optimal policy response with control parameter and intercept covariance

Comput Econ (2008) 31:1–20DOI 10.1007/s10614-007-9103-5

Optimal Policy Response with Control Parameterand Intercept Covariance

Fidel Gonzalez

Received: 19 December 2006 / Accepted: 15 July 2007 / Published online: 23 August 2007© Springer Science+Business Media, LLC 2007

Abstract Parameter uncertainty and the interaction between the uncertainparameters are important aspects of economic policy. In this work, I developan analytical one-state variable, one-control variable model with two uncertainparameters (the control parameter and the intercept) and a nonzero covariance.I characterize the effect of changes in each of the covariance components onthe optimal expected control variable. I found that the nature of the optimalpolicy maker’s response depends on the specific changing component of thecovariance, the sign of the correlation coefficient and the sign of the optimalexpected control variable when the covariance is zero. I obtain the conditionsunder which the effect of the covariance is considerable. This work comple-ments previous studies by providing a complete set of cases and conditions foran aggressive or cautionary optimal policy maker’s response to changes in eachcovariance component. Finally, the importance of the analytical results is shownfor the regulation of a stock pollutant leading to global warming.

Keywords Stochastic control · Multiplicative disturbances · Parametercovariance · Optimal policy response

JEL Classification C61 · E61 · Q5

F. Gonzalez (B)Department of Economics and International Business, Sam Houston State University,Huntsville, TX 77341, USAe-mail: [email protected]

2 F. Gonzalez

1 Introduction

Parameter uncertainty is an important aspect of economic policy and the effectof uncertainty on the optimal expected control has a long tradition of interest ineconomics. Previous work has shown a cautionary response of the policy makerto higher control parameter uncertainty. However, this result can change in thepresence of nonzero covariances between the parameters and variables in themodel. This work contributes to this field by characterizing the optimal policymaker’s response to changes in the value of each component of the covariancewhen the control parameter and the intercept are correlated.

Most studies dealing with the covariance between parameters and variableshave taken place in the field of macroeconomics. In general, these studiesconsider the effect of the covariance as potentially important. The seminalwork of Brainard (1967) shows that in a static model, the nature of the policymaker’s optimal response can change in the presence of large values of thecovariance. Recently, Amman and Kendrick (1999) show numerically that thepotential damages from ignoring parameter covariances are significant. Martinand Salmon (1999) explores the importance of parameter uncertainty includingcovariances of the parameters for the United Kingdom. Wieland (1998) findsthat the covariance between the intercept and the slope of the Phillips curvehas an important effect in a static model. Gonzalez (2006) analyzes the effectof parameter uncertainty for a stock pollutant on pollution taxes and stocks inthe presence of a positive covariance.

In this paper, I develop a finite horizon dynamic analytical one-state, one-control linear quadratic problem (LQP), with two uncertain parameters (thecontrol parameter and the intercept) and a nonzero covariance. Following thegeneral methodology of Mercado and Kendrick (2000), Gonzalez and Ro-driguez (2004) and Rodriguez (2004), I use the feedback matrices to characterizethe policy maker’s optimal response to changes in each of three covariance com-ponents: standard deviation of the control parameter, correlation coefficientand standard deviation of the intercept. The advantage of using an analyticalmodel comes at the cost of reducing the number of control and state variablesto one.

I find that the introduction of a nonzero covariance between the controlparameter and the intercept can have two effects: (i) change the sign of theoptimal expected control variable and (ii) alter the nature of the optimal policymaker’s response (cautionary or aggressive).

For the first effect, I show that in general Brainard (1967) results still hold:if the optimal control variable is positive (negative) in the presence of a zerocovariance, then the introduction of large enough positive (negative) covari-ance will change the sign of the optimal expected control variable. However, Icomplement previous work by showing that this result does not apply equallyfor all the components of the covariance. In some cases, there are no valuesof the standard deviation of the control parameter or the correlation coeffi-cient that would produce the aforementioned change in sign. For the standard

Optimal Policy Response with Control Parameter 3

deviation of the intercept there is always a value that changes the sign of theoptimal expected control variable.

The optimal policy maker’s response is defined as cautionary (aggressive)when an increase in the level of uncertainty produces a decrease (increase) inthe magnitude of the expected optimal control variable.1 I find that the natureof the optimal response (aggressive or cautionary) depends on the sign of thecorrelation coefficient and the sign of the expected optimal control variablewhen the covariance is zero. I characterize the optimal response to changes ineach covariance component, and I find two major results.

First, the optimal policy maker’s response to changes in the uncertainty ofthe control parameter is nonlinear. In particular, I show that the slope of theoptimal response is a quadratic function of the standard deviation of the controlparameter, with one positive and one negative root. Moreover, when the cor-relation coefficient is positive (negative) the positive root represents a relativeminimum (maximum) in the fourth (first) quadrant or an inflection point inthe first (fourth) quadrant. The concavity or convexity of the optimal responsecannot be determined analytically, and it will depend on the specific parame-ter values. However, I am able to determine the nature of the optimal policymaker’s response to changes in the control parameter uncertainty. If the optimalexpected control variable in the presence of a zero covariance and the correla-tion coefficient have different signs, the optimal response is first aggressive andlater cautionary. When they have the same sign the optimal responses can beeither: (i) always cautionary or (ii) first cautionary, later aggressive and finallycautionary. I provide the conditions for each optimal response in terms of thestandard deviation of the control parameter.

