Understanding Capital Utilisation in Business Cycles

1. Lecture 14: Capital Utilisation and the Business Cycle L11200 Introduction to Macroeconomics 2009/10 Reading: Barro Ch.9 24 February 2010

2. Introduction • Last time: • Developed model to incorporate labour supply decisions on part of the household • Predictions for procyclicality of employment and hours matched the data • Today • Model cannot explain non-employment of factors of production: today consider capital

3. The Problem, Again • ‘Equilibrium Business Cycle model’ • Based on idea that capital and labour markets are continually in equilibrium • Excess demand (shortages) or excess supply (unemployment) cannot occur • But unemployment of both labour and capital occur over the business cycle

4. K in the model • So far, K has represented both the stock and flow of capital • i.e. K units of capital held by firms were always used in the production • Now change this: firms could hold a stock of capital but choose not to use it • (just like households having a ‘stock’ of labour time available but choosing not to work)

5. Modifying K • Replace K in the production function with some proportional κK • κ is the ‘utilisation rate’: the proportion of capital held by households which they actually use • This is very similar to allowing L to vary and become Ls instead

6. Demand for Capital Services • So now compute how much of the stock of capital K is demanded: ‘capital services’ • Previously calculated demand for capital as a function of real rental cost R/P • If A increased, MPK increased so demand for capital shifted outwards • The same is true for some fixed utilisation rate κ

7. Supply of Capital Services • This is the element which now alters: households do not necessarily sell all of their capital to the capital market • Firms might not want to sell all capital if depreciation rate depends on intensity use • i.e. using machines more intensely raises depreciations

8. Supply Decision • So now net real income from supplying capital services given by: • Rate of return from owning capital is:

9. Supply Decision • So rental income is increasing in intensity, but so is depreciation rate • Rate of return is a linear function of κ, i.e. using machines more intensely matches 1:1 to higher rental income • Depreciation is a non-linear function of κ, i.e. more intensive use increases depreciation rate

10. Optimal Capital Supply • So optimal supply point maximises net capital rental income • i.e. maximum distance between the two curves • Can analyse impact of changing variables • Most importantly: if R/P increases, encourages more supply of capital • So have a upward-sloping relationship between R/P and capital supply

11. Implication • So should find that capital utilisation rate is procyclical • When economy faces positive technology shock, demand for capital services increases • Supply response raises rental rate (and so raises i) and increases capital supply • So capital utilisation and interest rate both move with growth in output

12. Labour Supply • With capital, we found that variable capital supply resulted in procyclical utilisation as well as procyclical price for capital • Same pattern as we found for labour supply • Bigger challenge in modelling labour supply is to explain unemployment • Need to introduce some friction in the model which prevents immediate adjustment to new equilibrium

13. Employment over the Business Cycle • From lecture 2, we need some basic distinctions • Labour force: everyone who wants to work • Employed: people who want to work and have a job • Unemployed: people who want to work and don’t have a job • Key: variations in employment over the business cycle are not explained by variation in labour force

14. Implications • Missing element must be unemployment • Not the case that fluctuations in employment are due to people wanting work more when economy is booming / less when economy is contracting • Instead, when economy is contracting people still want work but cannot find it • This is unemployment: how do we explain it?

15. Summary • Capital utilisation varies over the business cycle • Built a model of capital supply based on depreciation being a function of capital intensity • Generated upward-sloping supply and matched the aggregate data • Next time: unemployment, the missing element in the model