Welfare economic consequences
of a mandatory reduction of the work-week
Extra problem set, ECON4310
1 Introduction
1.1 Background and question
You are a young employee at the Ministry of Finance working on quantitative economic
analysis. One day a group of activists proposes that the government should enforce
regulation in order to mandatory reduce the work-week from 36 hours per week to 30 hours
per week. This proposal gets a lot of attention and creates a vivid debate in newspapers,
radio and TV.
In order to answer the questions from members of the Storting and the press, the Minister
asks you personally to answer the following question:
What are the welfare consequences of a mandatory reduction of the work-week
from 36hrs/week to 30hrs/week?
More specifically, she wants you to compute estimates of both the short-run effect and longrun
consequences of such a reform. In order to make the analysis tractable and transparent
she asks you to base your analysis on well established economic theory and consistency
with how national accounts are measured. Because the evidence on possible effects on
e.g. average hourly productivity and number of sick days so far is inconclusive, you are
asked to leave such potentially relevant aspects of a reform for further, later analysis. As
your identifying restriction you choose: In absence of work-time regulations the average
individual would choose to work 36hrs/week.
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2 Model economy
In order to answer the question you need a model of how individuals derive utility from
leisure as well as consumption, how individuals make choices over of savings and labor
supply over time, and how prices for capital and labor clear markets. The Minister asked
you specifically to base your estimate on well-established theory. The natural choice is
then to base your benchmark analysis on the detrended neoclassical growth model.
Specifically, individuals maximize the sum of discounted utility derived from consumption
and leisure
max
{ct,lt,it}∞
t=0
∞
t=0
βt
u (ct, lt) .
The utility function is given the following parametric form
u (ct, lt) = ln ct + ψ ln lt, ∀t. (1)
We assume the economy is closed and that consumption, ct, is both private and government
consumption. The expenditure approach to output in the model economy is
yt = ct + it, ∀t. (2)
Markets are competitive and the firms make zero profit. Output is paid out to labor and
capital. The income approach to output in the model economy is
yt = rtkt + wtht, ∀t. (3)
Output is produced from capital and labor input with Cobb-Douglas production technology.
The product approach to output in the model economy is
yt = f (kt, ht) = kα
t h1−α
t , ∀t, α ∈ (0, 1) . (4)
The law of motion for capital accumulation is
kt+1 = (1 − δ) kt + it ∀t, δ ∈ [0, 1] . (5)
Total time available for an individual can be spent on leisure or supplied in the market.
Without loss of generality we normalize total time available to 1
ht + lt = 1, ∀t. (6)
Consumption, capital stock, hours supplied in the market and leisure are all strictly posi-
tive
ct, kt, ht, lt ≥ 0, ∀t. (7)
2
The first welfare theorem holds so we can solve this problem as a social planner’s problem.
As we covered in class, our interest in the social planner’s problem is based on the fact
that the solution to the social planner’s problem is the competitive equilibrium allocation.
That is, there exists a set of prices such that the optimum solution can be decentralized as
a competitive equilibrium with a price system that has an inner product representation.
The social planner’s problem is much easier to solve since we get rid of the prices and the
individuals’ budget constraint (Eq. 3).
Since this problem obviously has a recursive structure, we reformulate the problem and
write up the Bellman equation.
v (kt) = max
kt+1,ht
{u (ct, lt) + βv (kt+1)}
For this problem we have two control variables, kt+1 and ht, and one (endogenous) state
variable, kt.
Problem 1. Take the first order condition of the Bellman equation with respect to the two
control variables and the envelope condition of the Bellman equation with respect to the
endogenous state variable to derive and show that the intertemporal optimality condition
is
βRt+1
u1 (ct+1, lt+1)
u1 (ct, lt)
= β (1 + f1 (kt+1, ht+1) − δ)
ct
ct+1
= 1
and the intratemporal optimality condition is
wt
u1 (ct, lt)
u2 (ct, lt)
=
f2 (kt, ht)
ψ
1 − ht
ct
= 1.
