Laffer Curves and Home Production

A representative individual chooses consumption (c_t), hours worked (n_t), capital stock (k_t), private investment (i_t), and government bond holdings (b_t) in order to maximize his or her discounted expected utility. Utility is derived from consumption and leisure (1 − n_t). The representative agent maximizes Ut=maxE0∑<!-- ? -->t=0∞<!-- 8 -->β<!-- ? -->tu(ct,1−nt) $$\displaystyle\ U_{t} = \text{max}\,\, E_{0}\sum\limits_{t=0}^{\infty}\beta^{t}u(c_{t} , 1 - n_{t})$$ (1) subject to (1+τ<!-- t -->tc)ctm+it+bt=(1−<!-- - -->τ<!-- t -->tn)wtntm+(1−<!-- - -->τ<!-- t -->tk)(rtk−<!-- - -->δ<!-- d -->)kt+δ<!-- d -->kt+(1+rtb)bt−<!-- - -->1+st+Π<!-- ? -->t, $$\begin{array}{} \displaystyle(1 + \tau_{t}^{c})c_{t}^{m}+ i_{t} + b_{t} = (1 - \tau_{t}^{n})w_{t}n_{t}^{m} \\ \qquad\qquad\,\,\,\displaystyle+ (1 - \tau_{t}^{k})(r_{t}^{k}-\delta)k_{t}\\ \qquad\qquad\,\,\,\displaystyle+ \delta k_{t} + (1 + r_{t}^{b})b_{t-1} + s_{t} + \mathit \Pi_{t}, \end{array}$$ (2)kt+1=(1−<!-- - -->δ<!-- d -->)kt+it, $$\displaystyle k_{t+1} = (1 - \delta)k_{t} + i_{t},$$ (3)

where β ∈ (0, 1) is the utility discount rate, ctm $\displaystyle c_{t}^{m}$ is the consumption of market goods, ntm $\displaystyle n_{t}^{m}$ denotes the labor supplied to the market, and τ<!-- t -->tc,τ<!-- t -->tn $\begin{array}{} \displaystyle \tau_{t}^{c},\,\tau_{t}^{n} \end{array}$ and τ<!-- t -->tk $\begin{array}{} \displaystyle \tau_{t}^{k} \end{array}$ denote the consumption, labor, and capital tax rates, respectively. On the income side, w_t denotes wage rate, s_t the government transfers, Π_t the profits of the firms, and rtk $\displaystyle r_{t}^{k}$ and rtb $\displaystyle r_{t}^{b}$ the interest rate applied to capital and government bonds, respectively. Profits equal zero in the equilibrium with perfect competition. Total hours worked is the sum of hours worked in the market sector and hours worked in home production: nt=ntm+nth. $\begin{array}{} \displaystyle n_{t} = n_{t}^{m} + n_{t}^{h}. \end{array}$ The capital depreciation rate is given by δ.

A representative agent derives utility from a composite consumption good: ct=(ω<!-- ? -->(ctm)k+(1−<!-- - -->ω<!-- ? -->)(cth)k)i/k, $$\begin{array}{} \displaystyle c_{t} = \bigg(\omega (c_{t}^{m})^{k} + (1 - \omega) \big(c_{t}^{h}\big)^{k}\bigg)^{i/k}, \end{array}$$ (4)

where the superscripts m and h denote the market goods and home-produced goods, respectively. The parameter ω denotes the share of market-produced goods in private consumption, and k measures the elasticity of substitution between home- and market-produced goods. Equation (4) is important in terms of results, and thus, a more detailed inspection of it is in order. Take a total differential of equation (4) and hold the composite consumption constant by setting d(c_t) = 0. The resulting equation is given as follows: d(ctm)ctm=−<!-- - -->(1−<!-- - -->ω<!-- ? -->)ω<!-- ? -->(cthctm)kd(cth)cth. $$\begin{array}{} \displaystyle\frac {d(c_{t}^{m})} {c_{t}^{m}} = -\frac{(1 - \omega)} {\omega} \bigg(\frac{c_{t}^{h}}{c_{t}^{m}}\bigg)^{k} \frac{d (c_{t}^{h})}{c_{t}^{h}}. \end{array}$$ (5)

