Phillips curve and the Tinbergen and Theil approach to economic policy

Class handout Phillips curve and the Tinbergen and Theil approach to economic policy Giovanni Di Bartolomeo University of Teramo 1. The Phillips menu In the 190s, econometricians estimated a negative relationship between inflation and unemployment (Phillips curve). 1 This relationship is depicted in the following figure. Inflation rate (π) 10 C 19 4 197 19 195 19 194 193 191 α 0 1 3 4 5 7 9 10 Unemployment rate (u) Figure 1 The Phillips curve in the 0s Formally, it can be represented as (1) π = αu + c where α and c and are estimated coefficients. The economists postulate that by controlling the monetary aggregates it is possible to control inflation then, by using (1), the unemployment rate. 1 Indeed, in his original work, Phillips estimated the relationship between the money wages and unemployment. The relationship between inflation and unemployment was later developed by Samuelson and Solow. 1

Note that in our simplified world, 1. assuming that the labor force is given ( l f ), there is a one-to-one (inverse) correspondence between unemployment and labor (l) by definition (i.e., u = 1 l/ lf );. assuming capital is given ( k ), there is a one-to-one (direct) correspondence between labor y = f k, l ). and output by the production function (i.e., ( ) It follows that there is a one-to-one correspondence between unemployment and output (Okun law). Thus we can use output, labor and unemployment as thesauruses.. The policy problem We can assume that the policymaker dislikes both inflation and unemployment and that he would like to obtain zero inflation and zero unemployment (first best solution). Moreover, we can assume that the marginal costs of both are increasing in their levels; it means, e.g., that a marginal increase of inflation costs more (in terms of unemployment) when inflation is high, and vice versa. The loss can be represented by a simple squared expression: u () L= a π + It is worth noticing that the loss is zero if and only if inflation and unemployment are zero otherwise is positive. The loss is increasing in both inflation and unemployment (check the signs of the first derivatives), with increasing marginal costs (check the signs of the second derivatives). Inflation rate (π) 10 L 3 L 4 L 1 First best L 0 0 1 3 4 5 7 9 10 Unemployment rate (u) Figure The policymaker s loss See Appendix A for a discussion.

The policy problem the is to minimize () subject to (1). Graphically, the problem is solved by finding the lowest loss possible given the Phillips curve constraint. Inflation rate (π) 10 L 3 E L First best 4 L 1 L 0 A B 0 1 3 4 5 7 9 10 Unemployment rate (u) Figure 3 The policymaker s problem Figure 3 shows that the policymaker would prefer to be e.g. in point A; however this point is unfeasible as the policymaker has to choose inflation taking account of the Phillip curve (he can only move along the Phillips curve). Point E and B are both feasible, but the former is associated to a lower loss than the latter. The lowest loss is achieved in point E that is the policymaker s problem solution. Therefore, the policymaker will set inflation at about 5% and about % unemployment rate will be observed. Analytically, it can be solved as follows by plugging (1) in () and differentiate for π, (3) π 1 c π L 1 c π L= a + = aπ α π α α We have used the relationship Optimal inflation is then (4) c π u = from (1). α 1 c * c 1 a+ π π a = = + α α α α Using (4) and (1), we find also the optimal unemployment rate: 1 (5) u * 1 c c 1 = a 3 + α α α 3

