

The hourly rate of working opportunities the agent could have at the two locations are w1 and w2

respectively. w1,w2 are not necessarily equal. The agent’s compensation function thus takes the

following form.

Π(tw) =w1twI{tw<t1}+wt1I{t1≤tw<t2}+w2twI{tw≥t2}

If t∗w = t1, then R< w.

The above result follows directly from Proposition 3. It captures settings in which an agent would

want to work longer hours at the initial wage rate but does not find the later wage sufficient to

compensate for the transition time.

5.2.3. Transportation Time as a Function of Working Time Lastly, we consider the

case of an agent having a choice of how long she wants to commute. Consider an agent weighing

two job offers with a short commute and one with a long commute. The more distant job allows a

longer working hours than the nearby one. Deciding which job offer to take, the agent solves the

following problem.

(Ptc) max(tl,cl,tw,cO)

U(tl, cl)

s.t. tl + tw +O(cO) + tc(tw)Itw>0 ≤ T

cl + cO ≤Π(tw)


where

tc(tw) = tc1I{tw<t1}+ tc2I{tw≥t1}

where tc1 < tc2 .

Proposition 9. Suppose the agent has a linear compensation structure, i.e. Π(tw) =wtw, with

same wage for each job. Denote the equivalent linear wage under the model with tc(tw) = tci to be

we(t1; tci), where i∈ {1,2}. We have,

Rtc < w for all w ∈ (we(t1; tc1),we(t1; tc2))

Rtc = w otherwise

16

Unlike the previous cases, the equality breaks down even though the compensation function is

linear in this setting. However, it is important to note that looking closely at the model, one would

conclude that, there is in fact also a discontinuity in the agent’s choice of working time introduced

by the working hours related commute time. The discontinuity again leads to the inequality between

marginal wage and cost of waiting, which make the wage an inaccurate estimator for the cost of

waiting.

6. Conclusion and Managerial Implications

We have examined whether hourly wages fairly estimate the cost of waiting under different com-

pensation structures. We have shown that under a linear compensation function, the hourly wage

is a good proxy for the cost of waiting. We found that whenever a kink or a discontinuity is present,

the equality breaks down. The kink or discontinuity point can come directly from the compensation

function or it can arise from other structures in the model, e.g. commute time. Under a continuous

but non-smooth compensation function, there exists cases when the agent do not want to work

more than the number of hours at the kink. This happens when the initial wage current the agent

receives is higher than her value of time while the next offer she would receive is not. She would

rather call it a day than working for longer hours.

Under a shift pay compensation function, the agent has only two options: not working at all

or working for the number of hours at the discontinuity point t1. Any other choice is suboptimal.

When the agent is offered a wage high enough to induce working but not so high that she would

choose to work at least t1 time units under a linear compensation function, she is time constrained,

i.e., she is working more than she really wants to. Her cost of waiting is then higher than the wage

offered. On the other hand, when the wage offered is high, the agent would be willing to work long

hours – more than the commitment required for employment. Now she has abundant time, but has

no chance to work for more money. Her wage now overestimates her waiting cost.

The major characteristics in the compensation functions that breaks the equality between wage

and waiting wage is whether the shape of the compensation functions allows the agent to make

granular decisions without sudden changes in the wage rate they are receiving. This smoothness

grants the agent a flexible working schedule and eliminates any mismatch between the cost of

waiting and the wage being offered.

Our results are robust to a discontinuous overhead structure and a fixed commute time. Again,

under theses variations to the model, there is no additional discontinuity added to the compensation

function, therefore the original results still hold. However, when we have a commute time dependent

working schedule, even if the compensation structure itself is smooth, equality breaks down. The

discontinuity in this case is introduced by the commute time. We have shown that there are wage

:17

levels that make the agent stuck with the job with less working opportunities but shorter commutes.

In this case, agent’s cost of waiting is smaller than the wage offered.

Through the analysis, we see that the mismatch in agents’ cost of waiting R and their wage w

is not a rare situation. It is disconcerting that wages over-estimates a well-compensated agent’s

cost of waiting, but under-estimates the cost of waiting of the low-paid agents. This result might

shed a light on how service providers should design their service system and price priority services.

Taylor (2016) substantiates the importance of understanding customer delay sensitivity, which is

measured by per-unit-waiting-time delay disutility, in on-demand service platform’s decisions on

setting its optimal prices to customers and wages to agents providing service. Additionally, it might

also offer a perspective to employees on how they should value a flexible working schedule.