Equation (5) states that, conditional on c_t, individual is willing to substitute market consumption with home production as a function of ω and k. A change in economic environment, for example, an increase in consumption tax, induces individual to substitute market-produced goods with home-produced goods. In Figure 1, an isoconsumption curve is drawn (holding c_t constant), illustrating the trade-off between ctm $\displaystyle c_{t}^{m}$ and cth $\displaystyle c_{t}^{h}$ .

Figure 1

Continuing with the model description, there is a production function, which defines the production technology of home-produced goods. Olovsson (2015) assumes a production function that uses home capital and home work as production inputs. Rogerson and Wallenius (2012), on the other hand, assume a functional form that combines market-purchased goods and home production time as inputs of home production. For simplicity and transparency, more along the lines of Olovsson (2009), a following form of home production function is assumed: cth=(nth)α<!-- a -->h. $$\begin{array}{} \displaystyle c_{t}^{h} = \bigg(n_{t}^{h}\bigg)^{\alpha^{h}}. \end{array}$$ (6)

The periodic utility function is increasing and concave in consumption and leisure and assumed to be of the following form: u(ct,1−<!-- - -->nt)=(ct)1−<!-- - -->σ<!-- s -->(1−<!-- - -->γ<!-- ? -->(1−<!-- - -->σ<!-- s -->)nt1+1/ϕ<!-- ? -->)σ<!-- s -->−<!-- - -->11−<!-- - -->σ<!-- s -->, $$u(c_{t},\displaystyle \ 1-n_{t})=\frac{(c_{t})^{1-\sigma}(1-\gamma(1-\sigma)n_{t}^{1+1/\phi})^{\sigma}-1}{1-\sigma},$$ (7)

where ϕ, σ and γ denote, respectively, Frisch elasticity of labor supply, measure of intertemporal elasticity of substitution (≠ 1), and a scale parameter for dis utility of labor. Utility function of this form features a constant Frisch elasticity of labor supply, which is convenient in this type of analysis, because the magnitude of the labor supply elasticity is important in terms of results. Particularly, this type of utility function is relevant, because it allows comparisons to the related research. A more in-depth treatment of a utility function of this specification is given in Trabandt and Uhlig (2011).

