Clément Malgouyres
email: clement.malgouyres@ipp.eu.

Thierry Mayer
email: thierry.mayer@sciencespo.fr.

Clément Mazet-Sonilhac

Who Benefits from State Corporate Tax Cuts? A Local Labor Markets Approach with Heterogeneous Firms: Comment *

Keywords: H22, H25, H32, H71, R23, R51 incidence, corporate income tax, discrete/continuous choice

Suárez Serrato and Zidar (2016) identify state corporate tax incidence in a spatial equilibrium model with imperfectly mobile firms. Their identification argument rests on comparative-statics omitting a channel implied by their model: the link between common determinants of a location's attractiveness and the average idiosyncratic productivity of firms choosing that location. This compositional margin causes the labor demand elasticity to be independent from the product demand elasticity, impeding the identification of incidence from the four estimated reduced-form effects. Assigning consensual values to the unidentified parameters, we find that the incidence share born by firm-owners is closer to 25% than 40%.

Introduction

The incidence of the corporate income tax is arguably one the most important questions in public finance and one that has defied clear-cut empirical answers until recently. In 2016, Suárez Serrato and Zidar (2016), SZ henceforth, published a paper in the American Economic Review which broke ground both empirically and theoretically.

Empirically, they estimate the effects of changes in business taxes on four key outcomes, namely wages, establishment growth, rental costs and local population, using variation in state corporate tax rates. Theoretically, they develop a spatial equilibrium model in which firms vary idiosyncratically in terms of how productive they are across locations, with workers also displaying idiosyncratic preferences across alternative states of residence. Within this model, they study how the reduced-form impacts of changes in business taxes relate to changes in the welfare of workers, firmowners and landowners. In particular, they show that, conditional on two parameters, their reduced-form results point-identify the incidence from changes in tax rates, i.e. the burden falling on each category of agents.

In this comment, we show that comparative statics computed by SZ ignore two channels implied by their own model. We derive the corrected expressions and discuss the implications of our corrections when mapping the reduced-form of the model with respect to a change tax rates-which can be transparently estimated through panel regressions-into incidence shares for workers, capitalists and landowners.

We first show that, since capital costs are not deductible in SZ setting, the corporate income tax increases the relative cost of capital. This relative cost effect implies that the sensitivity of business location choices to corporate tax will depend on the capital intensity of the technologies that are considered. This dependence is ignored in a key equation of SZ. Next, and more importantly, we show that comparative statics computed by SZ ignore the fact that changes in business tax rates in a given locationor other fundamentals affecting how attractive a location is for all business-ownerswill affect the conditional expectation of the idiosyncratic productivity draw of firms actually choosing to locate in that particular location. Technically, there is a missing term in SZ's derivation of labor demand elasticities with respect to local fundamentals that are commonly valued by firm owners. We show that correcting this omission has important implications for SZ's identification argument.

In SZ's model, firm (log) productivity is equal to the sum of a local component common to all firms in the same location and of an idiosyncratic firm-location specific productivity draw. In that setting, intuitively, a high tax state will tend to attract (or retain), ceteris paribus, firms with higher realization of their state-specific idiosyn-cratic productivity draws than if the business taxes were lower. Analogously, consider an exogenous decline in local wage (due for instance to an unexpected increase in local labor supply in a given state). SZ show that in their model this shock will affect total local demand for labor in two ways. First, low labor costs will attract new entrantswhat SZ refers to as the extensive margin. Second, for a given level of idiosyncratic productivity, it will make firms expand their scale of production and become more labor intensive-what SZ refer to as the scale and substitution effects respectively, the sum of which they call the intensive margin. We show that there is a third effect through which local labor demand will be impacted: the change in the average idiosyncratic productivity of firms actually choosing to be based in that area following the decline in local wages. This effect, which we refer to as the "compositional margin" (using terminology from the trade literature surveyed in Head and Mayer (2014)), will work in the opposite way as the first two: as labor cost decreases, the state becomes more attractive and newly arriving firms tend to display lower idiosyncratic productivity draws than before. Taking the compositional margin into account turns out to have important implications for the identification of the incidence of corporate tax cuts.

