Modelling the Services Sector
Stephen Millard
Bank of England, Durham University Business School
and Centre for Macroeconomics
Phil King
Bank of England
16 October 2013
Motivation
1. Recent poor UK productivity performance most obvious in
service sector. Is this a permanent feature or will service-sector
productivity recover as demand picks up?
2. Firms in standard macroeconomic models look like
manufacturers. Would better modelling of the service sector
improve our understanding of inflation dynamics in the economy
as a whole?
Motivation: Contributions of sectoral productivity to
aggregate productivity relative to trend
Motivation
• Standard firms combine labour and capital to produce output sold
in spot markets
• For the service sector:
– Output is hard to define/measure
– Intangible inputs are extremely important (maybe for manufacturing
too)
– How are wages determined given impossibility of measuring
productivity?
– Markets are rarely (if ever) ‘spot markets’
– So how are prices determined?
Roadmap
•
Motivation
•
Related literature
•
What we learnt from our firm visits
•
Model
•
Response of sectoral productivity to demand shocks
•
An experiment: Model response to a ‘financial crisis’ shock
•
Conclusions
Related literature
• Product market frictions
– Drozd and Nasal (2012): Need to build and maintain a customer
base can explain some international pricing ‘puzzles’
– Gourio and Roudanko (2011): Customer base acts as a form of
intangible capital
– Bai et al (2012): Search frictions mean that ‘demand’ shocks affect
productivity
– Nakamura and Steinsson (2011): Importance of brand loyalty
– Hall (2012): Procyclical marketing spend implies procyclical profit
margins
Related literature
• Intangible capital
– McGrattan and Prescott (2010): Addition of intangible capital to an
otherwise standard RBC model can help explain US 90s boom
– Corrado, Hulton and Sichel (2009): Add intangible capital to a
standard ‘sources of growth’ framework and find that capital
deepening takes over from TFP as the main source of US post-war
growth
– Goodridge, Haskel and Wallis (2012): Similar exercise for the UK –
intangible investment much more important than tangible investment
Related literature
• Increasing returns to scale
– Romer (1990): Once you’ve created the blueprint, replication is
costless. Price cannot equal marginal cost in this environment.
• Two-part tariffs
– Oi (1971): How do you price a ‘mickey mouse monopoly’ like
Disneyland?
– Schmalensee (1982): All you ever wanted to know about two-part
tariffs
– Laffont and Tirole (2000): Pricing telecommunications services
What we learnt from our firm visits
• We visited around 30 firms
• Size varied from ‘one man and his laptop’ consultants all the way
up to a major international financial corporation
• Even spread across private sector services (SIC Groups G, H, I,
J, K, M, N and R)
• Most visits carried out as part of the general ‘intelligence
gathering’ job of our Agents …
• The rest consisted of face-to-face interviews with smaller firms
What we learnt from our firm visits
• Output and price-setting
– Three types of service-sector firm
• Output produced using labour and capital at increasing marginal cost
and sold in spot markets (ie, ‘standard’ firms)
• Bespoke services where the value of the service depends on the match
between providers and buyers
• ‘Scaleable’ services (ie, high fixed cost and low – if not zero – marginal
cost)
• Inputs and investment
– Intangible inputs were important: especially ‘the brand’!
– Lots of emphasis on the importance of customer base and
marketing spend to maintain this
– So, important to model choice between using labour on production
of services vs. marketing
Bespoke services
• Complex bundles of services, unique to each customer.
– No two bundles are ever exactly the same
– Firms are multi-product firms
→ Firms don’t face a demand curve
– Instead, bilateral negotiation between firm and customer over
specification and value of the service
• So we model the matches between individual customers and
firms, and their bargaining over price
Bespoke services give rise to demand-side frictions
• We observed demand-side frictions
– Costly/time-consuming for a firm to build up its customer base via
marketing
– Customers are ‘sticky’ (ie, have brand loyalty, which is why the
‘brand’ is such an important intangible input)
• We think this follows from the bespoke nature of many services.
Bespoke nature
of services
Markets
characterised by
search-matching
Demand-side
frictions
• We observed that these frictions affect firms’ price-setting and
output. Evidence from price-setting surveys supports this
Increasing returns to scale
• Some services firms have high fixed costs and low (negligible)
marginal costs
• Examples include telecoms, publishing, software, finance,
insurance, musicians ...
• Although CRS may not be a bad assumption for service sector
as a whole
Source: Inklaar
(2007)
The model
• Closed economy
• Sticky wages and prices
• We split the private sector into scaleable services, bespoke
services (which we equate with business services), other
consumer services, and goods (ie, agriculture, production and
construction)
• Households own the capital stock and face costs of adjusting
capital
• Households can also decide how intensively their capital is used
The model: Households
• Households have Cobb-Douglas preferences over scaleable
services, other consumer services and non-services
• They minimise the cost of purchasing these
• Minimise
• Subject to
• Implying
P1,t c1,t  P2,t c2,t  at  P3,t c3,t
ct  
1 2  3  2  3
c2,t c3,t
c c1,t
 Pt ct  at
Pi ,t  
 ci ,t