The first-order conditions of the household’s optimization are as follows: ∂<!-- ? -->u(.)∂<!-- ? -->ntm=−<!-- - -->∂<!-- ? -->u(.)∂<!-- ? -->ct∂<!-- ? -->ct∂<!-- ? -->ctm(1−<!-- - -->τ<!-- t -->tn)(1+τ<!-- t -->tc)wt, $$\displaystyle \frac{\partial{u}(.)}{\partial n_{t}^{m}}=-\frac{\partial{u}(.)}{\partial c_{t}}\frac{\partial c_{t}}{\partial c_{t}^{m}}\frac{(1-\tau_{t}^{n})}{(1+\tau_{t}^{c})}{w_{t}},$$ (8)∂<!-- ? -->u(.)∂<!-- ? -->nth=−<!-- - -->∂<!-- ? -->u(.)∂<!-- ? -->ct∂<!-- ? -->ct∂<!-- ? -->nth, $$\displaystyle \frac{\partial{u}(.)}{\partial n_{t}^{h}}=-\frac{\partial{u}(.)}{\partial c_{t}}\frac{\partial c_{t}}{\partial n_{t}^{h}},$$ (9)1(1+τ<!-- t -->tc)∂<!-- ? -->u(.)∂<!-- ? -->ct∂<!-- ? -->ct∂<!-- ? -->ctm=β<!-- ? -->Et[1(1+τ<!-- t -->t+1c)∂<!-- ? -->u(.)∂<!-- ? -->ct+1∂<!-- ? -->ct+1∂<!-- ? -->ct+1m(1+(1−<!-- - -->τ<!-- t -->t+1k)(rt+1k−<!-- - -->δ<!-- d -->))], $$\begin{array}{} \displaystyle \frac{1}{(1+\tau_{t}^{c})}\frac{\partial{u}(.)}{\partial c_{t}}\frac{\partial c_{t}}{\partial c_{t}^{m}}= \\\displaystyle \beta E_{t}\bigg[\frac{1}{(1+\tau_{t+1}^{c})}\frac{\partial{u}(.)}{\partial c_{t+1}}\frac{\partial c_{t+1}}{\partial c_{t+1}^{m}}(1+(1-\tau_{t+1}^{k})(r_{t+1}^{k}-\delta))\bigg], \end{array}$$ (10)1(1+τ<!-- t -->tc)∂<!-- ? -->u(.)∂<!-- ? -->ct∂<!-- ? -->ct∂<!-- ? -->ctm=β<!-- ? -->Et[1(1+τ<!-- t -->t+1c)∂<!-- ? -->u(.)∂<!-- ? -->ct+1∂<!-- ? -->ct+1∂<!-- ? -->ct+1m(1+rt+1b)]. $$\begin{array}{} \displaystyle \frac{1}{(1+\tau_{t}^{c})}\frac{\partial{u}(.)}{\partial c_{t}}\frac{\partial c_{t}}{\partial c_{t}^{m}}=\\ \displaystyle\beta E_{t}\bigg[\frac{1}{(1+\tau_{t+1}^{c})}\frac{\partial{u}(.)}{\partial c_{t+1}}\frac{\partial c_{t+1}}{\partial c_{t+1}^{m}}(1+r_{t+1}^{b})\bigg]. \end{array}$$ (11)

Equation (8) characterizes the labor supply decision of an individual in the labor market, equation (9) determines the labor supply in home production, and finally, equations (10) and (11) determine the equilibrium rate of return for capital and guarantee that there are no arbitrage opportunities between the rate of return for capital and government bonds, that is, (1−<!-- - -->τ<!-- t -->tk)(rtp−<!-- - -->δ<!-- d -->p)=rtp. $\begin{array}{} \displaystyle (1 - \tau_{t}^{k})(r_{t}^{p} - \delta^{p}) = r_{t}^{p}. \end{array}$

There is a large number of identical final good firms that produce a homogeneous product by choosing k_t and ntm $\displaystyle n_{t}^{m}$ . The firms maximize their profits, which are given by the following: Π<!-- ? -->t=yt−<!-- - -->rtkkt−<!-- - -->wtntm $$\begin{array}{} \mathit\Pi_{t}=\displaystyle y_{t}-r_{t}^{k}k_{t}-w_{t}n_{t}^{m} \end{array}$$

Output, y_t, of a representative final good firm is given by yt=At(kt)α<!-- a -->(ntm)1−<!-- - -->α<!-- a -->, $$\begin{array}{} \displaystyle y_{t}=A_{t}\,(k_{t})^{\alpha}\,(n_{t}^{m})^{1-\alpha}, \end{array}$$ (12)

where A_t = (1 + g^A)A_t-1 is the total factor productivity, g^A denotes the trend growth of the total factor productivity, α and (1 − α) are the share parameters of private capital and labor, respectively. The rental rate of private capital and wage rate are, respectively, given by rtk=∂<!-- ? -->yt∂<!-- ? -->kt, $$\begin{array}{} \displaystyle r_{t}^{k}=\displaystyle \frac{\partial y_{t}}{\partial k_{t}}, \end{array}$$ (13)wt=∂<!-- ? -->yt∂<!-- ? -->ntm. $$\begin{array}{} \displaystyle w_{t}=\displaystyle \frac{\partial y_{t}}{\partial n_{t}^{m}}. \end{array}$$ (14)