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Appendix A: Sufficient Conditions for t∗w = tk

We start with extending the definition of equivalent linear wage and defining a new reference wage level, the

linear participation wage. Note that in the basic model, we assume a total time budget of T units and zero

savings, S = 0. We now assume, the agent has in total τ ≤ T units of time and also a non-negative savings,

S ≥ 0.

Definition 3 (Equivalent linear wage at t with (τ,S), we(t; τ,S)). Recall Definition 2, we(t; τ,S)

follows the same definition, except instead of problem (P ), it is defined under problem (Pτ,S), where

(Pτ,S) u∗ = max(tl,cl,tw,cO)

U(tl, cl)

s.t. tw + tl +O(cO)≤ τ

cl + cO ≤Π(tw) +S


The minimum participation wage is essentially the minimum payment at which the agent is better off to

work than doing nothing.

Definition 4 (Linear participation wage with with (τ,S), wp(τ,S)). The linear participation

wage wp(τ,S) is the wage rate that makes the agent starts working under (Pτ,S) with a linear compensation

function Π(tw) =wtw.

We are now ready to establish the sufficient condition.

Lemma 1. 1. Suppose there is only one kink in wage, that is Π(tw) = w1tw1t<t1 + (w2(tw − t1) +

w1t1)1t≥t1 . Then whenever, w1 >we(t1) and 0<w2 <wp(w1t1), t∗w = t1.

2. For piecewise linear compensation function with at least two kinks. Whenever, wk >we(tk − tk−1;T −

tk−1,wk−1tk−1) and 0<wk+1 <wp(T − tk,wktk), t∗w = tk ∀k.4

4 There can be the case that we(t) =∞. In this case, the optimal will always be at the interior point, for which wehave R= w.

:19

Notice that, Lemma 1 strengthens our conclusion in Proposition 4. Suppose, we find a piecewise linear

compensation function such that t∗w = tk and R = w, we can then fix wk+1 and increase wk. The sufficient

conditions are still satisfied and we have found compensation functions that give us a strict inequality R< w.

This suggests that continuous but non-smooth compensation function does not always preserve the equality

R = w. In particular, when the agent’s optimal working time exactly equals to one of the kinks in the

compensation structure, the equality breaks down.

Appendix B: Positive Savings

With a positive savings, the total budget the agent has is Π(tw) +S, where S > 0. Note that, in the proofs

of propositions 3, 4 and 5, results are independent of the fixed constant in the total budget. That is ∂S∂tw

= 0,

will not enter into further analysis. The only difference is in the magnitude of wage rate that will make the

agent start working. With a positive savings, it takes a larger w to get the agent participate in the workplace.

Also, the equivalent linear wage will increase. That is, the magnitude of the cutoff wage levels in inequalities

(4) will change, but the inequalities between R and w remain the same.

Appendix C: When Working Brings Positive Utility

Suppose the agent enjoys working. That is, we keep all other parts in problem (P) the same, we further

suppose U(tl, cl, tw) increases in tw. In general, the cost of waiting increases since the time wasted not working

now directly decreases the agent’s utility. The time taken away from the total available time bucket values

even higher than before. We will again look at the three types of compensation functions discussed in Section

4.1 and 4.2.

1. When Π(tw)∈ C1, R> w.

2. When Π(tw)∈ C0, R≤ w+ ∂U∂tw

/λ2.

3. When Π(tw) is of form (3),

R≤ w+∂U

∂tw/λ2 ∀w > we(t1)

R> w ∀w≤we(t1)

where we(t1) is again the equivalent linear wage under the new model. For case 1,

∂L∂tw

=∂U

∂tw−λ1 +λ2

∂Π(tw)

∂tw= 0

⇒λ1

λ2

= w+∂U

∂tw/λ2⇒R> w

since λ2 ≥ 0 and ∂U∂tw

> 0. By the same argument, R≥ w+ ∂U∂tw⇒R> tw. On the other hand for the other

two cases, R≤ w is not necessarily true.

Appendix D: Proofs

D.1. Proof of Proposition 1

Proof We prove the proposition in two steps. First we prove that, any underestimation of the delay

cost induced by using the wage rate as the cost, lead to a decrease in the threshold n∗, that makes the

20

customers are indifferent in whether joining the queue. Second, we calculate the revenue gap, given the drop

in throughput.