 i for i  1,2,3


The model: Households
• Maximise utility subject to a budget constraint, the demand for
their differentiated labour and sticky wages (Calvo parameter xw)
• Maximise
 ct1 c  1
 h 1 h 
E0   e 

ht 
t 0
 1 c 1  h


t  a ,t
• Subject to B j ,t  Pt k j ,t  1  it 1 B j ,t 1  W j ,t h j ,t  rk ,t Pt z j ,t k j ,t 1  1     z j ,t Pt k j ,t 1
2
k

 k t 1 
 k 
 k j ,t 1   Pt c j ,t   t
 Pt
k j ,t  

2k t 1 
 k t 2 


h j ,t
 W j ,t
 
 Wt




1 w 
w
ht
The model: Households
• First-order conditions imply:
• Is Curve
ct c
 Pt  a ,t 1  a ,t  c
  1  it Et 
e
ct 1
 Pt 1



• Marginal benefit of more intensive utilisation equals marginal cost
of so doing
 z t   rk ,t
• Marginal product of capital equals marginal cost
k
k


 1    r z   z     k t   k t 1   k t 
k ,t 1 t 1
t 1
k

k t 1   k t  k t 1 


1  it   Et 
k
 k



k
t
t 1


 
1 k
 
 k t 1  k t  2  










 Pt 1 
Pt 



The model: Households
• Household’s that cannot change their wages index them to
lagged inflation
W j ,t
 Pt 1
 
 Pt  2
w

 W j ,t 1

• This leads to the ‘Wage Phillips Curve’
Wt  EtWt 1   w t   w t 1

1  x w 1  xw 


 h hˆt   c cˆt  wˆ t 
 1  w  h 

x 1 
w


w


The model: Goods
• Goods producers maximise profits subject to their production
function, the demand for their goods and ‘menu costs’ a la
Rotemberg (ie, absolutely standard problem)
• First-order conditions imply
• Demand for labour Wt   1    y1,t
P1,t
• Demand for capital
Pt rk ,t
P1,t
• Production function
1,t
1
 1,t1

h1,t
y1,t
zt kt 1
y1,t  A1,t zt kt 1
• New Keynesian Phillips Curve

1 1
1
h1,t
1


 1,t 
 1,t 1 
Et 1,t 1 
ˆ1,t
1   1
1   1
x1 1   1 1   
The model: Bespoke service sector
Business Service Provider j
Produces bespoke services using
labour only
q j ,t  AB ,t hB , j ,t
Retailer r
Combines labour and service inputs to
produce output
q  A q~ 2 h1 2
r ,t
2,t r ,t
r ,t
Household demand
The model: Search and matching
• In order to trade, business service providers and retailers have to
match
• Once matched, they trade one unit per period
• Matches dissolve with exogenous probability δq
• Retailers search randomly over producers. There is a real cost
to search χ
• They match with certainty – the question is with whom
The model: Search and matching
• To attract customers, business service providers need to put
resources into marketing, sales, and advertising
– We model this as investment in marketing capital, m
• Business service providers increase the likelihood of matching
with customers by increasing their relative marketing capital
• Building up m requires labour. And it depreciates:
m j ,t  1   m m j ,t 1  hNB , j ,t
m j ,t
• New customers in period t =
st
mt
• Output is:
q j ,t  1   q q j ,t 1 
m j ,t
mt
st
• Number of searching retailers s will depend on search costs and
the marginal product of the business services they purchase
The model: Price of bespoke (business) services
• Once a business service provider and a retailer are matched, the
price of the service is determined by bilateral bargaining.
• Total surplus from a match:
S = net value of match for retailer
+ net value of match for service provider
= J(p) + λ(p)
• How to split the surplus?
• A Nash bargaining solution:
– Assume relative bargaining strength of producer is θ, and of
customer 1 – θ
– Solve:
q r ,t
– Solution:
Wt
pr ,t  2,t Pr ,t ~  1   
q r ,t
AB ,t
Marginal revenue product
of the service
Service provider’s
marginal cost
The model: Price of retail services
• Retailers operate in monopolistically competitive markets and
face menu costs a la Rotemberg
• They maximise the present discounted value of their profits,
where profit in period t is given by:
Pr ,t q r ,t  Wt hr ,t  p r ,t q~r ,t