The government collects taxes, T_t, and issues bonds (b_t) in order to finance expenditures for government consumption (g_t), investments ( itg $\displaystyle i_{t}^{g}$ ), transfers (s_t), and debt services: Tt=τ<!-- t -->tcctm+τ<!-- t -->tnwtntm+τ<!-- t -->tk(rtk−<!-- - -->δ<!-- d -->)kt, $$\begin{array}{} \displaystyle T_{t}=\tau_{t}^{c}c_{t}^{m}+\tau_{t}^{n}w_{t}n_{t}^{m}+\tau_{t}^{k}(r_{t}^{k}-\delta)k_{t}, \end{array}$$ (15)gt+itg+st+(1+rtb)bt−<!-- - -->1=bt+Tt. $$\begin{array}{} \displaystyle g_{t}+i_{t}^{g}+\mathrm{s}_{t}+(1+r_{t}^{b})b_{t-1}=b_{t}+T_{t}. \end{array}$$ (16)

The no-ponzi constraint of public sector debt must apply: limT→<!-- ? -->∞<!-- 8 -->(bT+1∏<!-- ? -->j=1T(1+rjb))=0. $$\begin{array}{} \displaystyle \lim_{T\rightarrow\infty}\bigg(\frac{b_{T+1}}{\prod_{j=1}^{T}(1+r_{j}^{b})}\bigg)=0. \end{array}$$ (17)

The no-ponzi condition states that the discounted stream of taxes must equal the current value of outstanding government debt plus stream of government expenditures. Public debt has no specific role in the analysis apart from making the government budget constraint more realistic.

It is necessary to have one “adjusting” or endogenous variable in the government budget constraint in order to have a well-behaving system of model equations. Following a standard practice in the literature, when taxes, government consumption, or debt is altered, government adjusts transfers (s_t) according to the government budget constraint.

Trabandt and Uhlig (2011) call this the s-Laffer curve, whereas the g-Laffer curve is the one where government consumption (g_t) is endogenous.

The endogeneity of s_t is important in terms of results and some observations are in order. First, the results of this paper do not change if the public debt (b_t) is made endogenous instead of s_t – the Ricardian equivalence typically holds in this type of models.

Second, the results change if the government consumption (g_t), which is assumed to be waste, is made the endogenous variable. Endogenizing government consumption shifts the Laffer peak to the right, because there is no (negative) income effect on labor supply via government transfers.

Third, and related to the previous point, the effect of utility producing government consumption or productive public capital is not explored in this paper. It is possible, however, that the inclusion of such mechanism would increase fiscal space, because the negative effect of a tax increase was mitigated by a positive effect on utility or production.

In the competitive (decentralized) equilibrium, individuals maximize their utility, firms maximize profits, all constraints are satisfied, and all markets are clear. Specifically, general equilibrium is the path of endogenous variables {yt,ct,ctm,cth,ntm,nth,kt,it,rtk,rtb,wt,Tt,st,Π<!-- ? -->t} $\begin{array}{} \displaystyle \{y_{t},\,c_{t},\,c_{t}^{m},\,c_{t}^{h},\,n_{t}^{m},\,n_{t}^{h},\,k_{t},\,i_{t},\,r_{t}^{k},\,r_{t}^{b},\,w_{t},\,T_{t},\,s_{t}, \mathit \Pi_{t}\} \end{array}$ that satisfies the individual budget constraint (2), law of motion for capital (3), equation defining composite consumption (4), individual first-order conditions (8)–(11), production technology for market goods (12) and home-produced goods (6), factor price equations (13) and (14), and the characterization of government (15)–(17), given the exogenous variables such as government consumption (g_t), government investment itg $\displaystyle i_{t}^{g}$ , government debt (b_t), and the tax rates τ<!-- t -->tc,τ<!-- t -->tn $\begin{array}{} \displaystyle \tau_{t}^{c},\,\tau_{t}^{n} \end{array}$ and τ<!-- t -->tk $\begin{array}{} \displaystyle \tau_{t}^{k} \end{array}$ .

The model is calibrated to match the essential features of the Finnish economy. The data used is of annual frequency, and the period of interest is post-2008 to capture the recent challenges in the economic environment, particularly the deteriorated fiscal position of the economy since 2009.

