39  All speeches are available online at www.bankofengland.co.uk/speeches  39  Appendix – estimation of firm-level mark-ups  To estimate firm-level mark-ups, we follow De Loecker and Eeckhout (2017, 2018) and Díez et al (2018), all of which are based on the original methodology proposed by De Loecker and Warzynski (2012). Theory  Consider a firm 𝑖 at time 𝑡 that produces output with a production function whose arguments are variable inputs, capital, and technology. Assume firms are cost-minimizers. Then, the Lagrangian associated with the cost minimisation problem (subject to the production function) leads to the following first-order condition for any variable input 𝑉 (free of adjustment costs):  𝑃𝑖𝑡𝑉 = 𝜆𝑖𝑡 (  𝜕𝑄𝑖𝑡 (⋅) ) 𝜕𝑉𝑖𝑡  (𝐴1)  where 𝑃𝑖𝑡𝑉 is the input price of variable input 𝑉 faced by firm 𝑖 at time 𝑡, 𝜆𝑖𝑡 denotes the marginal cost of production, and (  𝜕𝑄𝑖𝑡 (⋅) 𝜕𝑉𝑖𝑡  to marginal cost, 𝜇𝑖𝑡 ≡  ) denotes the marginal product of this input. Defining the mark-up as the ratio of price 𝑃𝑖𝑡 𝜆𝑖𝑡  , we have that the above equation can be rearranged to give  𝜇𝑖𝑡 =  𝑃𝑖𝑡 𝜕𝑄𝑖𝑡 (⋅) ( ) 𝑃𝑖𝑡𝑉 𝜕𝑉𝑖𝑡  (𝐴2)  We can rewrite this as  𝜇𝑖𝑡 =  𝑃𝑖𝑡 𝑄𝑖𝑡 𝜕𝑄𝑖𝑡 (⋅) 𝑉𝑖𝑡 ( ) = (𝛼𝑖𝑡𝑉 )−1 𝜃𝑖𝑡𝑉 𝜕𝑉𝑖𝑡 𝑄𝑖𝑡 𝑃𝑖𝑡𝑉 𝑉𝑖𝑡  (𝐴3)  where 𝛼𝑖𝑡𝑉 denotes the share of expenditures on input 𝑉𝑖𝑡 in total sales (𝑃𝑖𝑡 𝑄𝑖𝑡 ), and 𝜃𝑖𝑡𝑉 denotes the output elasticity of input 𝑉.
Money issued by central banks (monetary base) in Sweden, the euro area, USA, UK and Japan Per cent of GDP 140 120 100 80 60 40 20 0  01  05  09  The Riksbank ECB Federal Reserve  13  17  Bank of England Bank of Japan  Note. Monetary base here refers to the banknotes and coins held by the general public and banks’ assets with the central banks. Sources: The Bank of England, the IMF and Macrobond  35 [35]