Second, I find that the optimal policy maker’s response to changes in theintercept uncertainty is similar to the response to changes in the correlationcoefficient. In both cases, the optimal response takes the shape of a straight linewith a negative slope when the correlation coefficient is positive and a positiveslope when the correlation coefficient is negative. Thus, when the optimal con-trol variable in the presence of a zero covariance and the correlation coefficienthave different signs, the optimal response is always aggressive to changes ineither the standard deviation of the intercept or the correlation coefficient.When they have the same sign the optimal response is: (i) for changes in thestandard deviation of the intercept, first cautionary and later aggressive and (ii)for changes in the correlation coefficient, either always cautionary or cautionaryand later aggressive.

I apply the analytical results to the regulation of a stock pollutant leadingto global warming using a modified version of Hoel and Karp (2001) model. Ifind that in the relevant case of a negative covariance, higher parameter uncer-tainty produces: (i) higher expected emission taxes than when the covariance iszero and (ii) an aggressive and later cautionary response of expected optimal

1 In other studies the optimal response is defined cautionary (aggressive) when an increase thelevel of uncertainty produces a decrease (increase) in the magnitude of the optimal expected controlvariable with respect to its magnitude in the certainty equivalence case.

4 F. Gonzalez

emissions taxes. This shows the potential effect of the covariance on expectedoptimal emission taxes.

The paper is divided into eight sections. The next section presents the LQPand its analytical solution. The Sect. 3 analyzes the effect of the covariance onthe sign of the expected control variable. The Sect. 4 characterizes the optimalresponse of the policy maker to changes in the standard deviation of the controlparameter. The Sects. 5 and 6 analyze the optimal policy maker’s response tochanges in the correlation coefficient and the standard deviation of the inter-cept, respectively. The Sect. 7 shows the numerical results. In the last section,I present the conclusions.

2 Problem Statement

The framework for this analysis is a finite horizon LQP with an intercept andadditive noise. The control parameter and the intercept are uncertain and corre-lated. The weights on the state and control variable can be time varying or fixed.For simplicity, I assumed the weights of the state variable to be time varying andthe weights of the control variable to be time invariant. The state parameteris assumed to be constant and certain. Formally, the problem is expressed asfinding the controls (uk)

N−1k=0 in order to minimize a quadratic criterion function

J of the following form:

J = E

⎧⎨

⎩

12

x′NWNxN + w

′NxN +

N−1∑

k=0

(12

x′kWkxk + w

′kxk + 1

2u

′k�uk + λ

′xk

)⎫⎬

⎭

(1)

subject to

xk+1 = Axk + Buk + c + εk (2)

where E = expectation operator; xk = scalar state variable; uk = scalar controlvariable; � = scalar positive weight on the quadratic term of the control vari-able; λ = scalar weight on the linear term of the control variable; Wk = timevariant scalar weight on the quadratic term of the state variable; wk = time vari-ant scalar weight on the linear term of the state variable; A = a = scalar controlparameter, B = β = unknown scalar control parameter; c = unknown inter-cept; ε = additive noise with mean zero and variance σ . The control parameterand intercept are distributed with a mean and variance of (β, σ 2

β ) and (c, σ 2c ),

respectively. The covariance between them is represented as follows:

σβ,c = υσβσc (3)


where σβ,c is the covariance between the control parameter and the intercept;−1 < υ < 1 is the correlation coefficient; σβ is the standard deviation of β andσc is the standard deviation of c.

The solution to this optimisation problem is the feedback rule (see Kendrick1981, p. 17)

uk = Gkxk + gk (4)

By adapting Eq. (6.31) in Kendrick (1981) to this problem, the feedback gaincoefficients are the following:

Gk = − akk+1β

�+ kk+1(σ2β + β

2)

(5)

gk = −kk+1(σβ,c + βc)+ βρk+1 + λ

�+ kk+1(σ2β + β

2)

(6)

where k and ρ are the Riccati matrices (scalars in this case). The expressionsfor the Riccati matrices follow:

kk = Wk + a2kk+1

⎛

⎝1 − β2kk+1

�+ kk+1(σ2β + β

2)

⎞

⎠ with kN = WN (7)

ρk = aρk+1 + wk + akk+1β

⎛

⎝c

β− βρk+1 + λ+ kk+1(σβ,c + βc)

�+ kk+1(σ2β + β

2)

⎞

⎠

with ρN = wN (8)

Since WN and � are assumed to be positive kk is also positive.

3 Covariance and Optimal Control

The covariance between the control parameter and the intercept affects: (i) themagnitude and sign of uk and (ii) the optimal response of uk to changes in theuncertain parameters.