There are a number of parameters to be calibrated. Following the usual practice, as many parameters as possible are calibrated using evidence from existing research literature, and the rest are set to match certain ratios in the data. All the calibrated values of parameters and exogenous variables are reported in Tables 1 and 2.

The exogenous total factor productivity, γ^A, is assumed to be 0.9% in accordance with the long-run scenarios of the European Commission (2015). The labor share parameter in the production function, (1 − α), is calibrated to match the wage sum share of national income, which is, on an average, 0.611 in Finland between 2009 and 2014.

Deep preference parameters of the representative agent are σ, γ, and β, which represent, respectively, the measure of intertemporal elasticity of substitution, the consumption weight in utility function, and the time discount factor of the utility function. The utility discount factor, β, defines, in steady state, the real interest rate of the economy, which, in turn, is a function of capital–output ratio. Accordingly, β is calibrated to match the 2009–2014 capital–output ratio of the Finnish economy, which is equal to 2.617. The consumption weight in the utility function, γ, following, for example, Cooley and Soares (1999) and Papageorgiou (2012), is set so that the average working hours matches the data. According to the Finnish Time Use Survey 2009–2010, 19% of wake time is spent in gainful employment (ntm $\displaystyle n_{t}^{m}$ ), whereas 16% is spent in domestic work (nth $\displaystyle n_{t}^{h}$ ), which, in turn, is used to calibrate the share parameter, ω. The measure of intertemporal elasticity of substitution, or “the curvature parameter”, σ, is set to equal 2, which is in line with previous related literature.

There is a lively discussion upon the “correct” value of the labor supply elasticity (ϕ). Values used in macroeconomic literature are typically larger than those estimated from microdata. Keane and Rogerson (2012) raise a number of important points challenging the microelasticities and argue that elasticities between 1 and 2 can be credibly supported. On the other hand, in a recent survey, Chetty et al. (2012) conclude that the microeconomic evidence of Frisch elasticity points toward intensive margin elasticity of 0.54 and extensive margin of 0.28 and macroeconomic (cross-country) evidence points towards intensive margin elasticity of 0.54 and the extensive margin 2.3.

See Chetty et al. (2012) page 2, Table 1.

The substitution parameter, K, is a very important one with respect to the shape of consumption Laffer curve. There is also some empirical evidence on the value of this parameter. Aguilar et al. (2011) consider older individuals and find that the parameter value is between 0.5 and 0.6. McGrattan et al. (1997) report values between 0.4 and 0.44. Finally, Chang and Schorfheide (2003) estimate values between 0.44 and 0.6. Referring to this evidence, the substitution parameter, k, is set to 0.5 in the same spirit as, for instance, Rogerson and Wallenius (2012).

Exogenous variables are, as well as the parameters above, calibrated to match the 2009–2014 data, if possible. This implies that the government consumption-to-output and the debt-to-output ratios are set to, respectively, 0.243 and 0.509. Finally, the benchmark tax rates τ<!-- t -->tn,τ<!-- t -->tk $\begin{array}{} \displaystyle \tau_{t}^{n},\,\tau_{t}^{k} \end{array}$ and τ<!-- t -->tc $\begin{array}{} \displaystyle \tau_{t}^{c} \end{array}$ are specified using the method developed by Mendoza et al. (1994) wherein the idea is to relate relevant tax revenue to the relevant tax base. The tax rates are interpreted to be the average effective tax rates (AETR). Naturally, the method is not able to capture the complex nature of the tax system. On an average, however, it is presumably a reasonable approximation of the reality.

The essential steady-state values produced by the model are provided in Table 3. The baseline steady-state calibration fits the data reasonably well. The calibration given in Table 3 give rise to the Laffer curves depicted with dashed line in Figures 2–4. The regular line depicts the “traditional model,” the model without home production, in other words, with identical calibration method but setting ω = 1. Figures 2–4 depict aggregate tax revenue (T_t), which is normalized by the Laffer curve peak value of the model without home production. The gray vertical line marks the steady-state tax level, which is also the tax rate where the two Laffer curves cross. The figures tell a story of different mechanisms in taxation with and without home production.

