Abstract
These are two concepts used in the theory of economic policy. Although the subject of economic policy as one of the forms of applied economic theory is as old as economic science itself, the more systematic treatment meant by the phrase ‘theory of economic policy’ started much more recently, in close connection with the development of econometrics. Econometrics, as the combination of theory and observation in the area of intersection of economics, statistics and mathematics, introduced the possibility of dealing with economic policy not only qualitatively, but also quantitatively. This enables economists to formulate policy recommendations in the most concrete form conceivable, as they are needed by policy-makers – government, parliament and representatives of social groups. It seems appropriate to consider as the starting document of the theory of economic policy in this sense, Ragnar Frisch’s document (1949), written for the United Nations’ short-lived Employment Commission, ‘A memorandum of price–wage–tax–subsidy policies as instruments in maintaining optimal employment’.
These are two concepts used in the theory of economic policy. Although the subject of economic policy as one of the forms of applied economic theory is as old as economic science itself, the more systematic treatment meant by the phrase ‘theory of economic policy’ started much more recently, in close connection with the development of econometrics. Econometrics, as the combination of theory and observation in the area of intersection of economics, statistics and mathematics, introduced the possibility of dealing with economic policy not only qualitatively, but also quantitatively. This enables economists to formulate policy recommendations in the most concrete form conceivable, as they are needed by policy-makers – government, parliament and representatives of social groups. It seems appropriate to consider as the starting document of the theory of economic policy in this sense, Ragnar Frisch’s document (1949), written for the United Nations’ short-lived Employment Commission, ‘A memorandum of price–wage–tax–subsidy policies as instruments in maintaining optimal employment’.
The term ‘targets and instruments’ refer to economic variables in a special case of a more flexible version of the theory of economic policy, where the more general terms ‘aims and means of a policy’ are used, which may be qualitative as well as quantitative. An aim may then be the maximization (under possible restrictions) of social welfare, and among the means a reform may appear. Targets are numerical values of variables appearing in a social-welfare function and are supposed a priori to be the values that maximize social welfare. Instruments are quantitative values of means controllable by the policy-maker (cf. Preston and Pagan 1982).
Examples of target variables are employment, current balance-of-payment surplus, current government surplus, income, the rate of inflation, and others. Examples of instrument variables are direct and indirect tax rates, interest, total or specific public expenditures, working hours per week, working weeks per year, age of retirement, wage rates, and so on.
Problems of economic science may be subdivided into two categories: explanatory or analytical problems, and normative or policy problems. The complete mathematical formulation and solution of these problems require the introduction of two more categories of variables, to be called ‘exogenous’ (or ‘data’) variables and ‘other’ (or ‘irrelevant’) variables. In what follows, the four categories will sometimes be indicated by x (irrelevant), y (target), z (instrument) and u (data) variables. In addition, the mathematical formulation of the two types of problems requires the fulfilment of a number I of equations or relations, numbered i (=1, 2, …, I), a number of J of variables xj, a number K of variables yk, a number L of variables zl, and I variables ui.
The equations will be assumed to be linear, which for small variations is no restriction, but for large variations constitutes a limitation. They will be written:
Examples of relations are definitions, technical or legal relations, balance equations and behavioural relations such as demand or supply equations for either goods or factors of production (labour types, capital, etc.). The group of I equations (1) describes, in a simplified way, the operation of the economy studied and is called a ‘model’ of that economy, more particularly when all coefficients a, b and c have been given numerical values obtained from a series of values for all x, y, z and u over some observation period.
The mathematical-statistical (or econometric) methods of estimation will not be discussed here, but the choice of, in particular, the variables x and u will be such as to obtain reliable values of the coefficients. This implies that the coefficients of determination R2, corrected for the number of degrees of freedom (and then written \( {\overline{R}}^2 \)) as well as the so-called t-values satisfy certain conditions, usually \( \overline{R} \) should be not far below 1 and ts > 3, but this ideal is rarely attained.
A problem (and this applies to both types of problem mentioned) can be solved only if the number of unknowns N equals the number of equations’ I – this being a necessary but not a sufficient condition. The unknowns for each time unit are, for the explanatory problem, the target variables and the ‘other’ variables. So we must have:
For the political problem the unknowns are the instrument and the ‘other’ variable, and we must have:
From (2) and (3) we deduce that K = L must apply for the problems to be solvable; that is, the number of instruments must equal the number of target variables. Later we will discuss some exceptions to this thesis, but as a general rule our conclusion stands.
For a more concise treatment of our problems it is sometimes preferable to formulate the model in a simplified form by eliminating the irrelevant variables x. This elimination requires J equations and so we are left with N – J = K = L equations, in which only the y, z and u appear. In order to avoid confusion we will now use capital letters for the coefficients:
For simplicity’s sake we will discuss examples where K = L = 4. The explanatory problem’s solution is obtained by solving (4) for the target variables y:
The policy problem’s solution is found from solving (4) for the instrument variables z:
By use of matrix notation these equations might have been written more elegantly, but we shall refrain from doing so. It seems desirable, though, to express verbally the meaning of the coefficients used. Evidently pkl constitutes the change in yk caused by a unit change in zl and no change in the other zs or any u. If normalized variables had been used (i.e. variables with a mean equal to 0 and a standard deviation equal to 1, as is customary in sociologists’ path analysis) pkl becomes the partial elasticity of yk with respect of zl. In both cases p constitutes a measure for the impact of instrument variable zl on target variable yk, all other z and all u assumed constant. Inversely and similarly qlk measure the impact of a unit change in target yk on instrument zl, all other y and all data u assumed unchanged.