Note that, the consumers will not join the queue if they see the queue length is,

q∗ ≥ (r− θr(w))µ/c, where θr(w) = r− wbνr(w)c/µ.

nr(w) is the threshold the service provider would want to induce among the consumers coming to the system,

which is an integer. When w = c, we have the threshold q = wbνr(w)c/c|w=c = bνr(c)c, which is an integer.

Suppose, w= c−ε, q= b wcbνr(w)cc. Note the change in function νr(·) is continuous. Thus bνr(w)c= bνr(c)c,

n∗. Therefore, w < c leads to a one less unit after taking the floor function on the threshold value, that is

b wcbνr(w)cc= bνr(c)c− 1. We denote the blocking probability of the new arrivals to the system induced by

n∗ as pn∗ . While, under the underestimation of cost, we have argued that the new threshold is n∗ − 1, we

can then denote the mis-measured blocking probability as ˜pn∗ = pn∗ρn∗

ρn∗−pn∗

.

The revenue gap is then given by,

L(pn∗) =λ(1− ˜pn∗)θw−λ(1− pn∗)θc = λ(r−n∗c/µ)(pn∗ − ˜pn∗) +ελn∗

µ(1− ˜pn∗)

→ λ(r−n∗c/µ)(pn∗ −pn∗ρ

n∗

ρn∗ − pn∗) as ε→ 0

That is L(pn∗) is always a negative number regardless of ε.

This completes the proof.


Proof Denote (tl, cl, tw, cO) as the solution to (P ), while (t′l, c′l, t′w, c′O) as the solution to (P ′).

Denote the constant on RHS of the budget constraint as bc, i.e. cl + cO−Π(tw)≤ bc. In problem (P ), bc = 0.

The Lagrangian of problem (P ) is given by:

L=U(tl, cl) +λ1(T − (tl + tw +O(cO))) +λ2(bc− (cl + cO −Π(tw))) +∑i=l,w

µiti +∑i=l,O

δici

By envelope theorem, {∂U∗

∂T= ∂L∗

∂T= λ∗1

∂U∗

∂bc= ∂L∗

∂bc= λ∗2

⇒ dU∗ = λ∗1dT +λ∗2dbc

Thus after the perturbation on both constraints as in (P ′), we have,

U(t′l, c′l) = u∗−λ∗1δ+λ∗2εδ as δ→ 0 (5)

Now consider (P ′), at optimum, we want to show the third constraint is binding, i.e. U(t′l, c′l) = u∗. In words,

this means the post-perturbation utility remains unchanged. Suppose U(t′l, c′l)> u∗. Let d=U(t′l, c

′l)− u∗ >

0. Since U(tl, cl) increases in cl and it is smooth, ∃ cl < cl such that U(t′l, cl) − u∗ ∈ (0, d). Furthermore,

cl+c′O < c′l+c′O ≤Π(tw)+εδ. Therefore, we have found a feasible ε < ε∗, contradiction. Thus, U(t′l, c

′l) = u∗ at

optimal ε∗. Together with expression (5), we have u∗ = u∗−λ∗1δ+λ∗2ε∗δ. Rearranging the terms, we conclude,

λ∗1/λ∗2 = ε∗.

:21


Proof First, since Π(tw)∈ C1, ∂Π(tw)

∂twexists and is continuous on R+. The Lagrangian is:

L=U(tl, cl) +λ1(T − tl− tw−O(cO)) +λ2(Π(tw)− cl− cO) +∑i=l.w

µiti +∑i=l,O

δici

By the KKT conditions, optimal values necessarily satisfy:

∂L∂tw

=−λ1 +λ2

∂Π(tw)

∂tw|t∗w +µw = 0

µwt∗w = 0

Since t∗w > 0, µw = 0. Rearranging terms in the first equation, λ1/λ2 = ∂Π(tw)

∂tw|t∗w .


Proof Given that t∗w = tk is optimal on tw ∈ (0, T ), it is also optimal on tw ∈ (tk−1, tk]. Then, t∗w = tk is

also optimal on the following subproblem.