Exercise 1. Consider (4) and (5), then answer the following questions. 1. What are the effects on inflation and unemployment when the policymaker dislikes inflation more (higher a)? [compute: π * / a and u * / a]. Try to explain.. What are the effects on inflation and unemployment when the Phillips curve becomes steeper (higher α)? [compute: π * / α and u * / a]. Try to explain. Alternatively, the problem can be solved by using the function of a function derivative rule (i.e., g( f( x)) = g'( f( x)) f '( x) ) as u is a function of π: We can write our problem as () ( π ) π u max L= a + π c π s.t. u ( π ) = α The first order condition is then (7) L = aπ + u u = 0 aπ = u u π π marg. cost π marg. benefit i.e., it is optimal for the policymaker to equalize the marginal cost of increasing inflation (higher u 1 inflation) to its marginal benefit (lower unemployment). Note that = is negative, thus the π α r.h.s. of (7) is positive. Equation (7) can be written as: 1 () π = u aα Solving the two-equation system (1) and () in u and π, the solution (4) and (5) is found. Equations (1) and () have a nice interpretation. Equation (), in fact, is the optimal policy rule: Given any possible unemployment rate, it describes the optimal reaction (inflation choice) of the policymakers, whereas equation (1) is the description of the economy. Graphically they are described in the following figure. 4

Inflation rate (π) 10 Policy rule E E B First best 4 Phillips curve after an adverse shock 0 1 3 4 5 7 9 10 Figure 4 Optimal policy rule Unemployment rate (u) Given the Phillips curve the optimal policy is described in point E (which is the only optimal combination of u and π feasible). Now consider an adverse shock in the Phillips cure (e.g., an increase in c) due to an oil shock. The optimal policy for the policymaker is to increase the inflation rate to about 7% and then the economy will face an unemployment rate of about 3.5% (point E ). If the policymaker would not chance his policy (keeping an inflation rate at about 5%), after the adverse shock the unemployment rate will raise to about 5.5% (point B), which clearly is not efficient as it is associated to an higher loss than point E. Exercise. Using Figure 4, discuss the effect of a change in the policymaker s preferences (a reduction of a) and a change in the Phillips curve coefficient α. Exercise 3. Consider the following model expressed in logs. Note that W / P (the real wage) in logs becomes log( W) log( P), we indicates log with lowercase letters; i.e., w p We assume that nominal wage (w) is set by a trade union. The trade union preferences increase in the real wage (w p) and fall in the unemployment rate (u): u = The unemployment rate is defined as follow: (9) U α ( w p) (10) u = 1 l i.e., labor force is normalized to unity. The first best for the union is clearly an infinite nominal wage and zero unemployment. However this solution is not feasible, as the union faces the following constraint: (11) l = η ( w p) Equation (11) is simply the firm s labor demand. 5

The idea is that the union (as a monopolist) sets the wage (price of labor) taking account of the labor demand (11) as constraint. The union problem is then (1) maxu α ( w p) w ( 1 l) = l = η w p subject to ( ) Assuming that the price level, p, is given, solve the above problem and comment the solution obtained. How the employment change if α increases, why?

Appendix A Iso-losses The representation of the policymaker s loss: (13) ( ) ( u u) L= a π π + in the π and u space is similar to the consumer utility representation by the indifference curves. The loss is defined in deviations of π and u from some (desired) targets π and u. Indeed, now we have to consider iso-losses instead of the indifference curves, as we are dealing with loss functions instead of utilities. An iso-loss curve represents all the π and u combinations that assure a certain given constant loss. Assuming for the sake of simplicity that the iso-losses are centered on zero (i.e., π = u = 0 ), if a=1, the iso-losses are concentric circles (panel (a)) centered on the fist best (π and u ), otherwise they are elliptic curves. In particular, for a<1 they have the form described by panel (b); for a>1 they have the form described by panel (c). π π π u u u Panel (a) Panel (b) Panel (c) Figure 5 Iso-losses We can refer to the case described in panel (b) as that of the populist central banker, i.e., a central bank that dislikes unemployment more than inflation. Hence large deviation of inflation (vertical axis) are compensate with smaller deviation in unemployment (horizontal axis). By contrast, panel (c) describes the case of a conservative central banker. In this case the central bank is more interested in stabilize inflation (as inflation is usually its institutional primary target), hence its isoloss curves are more concentrated along the horizontal axis (low inflation) implying an higher cost of inflation in terms of unemployment deviations). Appendix B Tinbergen-Theil approach * The policy problem presented here is an example of the more general Tinbergen and Theil approach to the economic policy. In the 50s, Tinbergen addressed in formal terms the issue of the controllability of a fixed set of independent targets by a policymaker facing an economy represented by a system of linear equations and endowed with a given set of instruments. He stated a well-known condition for policy existence in terms of number of instruments and targets. A similar approach was developed by Bent Hansen in roughly the same years. Tinbergen s approach to the policy problem has been further generalized by Theil. In particular, by prescribing that the policymaker should maximize a preference function, Theil solves the difficulties facing the * This appendix is optional. 7

