The main indications of the present model can be inferred from the above Equations (4)-(7). By solving the equations for C ^ and Y ^ as a function of G ^ , we obtain C ^ = m C G ^ and Y ^ = m Y G ^ , where

m C ≡ α + α ϕ − ϕ − ψ 1 + ϕ (8)

m Y ≡ α + α ϕ − ψ + 1 1 + ϕ (9)

which are the same as the results of Tervala [13] when ψ = 0 .

These two equations imply that the impact of government spending on private consumption and output is determined by three parameters: α , ϕ , and ψ . First, as is clear from (7), m Y is larger than m C and they differ by one: m Y − m C = 1 . Moreover, the marginal changes of m Y and m C are equivalent, such that

μ α ≡ ∂ m C ∂ α = ∂ m Y ∂ α = 1

μ ψ ≡ ∂ m C ∂ ψ = ∂ m Y ∂ ψ = − 1 1 + ϕ

μ ϕ ≡ ∂ m C ∂ ϕ = ∂ m Y ∂ ϕ = ψ − 1 ( 1 + ϕ ) 2

As detailed in subsequent subsection, since we consider the case where ϕ ≥ 0 and ψ ∈ [ − 2,1 ] , it follows that μ ψ is negative and μ ϕ is equal to or lower than zero. Our main interest is in assessing the effects of α and ψ on the two multipliers. Evidently, as emphasized in previous works, both parameters play an important role. First, without relying on any parameters, the multipliers are linear-proportional to α . This implies that the productivity of government spending can not only positively enhance the response of output, but also private consumption, to a rise in government spending, as shown in Linnemann and Schabert [11] and Tervala [13] . Moreover, it can be ascertained that the multipliers are inversely proportional to ψ , implying that the complementarity (substitution) between private consumption and government spending can generate (inhibit) positive responses, as shown in Ganelli and Tervala [8] .

Up to this point, we have given a simple quantitative rating of the government spending multipliers. In what follows, we focus primarily on the qualitative aspect. For investigating whether or not private consumption and output respond positively to a rise in government spending, we only have to observe their sign.

From an aforementioned condition ( ϕ ≥ 0 ), (8) and (9), we readily obtain the following propositions.

Proposition 1. Government spending leads to a rise in private consumption if and only if

m C > 0 ⇔ α + α ϕ − ϕ − ψ > 0 .

Proposition 2. Government spending leads to a rise in output if and only if

m Y > 0 ⇔ α + α ϕ − ψ + 1 > 0 .

Proposition 1 asserts that private consumption can respond positively to a rise in government spending only when α is large or ψ is small, or both are satisfied. On the contrary, Proposition 2 suggests that the positive output response requires easier conditions.

We further explore how the government spending multipliers can be positive within some realistic parameter settings. Before considering possible parameter settings, we provide an overview of the estimates in previous empirical works, which are relevant to the following exploration.

There are three key parameters for determining the sign of government spending multipliers as shown in (8) and (9). We review the estimates of α and ϕ according to Tervala [13] . First, for the production elasticity of government spending, α , majority studies in the empirical literature report the estimates that are less than 40 percent (for example, see Aschauer [10] ; Glomm and Ravikumar [12] ). More importantly, it is noteworthy that they are not estimated on the basis of flow (i.e., government spending) as in the present model, but on the basis of stock (i.e., government capital, also called public capital). On narrowing our focus to the empirical papers that estimate production elasticity by using the flow data, as stated in Tervala [13] , we find that there is no such work except for Evans and Karras [14] . They use a panel of the U.S. states in the period from 1970 to 1986 and point out that among the various items of government spending, only the government educational services are productive, and the production elasticity is estimated to be approximately 0.04. Therefore, in our analysis, α should be considered to be quite small and government spending has mild productivity.

The second parameter is the Frisch elasticity of labor supply, denoted earlier by 1 / ϕ . Till date, many empirical studies have attempted to estimate this value (e.g., MaCurdy [15] ; French [16] ). While these estimates vary, we are aware of a notable study by Domeij and Flodén [17] who state that the elasticity for men is mostly estimated between 0 and 0.5 in the existing literature, and find that it is estimated to be 50 percent lower than the true value when standard econometric techniques are employed. In other words, there is a strong possibility that the true value can be up to one. This conclusion is not inconsistent with that of Rotemberg and Woodford [18] . As stated in Ganelli and Tervala [8] , Rotemberg and Woodford find that the value is between 0.47 and 1.66, and the straightforward average is calculated to be 1.065. This is supported by a relatively recent study by Kimball and Shapiro [19] who provide evidence that the elasticity is approximately one.