(Pk) u∗ = max(tl,cl,tw,cO)

U(tl, cl)

s.t. tw + tl +O(cO)≤ T

cl + cO ≤Π(tw) =

k−1∑i=1

Πi(ti− ti−1) + Πk(tw− tk−1)

tl, cl, cO ≥ 0

tk−1 < tw ≤ tk

Set up the Lagrangian and KKT conditions,

L=U(tl, cl) +λ1(T − tl− tw−O(cO)) +λ2(Π(tw)− cl− cO) +µltl +∑

i=l,OH

δici+

µwu(tk− tw) +µwl(tw− tk−1)

µwu(tk− tw) = 0

µwl(tw− tk−1) = 0

Given t∗w = tk, the optimal values necessarily satisfy:

∂L∂tw

=−λ1 +λ2

∂Πk(tw− tk−1)

∂tw|t∗w −µwu

+µwl= 0

µwu≥ 0, µwl

= 0

This implies,

−λ1 +λ2

∂Πk(tw− tk−1)

∂tw|t∗w ≥ 0

⇒λ1

λ2

≤ ∂Πk(tw− tk−1)

∂tw|t∗w =

∂−Π(tw)

∂tw|t∗w = w

That is, R≤ w.

We next prove R≥w by similar arguments. Instead of looking at the piece to the left of the kink, we now

look at the piece to the right of t∗w = tk. The Lagrangian function is the same as above, instead of the left

22

derivative, we now look at the right derivative at the kink. We have µw′u(tk+1− tw) = 0, and µw′l(tw− tk) = 0.

That is,

∂L∂tw

=−λ1 +λ2

∂Πk+1(tw− tk)∂tw

|t∗w −µw′u +µw′l= 0

µw′u = 0, µw′l≥ 0

⇒R=λ1

λ2

≥ ∂Πk+1(tw− tk)∂tw

|t∗w =∂+Π(tw)

∂tw|t∗w =w

Therefore, we have proved both directions. Further notice that by construction, we assume w 6= w, the

equalities R= w and R=w cannot hold simultaneously. Thus, R ∈ (w, w] or R ∈ [w, w). This completes the

proof.

D.5. Proof of Lemma 1

Let us start with the base case, when there is only one kink. Our aim is the search for the optimal working

time t∗w in the domain (0, T ). Note again, we can view the original problem as two sub-problems, (P1) and

(P2), which are defined as in the proof of Proposition 4. Denote the optimal working time of Consider (P1),

follows from Claim 1, t∗w ≥ t1. While by definition, wage is smaller than the rate the agent want to work

above tk, that is t∗w ≤ t1. Thus t∗w = t1. Suppose now there are more than one kink. We first rescale the time

and budget limit. Let t′w = tw − tk−1 and S = wktk. The above argument for the base case applies. Thus

t∗w = tk.


Proof We discuss case by case.

1. R< w ∀w > we(t1). Consider a compensation structure, Πa1(tw) = wtw1tw<t1 +wt11tw≥t1 as illustrated

in the left panel of Figure 3. Denote the shift pay compensation function by Πs(tw) = Π11tw≥t1 .

Claim 1. Suppose Π(tw) =wtw, where tw ∈ [0, T ], t∗w = τ and also O(cO) decreasing and concave in

cO. Now suppose we further restrict tw ∈ [0, t′], where t′ < τ < T , then t∗w = t′. On the other hand, if

tw ∈ [t′, T ], where τ < t′ <T , again we have t∗w = t′.

Since w > we(t1), we have t∗w = τ > t1 under the linear compensation function Π(tw) = wtw with non-

restricted tw. Thus follows from Claim 1, we have t∗w = t1 under Πa1(tw). From Proposition 4, Ra1 < w

under Πa1(tw).

Claim 2. Shadow price ratios under Πa1(tw) and Πs(tw) satisfy: Ra1 =Rs.

Following from the Claim above, we have Rs < w under Πs(tw) when w > we(t1).

2. R ≥ w ∀w ≤ we(t1). Again, consider an auxiliary compensation structure, Πa2(tw) = 0 ∗ 1tw<t1 +

wtw1tw≥t1 as illustrated in the right panel of Figure 3. Note when w < we(t1), t∗w < t1 under the unre-

stricted linear compensation function. Therefore, follows from Claim 1, we have t∗w = t1 under Πa2(tw).

Claim 3. Shadow price ratios under Πa2(tw) and Πs(tw) satisfy: Ra2 ≤Rs.

:23

Figure 3 Left: Illustration of auxiliary compensation functions used in proof when w > we(t1); Right: In proof of

w≤we(t1)

Now, write down the KKT conditions under Πa2(tw) on the second segment, in particular for [tw].

