To help inform our output gap estimate we have used a simple Cobb-Douglas production function. This enables us to look more closely at developments in labour productivity and to split it into capital deepening and total factor productivity (TFP).
In the Cobb-Douglas framework, output is represented by a combination of factor inputs, and their elasticity to output (a), multiplied by the level of technology in the economy or TFP (A). The factors we use are labour (L) and capital (K). The equation below (1) shows the function in logs.
logYt = log At + a log Lt + (1- a) log Kt (1)
An estimate of potential (P) output simply uses the structural component of those inputs – equation (2). However, since potential capital is the full utilisation of the capital stock there is no justification to de-trend or smooth the series. The difference between equations (1) and (2) represents the output gap.
logYtp = logAtp + a logLtp + (1 – a)Kt (2)
We use ONS data on labour supply and the capital stock. The labour input represents total hours worked in the economy while the capital input is the total amount (in £bn) of capital stock in the economy, excluding housing. Capital stock data is currently only available up to 2009, so we derive an estimate of the capital stock from 2009 to the third quarter of 2012 using a simple law of motion formula:
Kt = Kt-1(1 – δt) + It (3)
where δ is the depreciation rate, and I is investment. We assume the depreciation rate is equal to the average from 2000 to 2009. We assume the elasticity of labour input is equal to the historical average labour share of income. An estimate of TFP in the economy is derived by solving for A in equation 1.
log At = logYt – a log Lt – (1 – a) log Kt (4)
The production function approach can also be used to produce an alternative estimate for potential output and the output gap. However, like any other approach it depends on the judgements that underpin it, especially the trend level of TFP (AP). This is particularly difficult to estimate and a common approach is to apply a statistical filter to actual TFP.
As discussed in paragraph 3.9 we have used this framework to produce an implied trend TFP series (equation (5)). To do so we use our output gap (OG) estimate, from the cyclical indicator approach, to produce a trend output series (YP) (equation (6)) combined with our assumptions on labour market trend input (LP) and our estimate of the capital stock (Kt)
log Atp = logYtp – a log Ltp – (1 – a) log Kt (5)
logYtp = logYt – log( 1 + OGt) (6)