policymaker when endowed with a lower number of instruments than the number of targets. In so doing, Theil arrived at a solution of the policy problem formally very similar to that predicated by Ragnar Frisch, who had first conceived policy problems in terms of maximizing a social preference function, to be derived by interviewing policymakers. Formally, the Tinbergen and Theil approach to the economic policy is based on the minimization of a loss function defined for deviations of the relevant variables from their target values is the following quadratic-matrix form: (14) U = ( y y) Q( y y) where y( i ) ; q y is the vector policymaker s q target variables, whose generic element is denoted by q y is the vector policymaker s (desired) targets; and Q is an appropriate symmetric positive semi-definite matrix. We refer to y and Q as the parameters of policymaker s preferences. 3 The economy is described by the following linear equation system (structural form of the economy): (15) Ay = Bu + K m where u is the vector of the m policymaker s instruments, whose generic element is denoted by u( j ) ; A and B are appropriate parameter matrices (i.e., the target and instrument coefficient matrices); and K is an appropriate vector of constants. 4 We assume that A and B are full-rank matrices, i.e., all targets and instrument variables are linearly independent (independence assumption). Intuitively, there are q distinct targets and m distinct instruments set by the policymaker. The linear reduced-form model can be written in matrix form as: (1) 1 1 y A = Bu+ A K = Cu+ C provided A is non-singular, as it is from our rank assumptions. Matrix C is a matrix of multipliers C i, j indicates the effect on target i of and is sometimes called the Jacobian matrix. Element ( ) changes in instrument j; that is, y( i) / u( j). The policy problem is to minimize equation (14) with respect to the vector of instruments, subject to equation (1). The corresponding first order condition is: (17) CQCu = CQ ( y C) which is m a equation system in m unknown, i.e., u. Existence of a solution of (17) is ensured if rank [ : K ] rank [ ] Φ = Φ, i.e., left invertibility of, where C QC K = CQ y C. Uniqueness requires the non-singularity of. If a solution or more exists, 5 the policy design problem implies the following optimal policy: Φ= and Φ ( ) Φ 3 Quadratic functions are used not only for their mathematical tractability, but also for their useful economic properties. In fact, deviations from the target are associated to increasing costs and, therefore, the marginal rate of substitution between any couple of target variables is never constant but depends on the values of the two variables in the point where it is computed. In addition, quadratic forms can be obtained as second-order Taylor approximations of more complex functions. 4 Each component of which is a linear combination of constants, exogenous variables and/or white noise shocks. 5 Note that we are solving a linear equation system, it thus will admit a unique solution, infinite solutions or no solutions at all.

(1) u * ( CQC) 1 = CQ ( y A 1 K) and outcomes (obtained by using (1) into (1)): (19) * ( ) 1 ( 1 ) If q y = C CQC CQ y A K + C = m, the above policy (1) reduces to * 1 (0) u = B ( Ay K ) which plugged in (1) gives (1) * y = y In this case ( q = m), the policymaker is able to obtain his first best. We refer tot this case as that where the policymaker can control the system (1); i.e., he is always able to achieve any possible vector of desired targets. In general, the policymaker can achieve any vector of independent targets by an appropriate vector of instruments if and only if the number of independent instruments is equal to, or greater than, the number of targets, q m (Tinbergen s Golden rule). 9