−λ1 +λ2w+µtw = 0⇒−λ1 +λ2w≤ 0⇒ λ1/λ2 =Ra2 ≥ w⇒Rs ≥ w

The last step follows from Claim 3.

Proof of Claim 1 First notice that, maxtl,cl U(tl, cl) is equivalent maxtw,cO U(T − tw −O(cO),wtw − cO).

The claim is thus equivalent to say

U(tw, c∗O(tw)) =U(T − tw−O(c∗O(tw)),wtw− c∗O(tw)) (6)

has one unique optimal solution t∗w. That is there is only one stationary point such that∂U(tw,c

∗O(tw))

∂tw= 0.

That is, there is no other local maximum in the domain of tw. Note that, by construction, we know there exist

an interior solution t∗w. We only need to prove that, the solution is unique. We thus omit the two implicit

boundary constraints T − tw − O(c∗O(tw)) > 0 and wtw − c∗O(tw) > 0 here and focus on the unconstrained

problem (6). The FOC to the unconstrained problem is thus as follows.

(−1− ∂O(cO)

∂cO

∂c∗O(tw)

∂tw)∂U(tl, cl)

∂tl+ (w− ∂c∗O(tw)

∂tw)∂U(tl, cl)

∂cl= 0 (7)

Our goal is to show that the above equation has a unique solution t∗w. Fix tw, c∗O(tw) is the optimal expenditure

in overhead c∗O as a function of tw. It satisfies the FOC of maxcO U(T − tw−O(cO),wtw− cO), fix tw, which

is,

∂U(tl, cl)

∂tl(−∂O(cO)

∂cO)− ∂U(tl, cl)

∂cl= 0⇒−∂U(tl, cl)

∂tl

∂O(c∗O(tw))

∂cO− ∂U(tl, cl)

∂cl= 0 (8)

Substitute (8) into (7), we have − ∂U(tl(tw,cO),cl(tw,cO))

∂tl+w ∂U(tl(tw,cO(tw)),cl(tw,cO(tw)))

∂cl= 0. If we can prove,

ut(tw) and uc(tw) are both monotone in tw and in addition, we have the two functions are monotonic in

different directions, then we have proved the uniqueness of t∗w5. Therefore, proved the claim. That is, we want

5 Suppose f(x) and g(x) are monotone and also f ′(x)< 0, g′(x)> 0. Then let h(x) = f(x)− g(x), clearly h′(x)< 0.Thus h(x) is strictly decreasing. Suppose ∃c1, c2, such that h(c1) = 0, h(c2) = 0. This is contradictory to h(x) to bestrictly decreasing.

24

to prove that, ∂ut(tw,cO(tw)

∂tw, ∂uc(tw,cO(tw)

∂twis either positive or negative ∀tw. Additionally, they have opposite

signs.

∂ut∂tw

=∂ut∂tl

(−1− ∂O(c∗O(tw))

∂cO

∂c∗O(tw)

∂tw) +

∂ut∂cl

(w− ∂c∗O(tw)

∂tw) (9)

∂uc∂tw

=∂uc∂tl

(−1− ∂O(c∗O(tw))

∂cO

∂c∗O(tw)

∂tw) +

∂uc∂cl

(w− ∂c∗O(tw)

∂tw) (10)

Note that, by assumption, we have U(tl, cl) to be increasing and concave in both of its arguments and

positive cross derivatives. Furthermore, we also assume a convex and decreasing overhead structure. That is

we assume,

1. ∂U(tl,cl)

∂tl, ∂U(tl,cl)

∂cl> 0; ∂2U(tl,cl)

∂t2l

, ∂2U(tl,cl)

∂c2l≤ 0

2. ∂U(tl,cl)

∂tl∂cl≥ 0

3. ∂O(cO)

∂cO< 0, ∂2O(cO)

∂c2O≥ 0

We next present the proof of the proposition with or without imposing the separability condition on the

utility function.

