Abstract
To explain changes in outputs and inputs, and therefore changes in productivity, we need to know something about managerial behaviour. The existence of different sets and functions has few, if any, implications for behaviour. The existence of revenue functions, for example, does not mean that managers will choose outputs in order to maximise revenues, and the existence of cost functions does not mean they will choose inputs to minimise costs. Instead, different managers will tend to behave differently depending on what they value, and on what they can and cannot choose. For example, if managers value goods and services at market prices, then, if possible, they will tend to choose outputs and inputs to maximise profits. This chapter discusses some of the simplest optimisation problems faced by firm managers.
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Notes
- 1.
If \(Q(q_{it})=a_1q_{1it}+a_2q_{2it}\), then \(q_{2it}=Q(q_{it})/a_2-(a_1/a_2)q_{1it}\). This is the equation of a line with a slope of \(-a_1/a_2\) and an intercept of \(Q(q_{it})/a_2\). The term iso-output derives from the fact that all points on this line yield the same aggregate output, namely \(Q(q_{it})\).
- 2.
If \(X(x_{it})=b_1x_{1it}+b_2x_{2it}\), then \(x_{2it}=X(x_{it})/b_2-(b_1/b_2)x_{1it}\). This is the equation of a line with a slope of \(-b_1/b_2\) and an intercept of \(X(x_{it})/b_2\). The term iso-input derives from the fact that all points on this line yield the same aggregate input, namely \(X(x_{it})\).
- 3.
Here, the term ‘exogenous’ means that demand shifters (e.g., population, tastes) are not affected by the actions of the firm (or, more precisely, the firm manager).
- 4.
If \(R^t(x_{it},d_{it},z_{it})=p_1(\ddot{q}_{it},d_{it})\ddot{q}_{1it}+p_2(\ddot{q}_{it},d_{it})\ddot{q}_{2it}\), then \(\ddot{q}_{2it}=R^t(x_{it},d_{it},z_{it})/p_2(\ddot{q}_{it},d_{it})-[p_1(\ddot{q}_{it},d_{it})/p_2(\ddot{q}_{it},d_{it})]\ddot{q}_{1it}\). This is the equation of a line with a slope of \(-p_1(\ddot{q}_{it},d_{it})/p_2(\ddot{q}_{it},d_{it})\) and a vertical intercept of \(R^t(x_{it},d_{it},z_{it})/p_2(\ddot{q}_{it},d_{it})\). The term iso-revenue derives from the fact that if \(p(\ddot{q}_{it},d_{it})\) did not vary with \(\ddot{q}_{it}\), then all output combinations on this line would yield the same revenue. The term pseudo is used here because \(p(\ddot{q}_{it},d_{it})\) does vary with \(\ddot{q}_{it}\).
- 5.
If \(R_{it}=p_{1it}q_{1it}+p_{2it}q_{2it}\), then \(q_{2it}=R_{it}/p_{2it}-(p_{1it}/p_{2it})q_{1it}\). This is the equation of a line with a slope of \(-p_{1it}/p_{2it}\) and an intercept of \(R_{it}/p_{2it}\). The term iso-revenue derives from the fact that all points on this line yield the same revenue, namely \(R_{it}\).
- 6.
Here, the term ‘exogenous’ means that supply shifters (e.g., characteristics of production environments in upstream sectors) are not affected by the actions of the firm (or, more precisely, firm managers).
- 7.
If \(C^t(s_{it},q_{it},z_{it})= w_1(\ddot{x}_{it},s_{it})\ddot{x}_{1it} + w_2(\ddot{x}_{it},s_{it})\ddot{x}_{2it}\), then \(\ddot{x}_{2it}=C^t(s_{it},q_{it},z_{it})/w_2(\ddot{x}_{it},s_{it}) - [w_1(\ddot{x}_{it},s_{it})/w_2(\ddot{x}_{it},s_{it})]\ddot{x}_{1it}\). This is the equation of a line with a slope of \(-w_1(\ddot{x}_{it},s_{it})/w_2(\ddot{x}_{it},s_{it})\) and a vertical intercept of \(C^t(s_{it},q_{it},z_{it})/w_2(\ddot{x}_{it},s_{it})\). The term iso-cost derives from the fact that if \(w(\ddot{x}_{it},s_{it})\) did not vary with \(\ddot{x}_{it}\), then all input combinations on this line would yield the same cost. The term pseudo is used here because \(w(\ddot{x}_{it},s_{it})\) does vary with \(\ddot{x}_{it}\).