Figure 2

Figure 3

Figure 4

The Laffer curves, or aggregate tax revenue curves (see equation (15)), are calculated so that one tax instrument at a time is varied between 0% and 100%, while all the other parameters and exogenous variables (including the two other tax rates) in the model are held constant (the ceteris paribus assumption). Setting one tax rate to zero does not, thus, imply tax revenue of zero, because there are still two other strictly positive tax rates. The Laffer curves with home production are not of the expected form and do not follow the same pattern as the Laffer curves in the earlier literature (cf. Trabandt and Uhlig (2011)).

The consumption tax Laffer curves are depicted in Figure 2. The model without home production implies strictly increasing Laffer curve between 0% and 100% tax rates, whereas the consumption Laffer curve exhibits a hump shape when home production is included in the model. The peak of the consumption tax Laffer curve lies at 60% (100%) with (without) home production.

There is no apparent reason why the upper bound of consumption tax rate should be 100%. This upper bound is imposed somewhat arbitrarily for communicational reasons.

Not only the location of the Laffer peak but also the shape of the curve differs between these two specifications has significant effects on the tax revenue estimates. The average tax revenue elasticity

Tax revenue elasticity with respect to tax rate τ_i is approximated with ∂<!-- ? -->Tt∂<!-- ? -->τ<!-- t -->i≈<!-- ? -->T(τ<!-- t -->i)−<!-- - -->T(τ<!-- t -->i−<!-- - -->ε<!-- e -->)ε<!-- e --> $\begin{array}{} \displaystyle \frac{\partial T_{t}} {\partial \tau^{i}} \approx \frac{T(\tau ^{i})-T(\tau^{i}-\varepsilon)}{\varepsilon} \end{array}$ , where ε = 1%

The inclusion of home production, thus, lowers the aggregate tax revenue elasticity considerably, which implies that in order to achieve a given increase in consumption tax revenue, a larger increase in tax rate is needed. On the other hand, a decrease in the tax rate is not as detrimental to the public sector revenues as is in the case without home production.

The labor income tax Laffer curves are depicted in Figure 3. The Laffer curve with (without) home production is increasing up to 35% (57%), implying that the Finnish economy is in the “wrong” side of the Laffer peak with home production but on the “right” side without home production in the model. The recommended policy advice in terms of tax revenue, thus, depends crucially on whether or not home production is included in the model. In terms of maximizing the aggregate tax revenue, labor income tax rate should be decreased (increased) in order to maximize tax revenue with (without) home production in the model.

Finally, Figure 4 depicts capital tax rate Laffer curves. The two curves are of completely different form. The Laffer curve is strictly decreasing with home production, while without home production, it increases up to the tax rate of 29%, after which it decreases rather abruptly. In both the cases, the capital Laffer curve is flatter than labor income or consumption tax Laffer curve, implying lower tax revenue elasticity in the flat part of the curve. Increasing the capital tax rate from 0% to 29% would lead, in the steady-state equilibrium, to a -3.0% (1.7%) change in aggregate tax revenue with (without) home production. In general, the impact of capital taxation on aggregate tax revenue is clearly smaller than that of labor income or consumption taxation.

The reasoning behind the not so familiar looking Laffer curves is the following. With home production in the model, compared to a model without it, there is an additional margin of adjustment (see equation (9)). Suddenly individuals do not alter only labor supply (see equation (8)) in response to a tax change, but directly also consumption. Equation (9) ensures that the marginal utility of consuming market-produced goods and home-produced goods equalizes.

When the consumption tax rate increases, consumption of market goods becomes relatively more expensive and labor supply adjusts according to the intra-temporal Euler condition (see equation (8)), which lowers the disposable wage income and, consequently, has a negative effect on consumption. This is the traditional effect of a consumption tax change. Furthermore, also the marginal utilities from market consumption and home production must equalize. In this case, if there is a tax hike, home production becomes relatively more attractive and home production increases in detriment to market consumption. The outcome is that the consumption tax base deteriorates more quickly as the tax rate increases, thus, making a crucial difference in the shape of the Laffer curve. In other words, inclusion of home production brings forth a mechanism that accelerates the deterioration of the tax base. This second mechanism is not present in the model without home production.