• Suppose we further assume the agent has a separable utility function. Then (9), (10) reduce to,

∂ut∂tw

=∂ut∂tl

(−1− ∂O(c∗O(tw))

∂cO

∂c∗O(tw)

∂tw),

∂uc∂tw

=∂uc∂cl

(w− ∂c∗O(tw)

∂tw)

Taking derivative on both sides of (8) with respect to tw, we have,

∂cO(tw)

∂tw=

wucc−uttO′

utt(O′)2 +ucc> 0

Since utt, ucc < 0, in order to show ∂ut

∂tw, ∂uc

∂tware of opposite signs, it is sufficient to show (w−c′O(tw))(−1−

O′(cO)c′O(tw))< 0.

w− c′O(tw) =uttO

′(wO′+ 1)

utt(O′)2 +ucc

−1−O′(cO)c′O(tw) =ucc(wO

′+ 1)

utt(O′)2 +ucc

Note that, regardless of the sign of (wO′ + 1), we always have the above to be of opposite sign since

ucc < 0, uttO′ > 0. This completes the proof.

• Next we show the result for a general utility function.

However, we need to impose an additional assumption on the values of w. That is,

w ∈ (uttO

′+utcutO′′−utcO′−utt(O′)2

,utO

′′−utcO′−uccutc(O′)2 +uccO′

) (11)

Again, we start with showing that c∗O(tw) is monotone in tw, then∂c∗O(tw)

∂twtakes on a unique sign. Taking

derivative on both sides of (8) with respect to tw, we have,

((utt(−1− O(c∗O(tw))

∂cO

c∗O(tw)

∂tw))+utc(w−

∂c∗O(tw)

∂tw))(−∂O(cO)

∂cO)

−uct(−1− ∂O(c∗O(tw))

∂cO

c∗O(tw)

∂tw)−ucc(w−

∂c∗O(tw)

∂tw) = 0

(12)

⇒

∂c∗O(tw)

∂tw(utt(

∂O(c∗O(tw))

∂cO)2+utc

∂O(c∗O(tw))

∂cO+uct

∂O(c∗O(tw))

∂cO+ucc−ut

∂O2(c∗O(tw))

∂c2O)

=−utt∂O(c∗O(tw))

∂cO+wutc

∂O(cO(tw))

∂cO−utc +wucc

(13)

:25

Follows from the conditions above, we have∂c∗O(tw)

∂tw> 0. Thus, we manage to prove that both ut(tw)

and uc(tw) are monotone in tw. Now let us further show the direction of change of ut, uc in tw. Consider

expression (9) and (10). It is easy to see the ∂ut

∂twand ∂uc

∂tware having opposite signs if (w− ∂c∗O(tw)

∂tw) and

(−1− ∂O(c∗O(tw))

∂cO

∂c∗O(tw)

∂tw) are having opposite signs 6. That is, we have

w− ∂c∗O(tw))

∂tw=wuttO

′(1 +wO′) +utc(1 +wO′)−wutO′′

utt(O′)2 + 2utcO′+ucc−utO′′

(−1− ∂O(c∗O(tw))

∂cO

∂c∗O(tw)

∂tw) =−utcO′(1 +wO′) + (−ucc(1 +wO′)) +utO

′′

utt(O′)2 + 2utcO′+ucc−utO′′

We now apply the assumption on values of w (11)7. We then have,

w− ∂c∗O(tw))

∂tw> 0,−1− ∂O(c∗O(tw))

∂cO

∂c∗O(tw)

∂tw< 0

which implies,

∂ ∂u(tl,cL)

∂tl

∂tw> 0,

∂ ∂u(tl,cL)

∂cl

∂tw< 0

In all, we have shown that ut is monotonically increasing in tw and uc is monotonically decreasing in tw.

We can then conclude the two partials have at most one intersection, we know t∗w is one of them, which

means there is only one unique stationary point of equation (7). We have thus proved that problem (6)

has only one optimal (local and global) tw and it is in the interior. This completes the proof.

Proof of Claim 2 Recall, we proved in 2 that shadow price ratio is equivalent to the solution to problem

(P ′). We want to compare the shadow price ratios of problems with Πa1(tw) and Πs(tw). That is,

min(tl,cl,tw,cO,ε1)