As previously observed the conditions so far mentioned are necessary but not sufficient. Other conditions which must be fulfilled are that equations (4) be neither incompatible nor dependent nor overdetermined. Simple illustrations are the following. If of four unknowns three appear in only two of the equations and the fourth in the other two equations, then the first two equations are overdetermined and the other two are either incompatible or dependent. Overdetermination implies that there is not just one solution but an infinity of them. Incompatibility means that the solution of one of the two equations does not satisfy the other. Dependency of equations means that one equation can be deduced from the other. In that case they do have the same solution, and so the occurrence of one unknown in both equations does no harm, but the solution for the three other unknowns from the two remaining equations is impossible.
An example of a system of equations suffering from non-fulfilment of the conditions just discussed can be found in Tinbergen (1956), Problem 161, using Model 16. As a counterexample without this difficulty, Problem 162 has been added. In these examples the irrelevant variables had not been eliminated first and the complete model containing 17 variables is shown.
Some politicians think that the normal situation is that there is a one-to-one correspondence between particular targets and particular instruments, for example that a tax rate is used to equilibrate the government budget, an exchange rate to equilibrate the balance of payments, and a wage rate to create enough employment. As a rule this is not correct, for such a situation would imply that only one zl appears in each of the four equations (5) and only one yk in each of the four equations (6), implying in turn that equations (5) and (6) could be arranged so that only the diagonal elements of the matrices P and Q could be non-zero. The normal situation is that not all elements off the diagonal vanish.
There are however some elements equal to zero in most models. An interesting case is that where the equations can be ordered so that all elements above the diagonal are nought, or where blocks of elements are equal to zero. Connected with such coefficient matrices is H.A. Simon’s concept of the ‘order’ of an unknown, which in a policy problem corresponds to the instruments and the irrelevant variables. The concept indicates that the unknowns can be solved in a predetermined order only; the one with order 1 depends on one coefficient only, or if a group of unknowns has order 1, they depend on as many coefficients as appear in the group of equations in which these unknowns only appear. A next unknown or group of unknowns depends on the coefficients appearing in the equations containing groups 1 and 2 of the unknowns, and so on. Evidently an ordered system may be organized in a simpler way, because some decision-makers (say, government ministers) can decide quite independently of other decision-makers without deviating from the optimal policy.
Further deviations from the standard case discussed in illustrating equations (5) and (6) will occur if some instrument variables are subject to restrictions, such as the impossibility of negative values or of values less than a previous value. A large number of economic variables (for instance, production and consumption as well as prices) cannot be negative. In today’s industrial countries a reduction in nominal wages is almost impossible. If without such a restriction an impossible value of some instrument would be part of the solution, the restriction becomes active; that is, it becomes an equation instead of an inequality. Since the number of unknowns is then less than the number of equations, we have either to add an unknown (an additional instrument) or to omit one equation. A possible example is that a foreign loan may be introduced as an instrument in order to keep the balance of payments in equilibrium.
The introduction of several restrictions (non-negativity of several unknowns) may leave us, after using all the equations to eliminate unknowns, with, say, two unknowns and three restrictions, still permitting any point within a triangle. The latter is called the ‘admissible’ or ‘feasible’ area and the remainder of the policy problem may be presented as a problem of linear programming. A choice among the points within this feasible area is now possible by adding the condition that some function of the remaining two unknowns be maximized. The substitution of equations by inequalities need not be used only to express the necessity that a variable be non-negative; a production function may also be interpreted as yielding the maximum quantity of product obtainable from given inputs, any deviation from that maximum then representing waste or ‘X-inefficiency’.
Finally, a few words may be said about the use of target and instrument variables in interactive planning (J.A. Hartog, P. Nijkamp, J. Spronk). Frisch and his school built their policy-planning on a social-welfare function obtained by interviewing policy-makers on a universe of local trade-off rates between the variables that determined, in their opinion, the population’s level of satisfaction. If n such variables are thought to exist, hypersurfaces of n dimensions would be the hypersurfaces of constant satisfaction. The interaction planning school doubts whether the average policy-maker is able to describe such hypersurfaces. The method they propose is that the policy-planner starts with a given situation and subsequently shows the policy-maker what change in the targets is obtained by an assumed first set of changes in instruments, asking him whether that change in the targets constitutes an improvement. The policy-maker may propose a further change in the instruments and the planner will inform him on the consequences for the targets. Thus, step by step, in this dialogue, planner and policy-maker will approach a situation which does not admit any improvement as a consequence of changes in instruments to be proposed by the policy-maker. In this dialogue the policy-maker will have to compare a limited number of sets of instruments and targets, presumably a much lower number than was needed by the interview method (cf. also Hughes Hallett and Rees 1983).
Bibliography
Frisch, R. 1949. A memorandum of price–wage–tax–subsidy policies as instruments in maintaining optimal employment. United Nations Employment Commission. E/CN.1/Sub 2/13, April.
Hughes Hallett, A., and H. Rees. 1983. Quantitative economic policies and interactive planning. Cambridge: Cambridge University Press.
Preston, A.J., and A.R. Pagan. 1982. The theory of economic policy, statics and dynamics. Cambridge: Cambridge University Press.
Tinbergen, J. 1956. Economic policy: Principles and design. Amsterdam: North-Holland.
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Tinbergen, J. (2018). Targets and Instruments. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_1585
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