The third parameter is ψ . While Ganelli and Tervala [8] and Tervala [13] do not mention what value of ψ is empirically supported, the degree varies across countries. Karras [20] estimated ψ for 30 countries by supposing the utility function that is similar to our specification. Table 3 and Table 4 in Karras [20] reported that most of the estimates are in the range of −2 to 1. The average for 30 countries in Table 3 (Table 4) is −1.309 (−1.024) and the median is −1.050 (−0.901), and the approximate value of ψ is considered to be −1.

Summarizing the above-mentioned empirical evidence, it seems reasonable to assume that approximately, α is minimal, ϕ = 1 , and ψ = − 1 . In what follows, these are taken into account.

Figure 1 captures the main implication of our analysis. The figure presents a view of the threshold surface, on which α + α ϕ − ϕ − ψ = 0 holds, with responses of C ^ to a rise in G ^ . In the space below (above) the surface, α + α ϕ − ϕ − ψ > 0 (<0) holds, and accordingly, from proposition 1, government spending leads to

an increase (decrease) in private consumption. For comparison with Tervala [13] , who focuses only on ϕ and α , and does not consider ψ, several iso-ψ contours are also shown in a certain ( ϕ , α ) plane in the figure.

We begin by reconfirming the case of Tervala [13] , which corresponds to a contour passing through point ( ϕ , α , ψ ) = ( 0 , 0 , − 1 ) in Figure 1. To do this, we now assume that ψ = 0 , that is, government spending neither complements nor substitutes for private consumption. It turns out at once that if the Frisch elasticity of labor supply is in the vicinity of the value one, as suggested by Kimball and Shapiro [19] , the production elasticity of government spending must be higher than 0.4 ( α > 0.4 ) in order to satisfy the condition, α + α ϕ − ϕ > 0 . Contrary to the results of Linnemann and Schabert [11] , it is Tervala’s [13] sum and substance that the positive response of private consumption to government spending cannot occur solely in the presence of productivity, as long as government spending is mildly productive ( 0 < α < 0.4 ).

Next, we consider the case of Ganelli and Tervala [8] where although complementarity exists between private consumption and government spending, government spending is absolutely not productive ( α = 0 ). We observe in Figure 1 that even when certain strong complementarity is assumed, the condition ϕ + ψ < 0 is less likely to be satisfied in the absence of productivity. Specifically, when government spending is a perfect complement to private consumption ( ψ = − 1 ), the equation ϕ + ψ = 0 holds if productivity of government spending is zero and the Frisch elasticity of labor supply is equal to one ( ϕ = 1 ). As a result, the multiplier m C becomes zero and thereby, the positive consumption response is not explained. Additionally, if the Frisch elasticity of labor supply is less than one, ( ϕ > 1 ), then ϕ + ψ > 0 , and private consumption negatively responds to government spending. These examples suggest that the sole presence of complementarity also rules out the positive response of C ^ to a rise in G ^ , in this simple framework.

In summary, it is shown that neither productivity nor complementarity can solely generate positive consumption response in the present model based on Tervala [13] . Instead, we abstract from an alternative possibility that seems a plausible explanation for the positive response of C ^ to a rise in G ^ . To show this, we now consider the case where government spending has both productivity and complementarity, and highlight some of their combinations within the realistic parameter settings as before.

For simplicity, we again assume the plausible case that government spending is a perfect complement to private consumption ( ψ = − 1 ) and the Frisch elasticity of labor supply is equal to one ( ϕ = 1 ). Additionally, mildly productive government spending is introduced, and we assume that α runs from higher than 0 through 0.4 at most. Given such parameter settings, there is a definite possibility of the presence of ( ϕ , α , ψ ) in the space below the threshold surface in Figure 1. This illustrates that mildly productive government spending can serve as a crucial factor even if it takes very minimal values (e.g., α = 0.01 ). Likewise, from Figure 1, we can confirm that multiple combinations of ( ϕ , α , ψ ) satisfy the inequality sign such that α + α ϕ − ϕ − ψ > 0 , on allowing for mildly productive government spending and complementarity to some extent.

Incidentally, mildly productive government spending may have relevance to the positive output response. Similar to Figure 1, a view of the threshold surface with responses of Y ^ to a rise in G ^ is depicted in Figure 2. α + α ϕ − ψ + 1 > 0 (<0) holds in space below (above) the surface, and from proposition 2, government spending leads to an increase (decrease) in output. It follows from Figure 2 that even if α = 0 , the positive output response is explained for any value of ψ ( ∈ [ − 2,1 ] ) , except for the case of perfect substitution where ψ = 1 is satisfied. In other words, even if perfect substitution is assumed, the sole presence of mildly productive government spending could cause positive output response.