The analysis of the second effect for each element of the covariance takesplace in Sects. 4–6. The first effect can be observed by analyzing the expressionof the optimal expected control variable:

uk = −kk+1(aβxk + βc + σβ,c)+ βρk+1 + λ

�+ kk+1(σ2β + β

2)

(9)

Equation 9 shows that the introduction of a nonzero correlation between thecontrol parameter and the intercept can change the sign of the optimal expected

6 F. Gonzalez

(a) Positive Covariance (b) Negative Covariance

0ku

ku 0

0

000

Fig. 1 Trajectory of the optimal control response to σβ

control variable. To observe this effect, let u0k be the optimal expected control

variable when σβ,c = 0 (u0k = uk|σβ,c=0). If u0

k > 0, the numerator in the righthand side of Eq. 9 is negative when σβ,c = 0. Hence, the introduction of a largeenough positive covariance will change the sign of uk to negative. Similarly, ifu0

k < 0, the introduction of a large enough negative covariance will make thesign of uk positive. This potential effect of the covariance was first noted byBrainard (1967) for a one period problem.

Formally, define σβ,c as the covariance value that makes uk = 0. From Eq. 9.I obtain the following:

σβ,c = −(

βaxk + βc + βρk+1 + λ

kk+1

)

(10)

Hence, the introduction of a nonzero covariance will change sign(uk) tonegative if σβ,c > σβ,c and u0

k > 0 or to positive if σβ,c < σβ,c and u0k < 0.

4 Control Parameter Uncertainty

In this section, I analyze the optimal response of uk to changes in σβ whenthe control parameter and the intercept are correlated. I define a cautionary(aggressive) response as the reduction (increase) in the magnitude of the opti-mal expected control variable when the level of uncertainty increases. I showthat the optimal control variable response to changes in σβ when σβ,c �= 0 fol-lows the trajectories shown in Fig. 1 (with exception of the dotted lines thatrepresent the trajectory when σβ,c = 0). This provides enough information todetermine the nature of the response (aggressive or cautionary). I prove thatthe optimal response is concave up or concave down only around the criticalpoints, in case they exist, but not along the complete trajectory.

The dotted lines in both panels of Fig. 1 show the optimal control responseto changes in σβ when υ = 0. They show the traditional Brainard (1967) result:


the response is always cautionary and it does not cross the x-axis. In particular,when υ = 0, the response has a negative slope with respect to |uk| and it isinitially concave down and later concave up (see Appendix A for the completederivation and the value of the inflection point). Figure 1 shows that trajectorieswhere υ �= 0 can generate substantially different trajectories than where υ = 0.In the former trajectories can: (i) cross the x-axis, changing the sign of the opti-mal control variable; (ii) have positive slopes with respect to |uk|, producing anaggressive response and (iii) change the slope sign, switching the nature of theresponse. In all cases the response approaches zero as σβ → ∞.

In the next five steps, I analytically prove why the optimal response to σβwhen υ �= 0 follows the trajectories shown in Fig. 1.

Step 1 Show that when υ > 0 (υ < 0) the slope of the optimal response isalways negative (positive) in the first (fourth) quadrant.Step 2 Show that the slope of the optimal response is a quadratic functionof σβ with only one positive root.Step 3 Show that when υ > 0 (υ < 0) if a section or the complete optimalresponse is below (above) the x-axis the positive root represent a rela-tive minimum (maximum) in the fourth (first) quadrant. In addition, whenυ > 0 (υ < 0) if the optimal response is always above (below) the x-axis thepositive root can represent an inflection point in the first (fourth) quadrant.Step 4 Show that the optimal response crosses the x-axis once at most.Step 5 Show that uk → 0 as σβ → ∞.

Note that by definition u0k represent the y-intercept of the optimal response.

In other words, when u0k > 0 (u0

k < 0) the initial point of the optimal responseis above (below) the x-axis.

Step 1

Substituting Eq. 3 into Eq. 9 and taking the derivative of uk with respect to σβ ,I obtain the slope of the optimal response:

∂uk

∂σβ= −kk+1σcυ

�+ kk+1(σ2β + β

2)

−2σβkk+1(−kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ)

(�+ kk+1(σ2β + β

2))2

(11)

Using Eq. 9 and rearranging terms

∂uk

∂σβ= −kk+1

�+ kk+1(σ2β + β

2)

[vσc + 2σβuk

](12)

The fraction on the right hand side of Eq. 12 is negative. Hence, the slope will bealways negative as long as υ > 0 and uk > 0 (in the first quadrant when υ > 0).

8 F. Gonzalez

On the other hand, the slope will be always positive as long as υ < 0 and uk < 0(in the fourth quadrant when υ < 0). This implies that when sign(υ)=sign(uk),the optimal response is always cautionary. The more difficult cases arise whensign(υ) �= sign(uk).