x 
 2
2 

Pr , t
Pr , t 1
 
P2 , t 1  2
P2 , t  2
2


 1 P2,t y 2,t


• Optimisation leads to the New Keynesian Phillips curve:
 2 ,t
2



 2,t 1 
Et  2,t 1 
ˆ 2,t
1   2
1   2
x 2 1   2 1   
The model: Scaleable services
• Monopoly producer in a contestable market
y3,t  A3,t h3,t
• Production function:
• Fixed costs: Wt h
– Overhead labour
– Sunk costs: producing engineering designs, writing software,
producing films, recording CDs, writing general economic reports,
writing generic insurance contracts...
• With increasing returns to scale, we have the viability constraint:
– Price ≥ Average cost
• Fully linear pricing may not allow firm to cover fixed costs, given
demand
→ Two-part tariff
− A way to increase profits relative to fully linear prices → fixed costs
can be recouped
The model: Scaleable services
• We allow our firm to charge two-part tariffs a + p(q)
→ In a flexible price world, firm sets price equal to marginal cost, as
shown by Oi (1971)
P3,t 
Wt
A3,t
→ But we have costs of adjusting prices
→ So firm’s problem is to maximise present discounted profit flow:








y
P
x

3, t
3,t
  3 
E 0   t e a ,t ct c  at P3,t   P3,t y 3,t P3,t   Wt  h 
A3,t  2 
t 0




P3 , t
P3 , t 1
 
P3 , t 1  3
P3 , t  2
2




 1 P3,t y 3,t 




The model: Scaleable services
• The first-order conditions in this sector imply:
y3,t  A3,t h3,t
• Aggregate production
• New Keynesian Phillips curve
 3,t
3
1   3   32
3

 3,t 1 
Et  3,t 1 
 2 Et  3,t  2
1   3 1   3 
1   3 1   3 
1   3 1   3 

1
x 3 1   3 1   3

Wˆ  Aˆ

t
3,t
 Pˆ3,t

• a is set as high as possible
• Contestability implies zero profit
2

x 3  1   3,t
at  Wt h  h3,t  
 1 P3,t y3,t  P3,t y3,t


2  1   3,t 1  3

The model: Monetary policy and market clearing
• The central bank operates a Taylor rule:
it  i   i it 1  i   1   i   t   y yˆ t   m,t
• All markets clear:
ht  h1,t  hB,t  hNB ,t  h  h3,t
y1,t  c1,t  k t  1   k t 1
y 2 ,t  c 2 ,t
y 3, t  c 3, t
Calibration
• We set the consumption shares as follows:
– Goods (Agriculture, production and construction) 59%
– Scalable services (Information and communication, Finance and
insurance, Arts, entertainment and recreation) 18%
– Other consumer services (Retail, Repair of motor vehicles, Rail
transport, Air transport, Accommodation and food, Real estate, Vets)
29%
– Business services (Wholesale, Transportation and storage ex. rail
and air transport, Professional, scientific and technical ex. vets,
Admin and support)
• This implied values for 1, 2 and 3 of , 0.5513, 03016 and
0.1471, respectively
Calibration
• We set the employment shares as follows:
– Goods (Agriculture, production and construction) 28%
– Scalable services (Information and communication, Finance and
insurance, Arts, entertainment and recreation)
• Fixed labour 3%
• Variable labour 11%
– Other consumer services (Retail, Repair of motor vehicles, Rail
transport, Air transport, Accommodation and food, Real estate, Vets)
27%
– Business services (Wholesale, Transportation and storage ex. rail
and air transport, Professional, scientific and technical ex. vets,
Admin and support)
• Billable hours 25%
• Non-billable hours 6%
Calibration
•
•
•
•
•
•
•
Log utility, ie, c=1
Frisch labour supply elasticity of 2, implying h=0.5
Discount factor, , of 0.99
Steady-state wage mark-up of 1.5, implying w=0.5
Average duration of wages of 1 year, implying xw=0.75
Degree of wage indexation, w, of 0.3
Taylor rule
it  i  0.8it 1  i   0.21.5 t  0.125 yˆt   m,t
• Demand shock
 a,t  0.88 a,t 1  a,t
Calibration – Goods sector
•
•
•
•
•
•
•
•
Depreciation rate of 10% pa, implying =0.025
Elasticity of capital adjustment costs, k, set to 0.5
Scale of capital adjustment costs, k, set to 201
Elasticity of capital utilisation costs, z, set to 0.56
Elasticity of output with respect to capital input, 1, set to 0.395
Steady-state price mark-up of 1.1, implying =10/11
Average duration of prices of 1 year, implying x1=116.5501
Degree of price indexation, 1, set to 0.3
Response of productivity to demand shocks:
Goods
prod
prod
0.06
0.4
0.35
0.05
0.3
0.04
0.25
0.03
0.2
0.15
0.02
0.1
0.01
0.05
0
0
-0.01
-0.05
2
4
6
8
10
12
14
16
18
• Consumption risk premium
shock
20
2
4
6
8
10
12
14
16
• Monetary policy shock
18
20
Effects of a demand shock: Goods sector
• Response of productivity to a negative demand shock is positive
on impact in the goods sector
– This follows from the production function