We show this in two main steps. We first establish that, under the distributional assumptions made by SZ, the compositional margin exactly offsets the substitution and scale effects described above, so that the local wage elasticity of labor demand is entirely driven by the location choice of plants. This result causes SZ's exact identification argument to break down. In particular, we show that the term γ(1 + ε PD ), that is the labor elasticity of output (γ) times 1 plus the elasticity of product demand (ε PD ), is no longer identified through the combination of reduced form effects for which SZ obtain empirical estimates. When accounting for the compositional margin, the two parameters (γ, ε PD ) need to be calibrated in order to identify the incidence of the corporate tax cuts on firm owners from SZ reduced-form estimates. 1 Second, we calibrate the corrected incidence formulas using a range of values for the parameters that cannot be identified and assess the associated incidence. Given consensual values, taken from SZ's own baseline, we find that firm owners bear a smaller share of the burden (around 25%) than the magnitudes reported by SZ in the relevant part of their paper using reduced form estimates. 2 This comment is structured as follows. We recall the broad structure of SZ model in section 2. We highlight the role played by the compositional margin in the macro local labor demand in section 3. Implications regarding the incidence on workers and business owners and the exact identification of the incidence terms from the reduced-form moments are discussed in section 4. Section 5 presents new results using calibrated values for the two elasticities that are not identified when accounting for the compositional margin.

Overview of the model

We start by presenting the key building blocks of the model used in Suárez Serrato and Zidar (2016) (using the version published in the AER and its online appendix). Unless otherwise specified, we follow their notation exactly.

The goal of SZ is to characterize the incidence on wages, profits, and land rents of a change in the local business tax. Accordingly, their economy is populated by three types of agents: workers, business owners, and landowners. The effects of a change in the local business net-of-tax rate, denoted 1τ b c , on the welfare of each of these agents are characterized as functions of estimable elasticities (reduced-form effects) which themselves reflect structural parameters governing the supply and demand sides of the housing, labor, and product markets.

Workers choose their location to maximize utility, landowners supply housing units to maximize rental profits, and business owners choose the location of their production site and the price of their output so as to maximize after-tax profits. There are no trade costs when shipping the goods between regions.

Household location choice. SZ consider a standard environment in terms of workers' location choices (see e.g. Kline and Moretti, 2014b). Wages, rental costs, and amenities vary across locations and are commonly valued by all households. Each household however displays idiosyncratic preferences for each location. The household picks the location yielding the highest total utility, which is equal to the sum of the common component and the idiosyncratic term. Given that the idiosyncratic term follows an Extreme Value Type I distribution with dispersion parameter σ W , this set-up yields the classical multinomial logit functional form for the location choice probability. Housing costs enter indirect utility with a constant weight α.

Housing supply. Landowners supply housing units to maximize rental profits. The elasticity of housing supply-denoted η-determines how much an increase in labor supply-due for instance to an increase in local amenities-translates into rising prices or rising population.

Labor supply. The elasticity of housing supply η, together with the Cobb-Douglas weight on housing α, and the dispersion parameter σ W , determine an "effective" elasticity of labor supply: by how much does local labor force (N c ) increase following an increase in local wage w c (taking into account the fact that housing costs will go up following the arrival of new workers). This elasticity is denoted as ε LS .

Business owners' problem. When choosing the location of establishment j, business owners seek to maximize after tax profits π jc . The log of establishment j's productivity in location c (B jc ) is the sum of a location-specific productivity term common to all establishments ( Bc ), and of a location-establishment specific term ζ jc distributed i.i.d. Extreme Value Type I with dispersion parameter σ F . This set-up also gives rise to a multinomial logit model of the establishment's location choice probability. Firms act as competitive monopolists and face a firm-level elasticity of demand denoted by ε PD < -1. They operate a constant returns to scale Cobb-Douglas production function aggregating labor, capital and intermediates, with output elasticity respectively equal to γ, δ and 1δγ.

In the SZ setup, the establishment problem involves i) the above described location choice and, ii) conditional on location choice, a decision regarding inputs, in particular labor, in order to maximize profit once a location has been chosen. The location choice is discrete while the input choice is continuous. In that sense, the problem facing the establishment is formally very similar to what Hanemann (1984) refers to as "discrete/ continuous models" of consumer demand. 3 We will explore the implications of this formal similarity in the next section.

3 The "macro elasticity of local labor demand"

In this section, we derive a amended version of SZ's expression of local labor demand, which drives incidence in their model. There are two differences with the original expression derived in SZ: 1) we account for how corporate taxation affects the effective cost of capital, 2) we include the "compositional margin", arising from endogenous location choices by firms in this setup. As a convention, when referring to equations that are directly taken from SZ we recall their original number on the left side of the equation with the number in brackets followed by the letters "SZ". 4 The "macro" local labor demand L D c is defined as total labor demand by firms choosing to locate in a given location c. Without loss of generality, it can be written as the measure of firms locating in c-denoted E c -multiplied by the average of individual labor demands by firms located in c (denoted with l * jc ):

L D c ≡ E c × E ζ (l * jc |c). (1)
Following SZ, the conditioning on c (.|c) in the expected labor demand term of ( 1) is a notation shortcut denoting the event that c is the best location for firm j. The expectation is taken over the random draws of establishment-location productivity ζ,

hence the E ζ notation.