Step 2

Substituting Eqs. 3 and 9 into Eq. 12 and rearranging, I obtain the expressionthat makes the slope of the optimal response equal to zero:

∂uk

∂σβ= υσc + 2σβ · −kk+1βaxk − kk+1βc − kk+1σcσβυ − βρ − λ

�+ kk+1(σ2β + β

2)

= 0

∂uk

∂σβ= −kk+1σcυσ

2β − 2(β(ρ + kk+1axk + kk+1c)+ λ)σβ

+υσc(�+ kk+1β2) = 0 (13)

Next, set ψ ≡ −kk+1σcυφ ≡ −(β(ρ + kk+1axk + kk+1c)+ λ), and θ ≡ υσc(�+kk+1β

2). Since the term −ψθ is always positive, the magnitude of φ is less than

the magnitude of√φ2 − ψθ (the square root of the discriminant of the quadratic

equation). Hence, Eq. 13 will have one positive and one negative root.2

Step 3

Next, I consider the second derivative uk with respect to σβ :

∂2uk

∂2σβ= −2kk+1

�+ kk+1(σ2β + β

2)

[

σβ∂uk

∂σβ+ uk

]

(14)

Define σβ as the positive root of Eq. 13 and uk as the respective value of theoptimal expected control variable (uk = uk|σβ=σβ ). Only σβ is a suitable can-didate for a minimum or maximum of the optimal response. Evaluating thesecond derivative at σβ , I obtain the following:

∂2uk

∂2σβ

∣∣∣∣σβ=σβ

= −2kk+1uk

�+ kk+1(σ2β + β

2)

(15)

When uk < 0 the second derivative at the critical point (given by Eq. 15) ispositive and if υ > 0 then σβ is a relative minimum (the optimal response isconcave up around σβ). However, when uk < 0 and υ < 0 the critical value

2 Note that a change in the sign of υ only changes the sign of the roots, not the magnitude. This isbecause υ enters the discriminant in a square form and υ is not included in φ.


cannot represent a relative minimum because the slope of the optimal response(the first derivative) does not change sign when υ and uk are negative (seeStep 1), it can represent an inflection point.

On the other hand, when uk > 0 the second derivative at the critical pointis negative and if υ < 0 then σβ is a relative maximum (the optimal responseis concave down around σβ). However, when uk > 0 and υ > 0 the criticalvalue cannot represent a relative maximum because the slope of the optimalresponse (the first derivative) does not change sign when υ and uk are positive(see Step 1), it can represent an inflection point.

Summarizing, when υ > 0 (υ < 0) the positive root is a relative minimum(maximum) in the fourth (first) quadrant if a section or the whole optimalresponse takes place in such quadrant. Moreover, when υ > 0 (υ < 0) thepositive root will be an inflexion point in the first (fourth) quadrant if theoptimal response is always above (below) the x-axis.

Step 4

Define σβ as the value of the standard deviation of β that makes uk = 0. Substi-tuting Eq. 3 into the left hand side of Eq. 10 and solving for σβ , I obtain thefollowing:

σβ = −β(kk+1axk + kk+1c + ρ)+ λ

kk+1σcυ(16)

Since σβ > 0 it is possible that the optimal response does not cross the x-axis.Thus, a positive σβ will exist if either: (i) β(kk+1axk + kk+1c + ρ)+ λ > 0 whenυ < 0 or (ii) β(kk+1axk + kk+1c + ρ)+ λ < 0 when υ > 0. Note that if σβ > 0exists, its value is unique and the optimal response will cross the x-axis onlyonce. A positive value of σβ represents the point after which the sign of theoptimal expected control variable changes.

Step 5

The optimal response of the policy maker will tend to zero as the standarddeviation approaches infinity.

limσβ→∞ uk = lim

σβ→∞β(−kk+1axk − kk+1c − ρ)− λ

�+ kk+1(σ2β + β

2)

+ limσβ→∞

−kk+1υσcσβ

�+ kk+1(σ2β + β

2)

(17)

The first limit in the right hand side of Eq. 17 is equal to zero. ApplyingL’Hopital’s rule to the second limit, I obtain the following:

10 F. Gonzalez

limσβ→∞ uk = 0 + lim

σβ→∞−kk+1υσc

2kk+1σβ= 0 (18)

Equation 18 shows that caution will prevail for extreme high levels of controlparameter uncertainty.

The results of Steps (1–5) allow me to present the nature of the optimalpolicy maker’s response to higher control parameter uncertainty as follows:

σβ ∈ (0, max{0, σβ}) cautionary response (19)

σβ ∈ (max{0, σβ}, σβ) aggressive response (20)

σβ > σβ cautionary response (21)

The max operator in the previous conditions is necessary since by definition σβis nonnegative. The relevance of each case depends on the specific parametervalues. The policy maker can be interested in levels of uncertainty located insideof only one of the ranges shown above.

The optimal policy maker’s response can also be presented in terms of thesigns of υ and u0

k. When the sign(υ) �= sign(u0k) the optimal response is first

aggressive and later cautionary. On the other hand, when sign(υ) = sign(u0k)

the optimal response can be either: (i) always cautionary or (ii) first cautionary,later aggressive and finally cautionary. These results contrast with the optimalresponse of the optimal control variable to σβ when the covariance is zero. Inthat case, the standard results apply: the optimal policy maker’s response iscautionary for all degrees of σβ (see Appendix A for the proof).

5 Correlation Coefficient

In this section, I analyze the optimal response of uk to changes only in thecorrelation coefficient (υ). I show that the optimal expected control variableresponse to changes in the correlation coefficient can take any of four shapesdisplayed in Fig. 2. In all four cases, the optimal policy maker’s response tochanges in υ is linear with a negative slope that shifts along the y-axis. On theother hand, the responses differ on the sign of the y-intercept and whetherthey cross the x-axis. Thus, the nature of the optimal response (cautious oraggressive) depends on the quadrant where the response is located.