yˆ1,t  Aˆ1,t  1 kˆt 1  zˆt  1  1 hˆ1,t


 yˆ1,t  hˆ1,t  1 hˆ1,t  zˆt  0
– Since capital utilisation adjusts by less than labour input
• Once capital adjusts down, productivity falls
• Addition of labour hoarding can alter this result
• As can the presence of a fixed costs as we’ll see later
Calibration – Bespoke services sector
Steady-state price mark-up of 1.1, implying =10/11
Average duration of prices of 1 year, implying x2=116.5501
Degree of price indexation, 2, set to 0.3
Depreciation rate for marketing capital, m, set to 0.6
Depreciation rate for matches, q, set to 0.1
Elasticity of output with respect to business services input, 2, set
to 0.241
• Bargaining power of business services producers, , set to 0.5
•
•
•
•
•
•
Response of productivity to demand shocks:
Retail/Business services
bsprod
bsprod
2
5
0
0
-5
-2
2
4
6
8
10
12
14
16
18
2
20
4
6
8
14
16
18
20
12
14
16
18
20
12
14
16
18
20
12
10
csprod
csprod
0.01
0.5
0
0
-0.01
-0.5
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
prod
prod
0
2
-0.5
0
-1
2
4
6
8
10
12
14
16
18
• Consumption risk premium
shock
20
-2
2
4
6
8
10
• Monetary policy shock
Effects of a demand shock: Business services and
retail sectors
• Response of productivity to a negative demand shock is negative
in the business services sector
– Labour used for marketing is valuable and so is held on to, though it
is not measured as being productive
• Retail productivity is higher than base after a year in the case of
a consumption risk premium shock
– We’re still investigating what exactly is going on here
Calibration – Scalable services sector
• Average duration of prices of 1 year, implying x3=11.655
• Degree of price indexation, 3, set to 0.3
• Steady-state fixed charge, a, is equal to 0.0386 given
consumption and employment shares
Response of productivity to demand shocks:
Scaleable services
prod
prod
0
0
-0.1
-0.02
-0.2
-0.04
-0.3
-0.06
-0.4
-0.5
-0.08
-0.6
-0.1
-0.7
-0.12
2
4
6
8
10
12
14
16
18
• Consumption risk premium
shock
20
-0.8
2
4
6
8
10
12
14
16
• Monetary policy shock
18
20
Effects of a demand shock: 3
• Response of productivity to a negative demand shock is negative
in the scaleable services sector
– This follows given the increasing returns to scale in this sector …
– … as you’d expect given the above
Productivi ty 
y3,t
h  h3,t

A3,t h3,t
h  h3,t

h 
Percentagechange in productivi ty  1001  3 hˆ3,t  0
 h3  h 
Using the model to analyse the financial crisis
• Model the financial crisis as a negative demand (consumer risk
premium) shock
• Calibrate the shock based on rise in consumer credit spread
• Likely to understate the true size of the shock o/a
– Shock assumed to have no effect on investment in the model
– No net trade shock (unlike in the real world)
– No fiscal consolidation (ditto)
• Peak effect is to push down on GDP by 2% (vs. 7% fall in GDP in
the data)
Response of productivity to the financial crisis
Sectoral productivity relative to pre-crisis tred
Scalable services
2007=100
130
Business services
Consumer services
Other private sector
120
110
100
90
80
70
60
1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1
Response of productivity to the financial crisis
Simulated response of sectoral productivity to the negative
demand shock
Percentage deviation from trend
Other consumer services
1
Other private sector
0
Scalable services
-1
Business services
-2
-3
-4
-5
2007
2009
2011
2013
2015
2017
Response of productivity to the financial crisis
• Our model suggests that the demand shock resulting from the
financial crisis led to a peak fall in business services productivity
of 4.3% relative to trend, as firms allocated relatively more labour
to building and maintaining their customer base, as opposed to
direct production
• The shock also generates a fall in ‘scalable’ services productivity,
through the increasing returns channel, though it is small: 0.4%
relative to trend
• Of course, the shock we model does not shed any light on the
additional fall in productivity from 2010 onwards, and is also likely
to understate the true size of the demand shock experienced by
the UK economy in the wake of the financial crisis
Next steps
• More careful calibration and/or estimation
• More on the response of productivity to a demand shock
– How different is the response of aggregate productivity to the shock
compared with the same impulse response in, say, the Smets and
Wouters model?
• Simulate responses of inflation (in the aggregate and in each
sector) to a monetary policy shock
– Does the model provide new insights into inflation dynamics at a
sectoral and aggregate level?
– Again, how different is the response of aggregate inflation to a
monetary policy shock relative to the Smets and Wouters model?
• Ideas and comments welcome