2.1 Calibration and measurement
In order to answer the question about what will happen after the policy change, we want
to calibrate the policy-invariant (preference and technology) parameters of the model so
that the behavior of the model economy matches features of the measured data in as many
dimensions as there are unknown parameters. We observe over time that certain ratios in
actual economies are more or less constant.
Problem 2. Show that along a detrended balanced growth path (steady state) the following
ratios of endogenous variables can be expressed as functions of the structural parameters,
i.e. technology and preference parameters:
• the investment/capital ratio
i
k
= δ,
3
• the factor prices
w = (1 − α) kα
h−α
,
r = αkα−1
h1−α
,
• capital’s and labor’s shares of national income
wh
y
= (1 − α) ,
rk
y
= α,
• the capital/labor ratio
k
h
=
1
β − (1 − δ)
α
1
α−1
,
• the capital/output ratio
k
y
=
α
1
β − (1 − δ)
,
• the investment/output ratio
i
y
=
i
k
k
y
=
αδ
1
β − (1 − δ)
,
• and the consumption/output ratio
c
y
= 1 −
αδ
1
β − (1 − δ)
.
We must calibrate the following structural, time-invariant parameters from annual Norwegian
national accounts:
• time-preference rate β,
• consumption-leisure trade-off ψ,
• capital’s share of output α,
• depreciation rate δ.
4
Problem 3. Download Norwegian National Accounts for the period 1970-2005, either
accessed from Statistics Norway
http://www.ssb.no/english/subjects/09/01/nr en/
or Source OECD,
http://new.sourceoecd.org/vl=6237706/cl=11/nw=1/rpsv/ij/oecdstats/16081188/v149n1/s1/p1
to compute the following average ratios1 (see the Appendix for some comments on useful
approximations given the SSB presentation of the data):
• labor’s share of national income,
• investment to capital ratio, and
• capital to output ratio.
Attach spreadsheets with the time series.
In addition is reasonable to assume that the average working-ago individual has 15 × 7
hours available per week net of sleep and personal care, i.e.
h =
36
105
≈ 0.343. (8)
Problem 4. Calibrate β, ψ, α and δ using steady-state ratios, the measured moments from
Norwegian national accounts, and the fraction of available time at work. Without loss of
generality we might normalize total steady-state output to 1.
2.2 Welfare prior to a reform
Given the ratios along the detrended balanced growth path (steady state), the calibrated
parameters, and the normalization of output to 1, we can now compute detrended balanced
growth path values of the endogenous variables of the model prior to the policy reform.
Problem 5. Compute the balanced-growth-path values for
• consumption c,
• leisure l,
• output y,
• investment i,
• capital stock k,
• labor income wh, and
• welfare, u (c, l).
1
Remember to compute the average of the ratios and not the ratio of the averages....
5
3 Policy reform
In the model economy, a government mandatory reduction in the work week would amount
to adding a constraint, h ≤ ¯h = 30
105 = 2
7 , to the social planner’s problem.
max
{ct,lt,it}∞
t=0
∞
t=0
βt
u (ct, lt)
subject to
ct + it ≤ yt, ∀t
kt+1 = (1 − δ) kt + it ∀t, δ ∈ [0, 1]
ht + lt = 1, ∀t
ht ≤ ¯h =
2
7
, ∀t
ct, kt, ht, lt ≥ 0, ∀t
k0 > 0. given
Functional forms
u (ct, lt) = ln ct + ψ ln lt, ∀t, σ > 0
yt = f (kt, ht) = kα
t h1−α
t , ∀t, α ∈ (0, 1) .
3.1 Comparison of long-run welfare
Problem 6. Solve the constrained optimization problem. Denote all the new steadystate
values with ˜. Show that the new steady-state values of the endogenous variables as
functions of the policy-invariant preference and technology parameters will be
˜l = 1 − ¯h
˜k =
1
β − (1 − δ)
α
1
α−1
¯h
˜y =
1
β − (1 − δ)
α
˜k
˜c = 1 −
αδ
1
β − (1 − δ)
˜y
˜ı =
αδ
1
β − (1 − δ)
˜y
˜w = (1 − α) ˜kα¯h−α
˜r = α˜kα−1¯h1−α
.