In Fig. 2 the y-axis represents the optimal expected value of the control var-iable (uk) and the x-axis the value of υ ∈ (−1, 1). Thus, the y-intercept of theresponse is the optimal expected value of the control variable when the covari-ance between the control parameter and the intercept is zero (u0

k). Moreover,since υ ∈ (−1, 1) the degree of uncertainty is given by |υ|. Hence, movementsof υ on the x-axis away from the origin represent an increase in |υ| and ahigher level of uncertainty, everything else equal. Conversely, movements of υon the x-axis toward the origin imply a decrease in |υ| and a lower degree ofuncertainty, everything else the equal.


ku

1-1 0

Fig. 2 Optimal control response to the correlation coefficient

Since the correlation coefficient appears only in gk, the response of the opti-mal expected control variable is:

∂uk

∂υ= ∂gk

∂υ= −kk+1σβσc

�+ kk+1(σ2β + β

2)< 0 (22)

Equation 22 shows that the optimal response takes the shape of a straight linewith a negative slope for all real values of uk and υ. Graphically, the optimalresponse of uk and gk are parallel to each other while Gkxk is a horizontal linethat shifts uk along the y-axis.

Since uk and υ can take positive or negative values, it is necessary to considerthe magnitude of uk to determine the nature of the optimal response:

|uk| = | − kk+1(aβxk + βc + υσβσc)− βρk+1 − λ|�+ kk+1(σ

2β + β

2)

(23)

The derivative of |uk| with respect to the correlation coefficient is the follow-ing (see Appendix B for the derivation):

∂|uk|∂υ

= −ukkk+1σβσc

| − kk+1(aβxk + βc + σ 2β,c)− βρk+1 − λ| (24)

From Eq. 24 I can conclude that when υuk < 0 (in the second and fourthquadrant) the optimal policy maker response is aggressive, i.e. an increase inthe magnitude of υ produces an increase in the magnitude of uk. On the otherhand, when υuk > 0 (in the first and third quadrant) the optimal policy maker’sresponse is cautious, i.e. an increase in the magnitude of υ produces a decreasein the magnitude of uk. Note that the optimal response is always aggressive onlyin the unlikely event that u0

k = 0. When sign(u0k) = sign(υ), the optimal response

12 F. Gonzalez

c

(b) Negative Covarianceku

0

c

(a) Positive Covarianceku

0

Fig. 3 Optimal control response of to changes in σc

is either: (i) first cautionary and later aggressive or (ii) always cautionary. Inaddition, when sign(u0

k) �= sign(υ) the optimal response is aggressive.

6 Intercept Uncertainty

In this section, I characterize the optimal response of uk to changes in theuncertainty of the intercept represented by σc. I show that the response takesthe shape shown in Fig. 3.

The optimal response of uk to σc is given by the following expression:

∂uk

∂σc= ∂gk

∂σc= −kk+1σβυ

�+ kk+1(σ2β + β

2)

(25)

Equation 25 shows that the optimal response takes the shape of a straightline with a positive slope when υ < 0 and a negative slope when υ > 0. Takingthe derivative of Eq. 23 with respect to σc, I obtain the nature of the optimalresponse:

∂|uk|∂σc

= −ukkk+1σβυ

| − kk+1(aβxk + βc + σ 2β,c)− βρk+1 − λ| (26)

Equation 26 shows that the nature of the optimal response depends on thesign of υuk. Hence, when υuk > 0, an increase in the uncertainty of the interceptproduces a cautionary response (Eq. 26 is negative). Similarly, when υuk < 0the optimal policy maker’s response is aggressive (Eq. 26 is positive).

In addition, the lim uk = lim gk = +∞ when σc → +∞ and υ > 0. Sim-ilarly, the lim uk = lim gk = −∞ when σc → +∞ and υ < 0. Therefore, ifsign(u0

k) = sign(υ) the optimal response will cross the x-axis (uk will changesign), and it will be first cautionary and later aggressive. This last result is


asymptotic. It is possible that for reasonable values of σc, the policy maker isalways cautious or aggressive. Finally, the optimal response is always aggressivewhen sign(u0

k) �= sign(υ).