As seen in the above, the effects of home production are not limited only to the consumption Laffer curve. Also income Laffer curves exhibit different behavior with home production. The intuition behind the result is similar to that of the consumption Laffer curve. An increase in income tax rate, be it capital or labor tax, lowers the disposable income and, thus, has a direct effect on labor supply. At the same time, the relative prices of consumption and leisure change induce a decrease in market-based consumption and an increase in home production. The mechanism is such that it amplifies the negative tax revenue effect of taxation. Another angle of the same mechanism is that inclusion of home production makes labor supply more responsive to taxation. This is the argument also made by Rupert et al. (2000).

It’s unclear why Vogel (2012) doesn’t basically find any effects (in the benchmark calibration) because of home production compared to a model without this mechanism. The particular model is much more complicated, and therefore, the relevant mechanism is somewhat blurred by other effects. This is one of the contributions of this paper: to build a model comparable to the Trabandt and Uhlig (2011) model and augment the model only with home production, thus, isolating away all the other potentially intervening effects.

The previous subsection calculated the Laffer curves when one tax rate at a time was varied. In this section, a more general approach is taken; all tax rates are allowed to vary between 0% and 100%, and the tax revenue maximizing tax mix is calculated. Once again, the analysis is of tax revenue doesn’t take welfare implications into consideration.

First, the tax revenue maximizing capital tax rate is zero if other tax rates are free to adjust. This observation is verified by numerical calculations.

Next, the Iso Tax Revenue Curves are plotted in Figure 5 when capital income tax rate is set to zero. Revenue maximizing tax mix with home production is found to be {τ^c, τⁿ, τ^k = 100%, 9%, 0%}. The corresponding tax mix without home production is τ^c, τⁿ, τ^k = 100%, 42%, 0%}. The introduction and modeling of home production, thus, implies lower labor income tax in the tax revenue maximizing tax mix, and therefore, less fiscal space. The revenue maximizing tax mix reflects the somewhat common result that consumption taxation is the most efficient tax form to collect taxes in this type of models. The “efficiency of consumption taxation” can also be seen using equation (8). Assuming exogenous w_t, taxation distorts the optimal labor supply decision by the factor, 1−<!-- - -->τ<!-- t -->n1+τ<!-- t -->c, $\begin{array}{} \displaystyle \frac {1 - \tau^{n}}{1 + \tau^{c}}, \end{array}$ which, in turn, implies that, for example, 30% labor income tax rate is as distorting as 43% consumption tax rate and so forth. 1−<!-- - -->30%1+0%≈<!-- ? -->1−<!-- - -->0%1+43% $\begin{array}{} \displaystyle \frac {1 - 30\text\%}{1 + 0\text\% } \approx \frac {1 - 0\text\%}{1 + 43\text\% } \end{array} $

Figure 5

The Iso Tax Revenue Curves in Figure 5 are plotted so that each curve to the right is at 10% lower aggregate tax revenue level. This illustrates the trade-off in tax revenue between the two plotted tax rates. An identical aggregate tax revenue, 0.7 of maximum, for instance, can be collected when {τⁿ, τ^c} = {0%, 55.8%}, when {τⁿ, τ^c} = {43.0%, 16.4%}, or when {τⁿ, τ^c} = {50.4%, 100.0%}.

Figure 5 also reveals that given a consumption tax rate of 20%, for instance, a 0.6 aggregate tax revenue can be achieved by setting Tⁿ to approximately 20% or to 61%. This implies that the {τⁿ, τ^c} = {20%, 61%} tax mix is on the slippery side of the Laffer curve and a decrease in labor income tax rate would lead to, ceteris paribus, an increase in aggregate tax revenue. Furthermore, the Figure 5 actually reveals all tax rate combinations that are on the slippery side of the Laffer curve.