Comment briefly on these results.
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As economists we are not primarily interested in changes in real variables as such, but
what effect is has for welfare.
Problem 7. First compute the welfare after the policy reform
u ˜c, ˜l = ln ˜c + ψ ln ˜l.
Then compute the compensating variation between the two policies – how much you must
compensate the representative individual under the new policy compared to the first policy,
i.e. find λ such that
u (c, l) = u λ˜c, ˜l
ln c + ψ ln l. = ln λ˜c + ψ ln ˜l.
Comment on your results and compare briefly with the results above
3.2 Dynamic welfare analysis
In the long run we are all dead.
John Maynard Keynes A Tract on Monetary Reform, ch. 3(1923).
The minister had also asked you to compute the short-run consequences of the reform. In
order to answer that question, we must compute the transition between the two detrended
balanced growth paths. In order to compute this transition, we need to solve for the
individuals’ decision rules.
In order to solve for the constrained equilibrium after a potential reform we can model ¯h
as an exogenous state variable
v kt, ¯ht = max
kt+1
u ct, 1 − ¯ht + βv kt+1, ¯ht+1 .
The problem has now one control variable (kt+1), one endogenous state variable (kt) and
one exogenous state variable (¯ht).
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Problem 8. Compute the time-consistent decision rules for the individuals as functions
of the state variables
kt+1 = d kt, ¯ht
ct = f kt, ¯ht + (1 − δ) kt − d kt, ¯ht
using discrete value function iteration. Plot the decision rules. Attach your Matlab
code.
Now that we have the time-consistent decision rules of the individuals we can also make
predictions for the transition between the long-run growth paths.
Problem 9. Let the first k be equal to the long-run growth path value before the reform
and ¯h be equal to value given by the proposed reform. Simulate 50 periods using the timeconsistent
decision rules you computed in the previous problem. Plot the transition for
the following variables
• marginal product of labor, {wt}50
t=1,
• marginal product of capital, {rt}50
t=1,
• output, {yt}50
t=1
• consumption, {ct}50
t=1,
• investment, {it}50
t=1,
• capital stock, {kt}50
t=1,
• labor income, {wtht}50
t=1, and
• welfare, u ct, 1 − ¯ht
50
t=1
.
Comment on your findings. How do they modify your findings from the comparison of
long-run growth paths?
4 Conclusion
Problem 10. The last thing you need to do is to write a short and well-written “executive
summary” which can be circulated to all the other ministries.
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A Measurement
A.1 Download time series for Statistics Norway
For some reason, SSB gives only time series for 1998-2006 in html format. In order to get
the time series for 1970-2006 (which you should!) you have to download the csv-file. The
csv-file can then be e.g. opened in Excel or imported into Stata.
csv (also known as comma-separated values, comma-separated list or comma-separated
variables) file format is a file type that stores tabular data. The format dates back to the
early days of business computing. For this reason, csv-files are common on all computer
platforms.2
A.2 The factors’ share of output
Ideally, taxes and subsidies should have been allocated to capital or labor in the presentation
of the income approach to output. If this had been a serious research project,
you should have done this allocation yourself. However, for this paper you might do the
following approximation
Labor’s share of output ≈
Compensation of employees
Operating surplus + Compensation of employees
.
The time series on the right hand side are available from “4 Gross domestic product by
income components. Million kroner”.
As you see, primarily due to the increase in revenue from the petroleum sector during the
last 10 years or so, this time series does not seem stationary. Also “taking the model to the
measurement”, given the question we want to answer we are interested in the structural
parameters for the mainland economy. In order to rather calculate this you might rather
use the time series for mainland Norway from Tables 12 and 19.
A.3 Investment
As an approximation for investment, you might use “Gross fixed capital formation” at
current prices from Table 26.
A.4 Capital stock
As an approximation for total capital stock, you might use “Fixed assets, total” at current
prices from Table 35.
2
Source: http://en.wikipedia.org/wiki/Comma-separated values
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