7 Numerical Results

An application will help illustrate the previous findings and show the impor-tance of the covariance between the control parameter and the intercept. I usea finite horizon version of the stock pollutant problem of Hoel and Karp (2001)as modified by Gonzalez (2006). The state variable (xk) is the pollution stockof greenhouse gases in billions of tons, the control variable (uk) is the emissiontax per ton and the intercept (ck) is the baseline emissions level of carbon inbillions of tons (emissions in the absence of regulation). All parameter valuesare obtained from Hoel and Karp (2001) and the initial year is 1990. Since theestimated stock of carbon in 1990 is 800 billions of tons then x0 = 800. Thevalues of W = 0.000688 and w = −0.55 represent a decrease of 1% in annualGross World Production (GWP) from its 1990 level ($22 trillion dollars) dueto doubling the world atmospheric stock of carbon from its 1990 level. Theintercept c = 6 is the estimated of annual emissions in the initial year equal to 6billions of tons and it is considered as the business-as-usual emissions level. Thevalue of β = −0.02045 is the negative of the expected marginal abatement costsimplied by a 1% loss of GWP due to a 50% reduction in emissions.3 The valueof a = 0.995 is the fraction of the carbon stock that persists until the next periodimplied by a decay rate of 0.005 (a = e−0.005). Moreover, � = 0.02045 repre-sents the expected marginal abatement cost of emissions. In order to observethe dynamics of the model I choose N = 10 where k represents 1 year. Notethat β and c represent the slope and the y-intercept of the marginal abatementcost curve, respectively. In order to illustrate all the analytical results obtainedin this study I show the numerical solutions for positive and negative values ofthe covariance. However, it is important to notice that in the models of Hoeland Karp (2001) and Gonzalez (2006) the covariance between the baselineemissions and the response of emissions taxes is negative.

First, I show the numerical results of the findings in Sect. 4 and comparethem to the case when the correlation coefficient is zero. In particular, I setσc = 1.7 and σβ ∈ (0, 20). Figure 4 shows the response of the optimal expectedemissions tax in the first period (u1) when υ = 0.8 and υ = 0 in Panel (a), andwhen υ = −0.8 and υ = 0 in Panel (b).

Substituting the relevant parameters I obtain the roots of σβ from Eq. 13and σβ from Eq. 16. All optimal responses start at u0

1 = 0.07622. When υ = 0.8the two roots are −1.59 and 1.95, and σβ = 0.174. The sign of the optimalexpected emissions tax changes (becomes a subsidy) for values of σβ > 0.174.The minimum value of the optimal expected emissions tax takes place in the

3 In principle βuk − ck can be treated as the level of emission. However, in the presence ofuncertainty it may not be possible to combine them into one control variable.

14 F. Gonzalez

0 5 10 15 20−0.4

−0.3

−0.2

−0.1

0

0.1(a) Positive Covariance

σβ

first

per

iod

emis

sion

s ta

x

u1 (υ=0.8)

u1 (υ=0)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5(b) Negative Covariance

σβ

first

per

iod

emis

sion

s ta

x

u1 (υ=−0.8)

u1 (υ=0)

Fig. 4 First period optimal emissions tax response to changes in σβ

fourth quadrant at $0.349 per ton of emissions when σβ = 1.95. The policymaker starts with a positive emissions tax. As the uncertainty increases the taxdecreases until it becomes a subsidy. The subsidy reaches a high point (min-imum tax), and then it decreases approaching zero. The policy maker has acautionary response for σβ ∈ (0, 0.174) and σβ > 1.95. An aggressive responsetakes place for σβ ∈ (0.174, 1.95).

On the other hand, when υ = −0.8 the two roots of Eq. (13) are 1.59and −1.95. Thus, the maximum optimal expected emissions tax takes place inthe first quadrant at $0.425 per ton of emissions and σβ = 1.59. Moreover,since σβ = −0.174 the optimal response does not cross the x-axis (the opti-mal expected emissions tax does not change signs). The optimal response isaggressive for σβ ∈ (0, 1.59) and cautionary for σβ > 1.59.

The results for υ = 0.8 and υ = −0.8 contrast with the results when υ = 0.In the latter, as shown in both panels of Fig. 4, increases in σβ produce thetraditional cautionary optimal policy maker’s response for all values of σβ .Moreover, the optimal expected control variable does not change sign andapproaches zero as σβ → ∞.

As mentioned at the beginning of this section in the model of Hoel and Karp(2001) model as modified by Gonzalez (2006) the sign of the covariance is neg-ative. Hence, the relevant comparison is between the two trajectories shown inPanel (b) of Fig. 4. In this case the existence of a negative covariance produceshigher expected optimal emissions taxes than in the case of zero covariance forall levels of σβ . In addition, for σβ ∈ (0, 1.59) the policy maker will increaseexpected emissions taxes when σβ increases if the covariance is negative butdecrease it if the covariance is zero. These results show the potential importanceof the covariance in the optimal expected emission tax. On the other hand, forσβ > 1.59 the optimal policy maker’s response is to decrease expected emissionstaxes when σβ increases in the case of zero and negative covariance.

From the estimation point of view a common situation given our specificparameter values takes place when σβ ∈ (0, 0.04). In the case of υ = −0.8 andσβ ∈ (0, 0.04), the optimal policy response is cautionary for this range of σβ andit does not cross the x-axis. This implies that the optimal expected emissions


−1 −0.5 0 0.5 1−5

−2.5

0

2.5

5

υ

first

per

iod

emis

sion

s ta

x u1

g1

G1 x

1

Fig. 5 First period optimal emissions tax response to changes in υ

tax does not change sign, it is between $0.07622 and $0.0587 per ton whichrepresents a reduction of 30%. In the case of υ = 0 and σβ ∈ (0, 0.04) thevalue of the optimal expected emissions tax decreases 0.03% from $0.07622 to$0.07620 per ton. That is, for σβ ∈ (0, 0.04) the response of the policy maker ismore cautionary when υ = −0.8 than when υ = 0. In the case of σβ ∈ (0, 0.04)and υ = 0.8, the optimal policy response is aggressive and it does not crossthe x-axis. The optimal expected emissions tax increases 29% from $0.07622to $0.098 per ton. This contrasts with the cautionary response of the optimalexpected emissions tax for the same range of σβ when υ = 0.