ε1

s.t. tl + tw +O(cO)≤ T − δcl + cO ≤wtw1tw<t1 +wt11tw≥t1 + δε1

u(tl, cl)≥ u∗



ε2

s.t. tl + tw +O(cO)≤ T − δcl + cO ≤wt11tw≥t1 + δε2

u(tl, cl)≥ u∗


First, suppose t∗w < t1 for problem on the right, the total available budget will be δε2 only, which is obviously

suboptimal to work for some t∗w ≥ t1. Thus, we know t∗w ≥ t1 for the problem on the right. Suppose we can

prove the problem on the left will also have t∗w ≥ t1, then the problems are essentially the same, thus ε1 = ε2

follows naturally, that is Ra1 =Rs. Thus, we want to prove t∗w = t1 for the problem on the left. We proceed

with the proof as follows. Fix an ε, suppose we can prove tw,1 = t1 for small δ, then we can conclude, at the

minimum value of ε, the result also holds. Note that, when we fixed ε, the minimum compensation problem

above is now equivalent to solve the utility maximization problem with nonzero δ, that is savings equal δε1

and time budget T − δ. By construction, the wage rate w here will make the utility maximization agent

work more than t1 units of time under linear compensation function Π(tw) =wtw. Thus, we are only left to

show that under nonzero but small δ, t∗w > t1 still holds under the linear compensation structure. This can

be shown easily via KKT. Then it follows from Claim 1, the modified problem has t∗w = t1 under Πa1(tw).

Thus, Ra1 =Rs. This completes the proof.

6 This is a sufficient condition but not necessary.

7 The set is nonempty

26

Proof of Claim 3 Follow the same arguments as above, we can write down the two problems that

minimize the compensation ε as follows:


ε1

s.t. tl + tw +O(cO)≤ T − δcl + cO ≤wtw1tw≥t1 + δε1

u(tl, cl)≥ u∗


tw ≥ t1


ε2

s.t. tl + tw +O(cO)≤ T − δcl + cO ≤wt11tw≥t1 + δε2

u(tl, cl)≥ u∗


tw ≥ t1

Notice that we do not need to consider the case when tw < t1, since it is obvious that without any

contribution to budget from working, the εi for both problems above is expected to be large, i.e. suboptimal.

We only need to consider the case when tw ≥ t1. In this case, note that the only different constraint is the

second one, where wtw ≥wt1 at any choice of tw. Thus, the problem on the left has a larger feasible region,

therefore, ε1 ≤ ε2, which is Ra2 ≤Rs. This completes the proof of Claim 3.


Proof We discuss case by case.

1. O(cO) ∈ C1, we can directly write the KKT condition, since both U(tl, cl) and the constraints are

continuously differentiable. It is easy to check that R= Π(tw)

tw|t∗w = w.

2. O(cO)∈ C0, we can first split the original optimization problem into small problems, solve each subprob-

lems and then compare all subproblem optimal solutions to get the global optimal solution. On each

subproblem, in addition to the original constraints, there is another new set of boundary constraints

on cO, cO ∈ xi, xi+1. Now we could write down the KKT conditions on each subproblem. Whichever

subproblem the optimal solution happens to fall onto, it necessarily satisfy the KKT condition, which

again gives R= Π(tw)

tw|t∗w = w.

3. O(cO) is discontinuous can be proved in the same way as in O(cO)∈ C0.


Proof Denote the wage rate that makes the agent to participate in the workplace as wp(tci). It is easy

to see that we have the following inequalities:

wp(tc1)<wp(tc2)

we(t1; tc1)<we(t1; tc2)

wp(tci)<we(t1; tci)

When wp(tc1) < w < wp(tc1), the agent’s optimal working time 0 < t∗w < t1. It is an interior solution of

the subproblem with T − tc1 and tw ∈ (0, t1]. Thus, R = w. Similarly, when w > we(t1; tc2), t∗w > t1. It is

an interior solution of the subproblem with T − tc2 and tw ∈ [t1,∞). Thus, R = w as well. While when

w ∈ (we(t1; tc1),we(t1; tc2)), t∗w = t1 on the subproblem with the shorter commute. t∗w = t1 follows again from

Claim 1. Also the utility of working for t1 hours at the job with longer commute is definitely lower than the

:27

one with shorter commute. Thus, now we can write down the Lagrangian of the subproblem with the shorter

commute. By the same argument as in Appendix D.4, we have

L=U(tl, cl) +λ1(T − tc1 − tl− tw−O(cO)) +λ2(Π(tw)− cl− cO) +µltl +∑i=l,O

δici+

µwu(t1− tw) +µwl

(tw)

µwu≥ 0

µwl= 0

⇒ ∂L∂tw

=−λ1 +λ2w−µwu +µwl= 0 ⇒Rtc =

λ1

λ2

≤ w

Again, with w>we(tc1) the strict inequality, we have Rtc <w= w.

paying by the hour: are wages the cost of waiting?

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