- 8.
If \(C_{it}=w_{1it}x_{1it}+w_{2it}x_{2it}\), then \(x_{2it}=C_{it}/w_{2it}-(w_{1it}/w_{2it})x_{1it}\). This is the equation of a line with a slope of \(-w_{1it}/w_{2it}\) and an intercept of \(C_{it}/w_{2it}\). The term iso-cost derives from the fact that all points on this line yield the same cost, namely \(C_{it}\).
- 9.
Here, the term ‘exogenous’ means that demand and supply shifters are not affected by the actions of the firm (or, more precisely, the firm manager).
- 10.
If \(\varPi ^t(s_{it},d_{it},z_{it})= P(\mathring{q}_{it},d_{it})Q(\mathring{q}_{it})-W(\mathring{x}_{it},s_{it})X(\mathring{x}_{it})\), then \(Q(\mathring{q}_{it})=\varPi ^t(s_{it},d_{it},z_{it})/P(\mathring{q}_{it},d_{it})+[W(\mathring{x}_{it},s_{it})/P(\mathring{q}_{it},d_{it})]X(\mathring{x}_{it})\). This is the equation of a line with a slope of \(W(\mathring{x}_{it},s_{it})/P(\mathring{q}_{it},d_{it})\) and a vertical intercept of \(\varPi ^t(s_{it},d_{it},z_{it})/P(\mathring{q}_{it},d_{it})\). The term iso-profit derives from the fact that if \(W(\mathring{x}_{it},s_{it})\) and \(P(\mathring{q}_{it},d_{it})\) did not depend on \(\mathring{x}_{it}\) and \(\mathring{q}_{it}\), then all points on this line would yield the same profit. The term pseudo is used here because \(W(\mathring{x}_{it},s_{it})\) and \(P(\mathring{q}_{it},d_{it})\) do depend on \(\mathring{x}_{it}\) and \(\mathring{q}_{it}\).
- 11.
If \(\varPi _{31}=P(q_{31},p_{31})Q(q_{31})-W(x_{31},w_{31})X(x_{31})\), then \(Q(q_{31})=\varPi _{31}/P(q_{31},p_{31})+[W(x_{31},w_{31})/P(q_{31},p_{31})]X(x_{31})\). This is the equation of a line with a slope of \(W(x_{31},w_{31})/P(q_{31},p_{31})\) and a vertical intercept of \(\varPi _{31}/P(q_{31},p_{31})\). The term iso-profit derives from the fact that if \(P(q_{31},p_{31})\) and \(W(x_{31},w_{31})\) did not vary with \(q_{31}\) and \(x_{31}\), then all points on this line would yield the same profit. The term pseudo is used here because, except in restrictive special cases (e.g., there is only one input and only one output), \(P(q_{31},p_{31})\) and \(W(x_{31},w_{31})\) do vary with \(q_{31}\) and \(x_{31}\).
- 12.
The equation of the dashed line through point A is \(Q(q_{it})={ {TFP}}(x_{it},q_{it})X(x_{it})\). This is the equation of a line with a slope of \({ {TFP}}(x_{it},q_{it})\) and an intercept of zero. The term iso-productivity ray derives from the fact that all points on this ray map to the same level of TFP, namely \({ {TFP}}(x_{it},q_{it})=Q(q_{it})/X(x_{it})\).
- 13.
To avoid clutter, the axis label for \(X(\breve{x}_{it})\) has been omitted from Fig. 4.12. In this particular example, \(X(\breve{x}_{it})=X(x^*_{it})\).
- 14.
If \(E(q_{it})=\pi _{1it}q_{1it}+\pi _{2it}q_{2it}\), then \(q_{2it}=E(q_{it})/\pi _{2it}-(\pi _{1it}/\pi _{2it})q_{1it}\). This is the equation of a line with a slope of \(-\pi _{1it}/\pi _{2it}\) and an intercept of \(E(q_{it})/\pi _{2it}\). The term iso-expected-output derives from the fact that all points on this line yield the same expected output, namely \(E(q_{it})\).
- 15.
It is not entirely clear what Olley and Pakes (1996) mean by the term ‘productivity’. They appear to use the terms ‘productivity’ and ‘efficiency’ interchangeably.
- 16.
See, for example, Tversky and Kahneman (1974, p. 1130).
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O’Donnell, C.J. (2018). Managerial Behaviour. In: Productivity and Efficiency Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-2984-5_4
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DOI: https://doi.org/10.1007/978-981-13-2984-5_4
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