To illustrate the findings in Sect. 5, I set σβ = 0.6 and υ ∈ (−1, 1). Figure 5shows the first period numerical results (G1x1, g1 and u1).

The optimal expected emissions tax changes sign at υ = 0.23. For υ > 0.23,the optimal emissions tax is a subsidy. Hence, the optimal response of the pol-icy maker is cautious for υ ∈ (0, 0.23) and aggressive for υ ∈ (0.23, 1) andυ ∈ (−1, 0). The value of the optimal expected emissions tax in the first periodis positive when the correlation coefficient is zero, in particular u0

k = 0.068.In addition, g1 and u1 are parallel to each other, whereas G1x1 is a horizontalline at $4.69 per ton. Since G1x1 is positive from Eq. 4 is clear that u1 > g1.Restricting the analysis to the negative values of the covariance, i.e. υ ∈ (−1, 0),a stronger negative correlation produces higher expected emission taxes.

Finally, I show the numerical results of Sect. 6. I set σβ = 0.6 and σc ∈ (0, 7).Figure 6 shows the first period optimal response of the optimal expected emis-sions tax when υ = 0.8 in Panel (a) and when υ = −0.8 in Panel (b).

The optimal expected emissions tax is always positive and increasing whenυ = −0.8 because u0

1 is positive (u01 = 0.068). In the relevant case where the

covariance is negative, the optimal response to higher uncertainty about thebaseline emissions level is always to increase expected emissions taxes.

On the other hand, when υ = 0.8, the sign of the optimal expected emissionstax changes to negative at σc = 0.5. In this case, since u0

1 = 0.068 the optimalresponse is cautious for σc ∈ (0, 0.5) and aggressive for σc > 0.5. In both cases,

16 F. Gonzalez

0 1 2 3 4 5 6 7−6

−4

−2

0

2

4

6

8(a) Positive Covariance

σc

first

per

iod

emis

sion

s ta

x

u1

g1

G1 x

1υ=0.8

0 1 2 3 4 5 6 7

−4

−2

0

2

4

6

8(b) Negative Covariance

σc

first

per

iod

pollu

tion

tax

u1

g1

G1 x

1υ=−0.8

Fig. 6 First period optimal emissions tax response to changes in υ

u1 and g1 are parallel to each other while G1x1 is a horizontal line at $4.69 perton of emissions.

8 Conclusions

In this work, I develop an analytical one-state, one-control variable modelwith two uncertain parameters (the control parameter and the intercept) and anonzero covariance.

I found that when characterizing the optimal response of the policy maker tochanges in the covariance it is important to distinguish between the changes ofeach component of the covariance. I complement previous research by showingthat when either the standard deviation of the control parameter or the correla-tion coefficient change, it is possible that the optimal expected variable does notchange signs. Moreover, in the case of the standard deviation of the intercept,there is always a value that would change the optimal expected control variable.

Regarding the nature of the optimal policy maker’s response to changes inthe covariance components, I complement Brainard (1967) result by consid-ering all the possible cases. In particular, I find that the optimal response tochanges in the uncertainty of the control parameter is nonlinear. The slope ofthe optimal response is a quadratic function of the standard deviation of thecontrol parameter with one positive and one negative root. I provide the condi-tions to determine if the positive root is a minimum, maximum or neither. Onthe other hand, the optimal response to changes in the intercept uncertaintyor the correlation coefficient takes the shape of a straight line with a negative(positive) slope when the correlation coefficient is negative (positive).

The numerical results for the regulation of a stock pollutant leading to globalwarming show the potential importance of the covariance between the slopeand the intercept of the marginal abatement cost curve. In the model consid-ered the most relevant case takes place when the covariance is negative. Inthis case, higher uncertainty about the slope of the marginal abatement costproduces: (i) higher expected emissions taxes than when the covariance is zero


and (ii) an aggressive and later cautionary optimal response of emissions taxes.The last result contrast with the traditional cautionary response to higher con-trol parameter uncertainty when the covariance is zero. In addition, higheruncertainty about the intercept of the marginal abatement cost curve or thecorrelation coefficient increase expected emissions taxes monotonically.

These results complement previous research, and some other economic fieldscould fit into the analytical framework. Future research can consider analyticalmodels of higher dimensions.

Acknowledgments I thank David Kendrick, Pedro Gomis, Arnulfo Rodriguez and other par-ticipants at 12th International Conference on Computing in Economics and Finance for usefulcomments on earlier versions of this paper. Any remaining errors and omissions are entirely myresponsibility.

Appendix A

In this appendix, I show that when the control parameter and the intercept areuncorrelated (υ = 0), the response to higher σβ is: (a) always cautionary, (b)initially concave down and later concave up and (c) the control variable doesnot change sign.

In this case, the magnitude of uk is the following:

|uk| = | − kk+1(aβxk + βc)− βρk+1 − λ|�+ kk+1(σ

2β + β

2)

(A.1)

The derivative of the |uk| with respect to σβ takes the following form:

∂|uk|∂σβ

= uk

|uk| · ∂uk

∂σβ(A.2)

The first derivative is given by:

∂|uk|∂σβ

= − 2σβkk+1u2k

| − kk+1(aβxk + βc)− βρk+1 − λ| < 0 (A.3)

Equation A.3 is a version of the standard Brainard (1967) result: an increasein the uncertainty of the control parameter produces a cautionary responsefrom the policy maker for any degree of uncertainty. The second derivative isgiven by the following:

∂∂|uk|∂σβ

∂σβ= 2kk+1u2

k

| − kk+1(aβxk + βc)− βρk+1 − λ|

·⎡

⎣4kk+1σ

2β

�+ kk+1(σ2β + β

2)

− 1

⎤

⎦ (A.4)

18 F. Gonzalez

The second derivative will take the sign of the square bracket on the right-hand side of Eq. A.4. Defining σ ∗

β as the value of σβ that makes the squarebracket on the right-hand side of Eq. A.4 equal to zero I obtain the following:

σ ∗β =

(λ+ kk+1β

2

3kk+1

)1/2

(A.5)

Thus, the second derivative is positive when σβ > σ ∗β and negative when σβ <

σ ∗β . Therefore, increases of σβ ∈ (0, σ ∗

β ) generate a decrease in |uk| at an increas-ing rate whereas an increase of σβ ∈ (σ ∗

β , ∞) generates a decrease in |uk| at adecreasing rate. Note that σ ∗

β is always positive and it represents an inflexionpoint because σβ = 0 is the critical value given by Eq. A.3. Finally, remov-ing the absolute value operator of Eq. A.1, it is clear that lim uk = 0 whenσ 2β → ∞. That is, the optimal expected control variable does not change signs

as a response to changes in σβ .

Appendix B

In this appendix, I show the complete derivation of Eqs. 24 and 26 in the textThe derivative of the |uk| with respect to υ takes the following form:

∂|uk|∂υ

= uk

|uk| · ∂uk

∂υ(B.1)

Substituting,

∂|uk|∂υ

= −kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ

�+ kk+1(σ2β + β2)

· �+ kk+1(σ2β + β2)

| − kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ|· −kk+1σβσc

�+ kk+1(σ2β + β2)

(B.2)

Simplifying and rearranging,

∂|uk|∂υ


�+ kk+1(σ2β + β2)

(B.3)

· −kk+1σβσc

| − kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ| (B.4)


The first fraction in the left hand side of Eq. B.4 is equal to uk (see Eq. 9).Hence, I arrive to the final expression:

∂|uk|∂υ

= −ukkk+1σβσc

| − kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ| (B.5)

Equation B.5 is referred to as Eq. 24 in the text.The derivation of Eq. 26 follows a similar procedure

∂|uk|∂σc


�+ kk+1(σ2β + β2)

· �+ kk+1(σ2β + β2)

| − kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ|· −kk+1σβυ

�+ kk+1(σ2β + β2)

(B.6)

∂|uk|∂σc

=−kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ

�+ kk+1(σ2β + β2)

· −kk+1σβυ

| − kk+1(aβxk + βc + σβσcυ)− βρk+1 − λ|(B.7)

∂|uk|∂σc

= −ukkk+1σβυ

| − βρk+1 − kk+1(aβxk + βc + σβ,c)− λ| (B.8)

Equation B.8 is referred to as Eq. 26 in the text.

Appendix C

In this appendix, I show the additive noise used for each period in the numericalresults of Sect. 7 (Table C.1).

Table C.1 Additive noise

Period 1 2 3 4 5 6 7 8 9 10

εk −1.21 1.86 −0.38 3.57 −0.16 −3.52 0.89 3.56 0.11 2.46

20 F. Gonzalez

Appendix D

In Fig. D.1 of this Appendix, I present the numerical value of the slope of thefirst period pollution tax response to changes in σβ .

−4 −3 −2 −1 0 1 2 3 4−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04 Positive Covariance

σβ

slop

e of

the

first

per

iod

emis

sion

s ta

x re

spon

se

σβ= 1.95σ

β= −1.59

−4 −3 −2 −1 0 1 2 3 4−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14 Negative Covariance

σβ

slop

e of

the

first

per

iod

emis

sion

s ta

x re

spon

se

σβ= −1.95 σ

β= 1.59

(a) (b)

Fig. D.1 Slope of the first period emission tax response to changes in σβ

The slope is a quadratic function of the standard deviation of the controlparameter with one positive and one negative root.

References

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Kendrick, D. (1981). Stochastic control for economic models, 2nd Edn. 2002 McGraw-Hill.Martin, B., & Salmon, C. (1999). Should uncertain monetary policy-makers do less? Bank of England

Working Paper, 99.Mercado, R., & Kendrick, D. (2000). Caution in macroeconomic theory: Uncertainty and the rela-

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optimal policy response with control parameter and intercept covariance

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