We start by deriving expressions for each of the two components of the macro local labor demand in subsection 3.1. We then highlight the differences between the resulting expressions and the corresponding ones in SZ in subsection 3.2.

Deriving the macro local labor demand.

Prices, quantities and profits. Under monopolistic competition, optimal pricing of firm j in c involves a constant markup over marginal cost (also unit cost u jc because of the constant returns to scale assumption):

p jc = -ε PD -ε PD -1 u jc , with u jc = w γ c ρ δ c P 1-γ-δ exp( Bc + ζ jc ) , ( 2
)
where P is the price index of intermediates, and the term ρ c = ρ/(1τ b c ) refers to the local cost of capital.5 CES demand implies that the quantity produced by j is y jc = p ε PD jc × IP -ε PD , where I is national real income and P is the CES price index over all available varieties. Combining CES demand with monopolistic competition, profits are equal to revenue divided by the CES parameter. After-tax profits are therefore equal to: π jc = (1τ b c )p jc y jc /(-ε PD ). Using the expression for unit cost (2), and CES demand y jc , we can write the non-stochastic component of after-tax profits (π c ) as:

π c = κ 1 × (1 -τ b c ) × w γ(1+ε PD ) c ρ δ(1+ε PD ) c (exp( Bc )) (1+ε PD ) , (3)
where a constant κ 1 accounts for all profit determinants that do not depend on c.

The establishment (discrete) location choice. The establishment problem involves a location choice giving rise to a probability for each location c to be chosen. This probability takes the familiar multinomial logit form given that ζ jc , i.e. the idiosyncratic productivity of establishment j in location c follows a Type I extreme value distribution (with dispersion parameter σ F ), with draws across location being i.i.d. Denoting E c the probability of a firm to locate in c, it can be expressed as:

(7SZ) E c = P V jc = max c V jc = exp(v c /σ F ) ∑ c exp(v c /σ F ) , (4)
where V jc = v c + ζ jc refers to the value function of establishment j when choosing location c which is itself the sum of the idiosyncratic productivity draw ζ jc and a value common to all firms denoted v c . Following SZ definition, the term v c is equal to the non-stochastic component of establishment log profit in location c-as given in levels in equation ( 3)-divided byε PD + 1 > 0 and writes as follows6 :

v c ≡ ln π c -(ε PD + 1) = ln 1 -τ b c -(ε PD + 1) + Bc -γ ln w c -δ ln ρ c + ln κ 1 -(ε PD + 1) , (5)
The intensive margin of labor demand. The labor demand for a given establishment j, located in c, is obtained using the Cobb-Douglas production technology as-sumption, which ensures that the share of labor in total costs (unit cost u jc , times output y jc ) is constant and equal to γ. Hence, we have l * jc = γ u jc y * jc w c . Using equilibrium output of the firm y * jc , we obtain

l * jc (ζ jc ) = w (γε PD +γ-1) c ρ (1+ε PD )δ c κ 0 e Bc( -ε PD -1) ≡l i,c exp -ε PD -1 ζ jc , (6)
where κ 0 combines determinants that are constant across establishments and locations. In equation ( 6), we see that that firm-level labor demand is a function of both l i,c , a term that captures determinants common to all firms located in c, and the idiosyncratic draw ζ jc . An important point to note here is that firms in this model choose an optimal location before deciding how much labor to hire in that location. Accordingly, labor demand by firm j in city c as expressed in equation ( 6) is latent in the sense that it will only be realized if c happens to be j's profit maximizing location choice.

Average labor demand in location c (which SZ refer to as the intensive margin) writes as:

E ζ l * jc ζ jc | c =w (γε PD +γ-1) c ρ (1+ε PD )δ c κ 0 e Bc( -ε PD -1) × E ζ exp -ε PD -1 ζ jc | c . ( 7
)
Macro labor demand. Total labor demand in c is equal to the share of firms locating in c multiplied by optimal labor demand conditional on choosing c. Combining (4) and ( 7):

L D c = E c × E ζ l * jc ζ jc | c = E c × l i,c × z c = exp(v c /σ F ) ∑ c exp(v c /σ F ) × w (γε PD +γ-1) c ρ (1+ε PD )δ c κ 0 e Bc( -ε PD -1) × E ζ exp -ε PD -1 ζ jc | c ≡z c . ( 8
)
The compositional margin in labor demand. Key to our understanding of labor demand is the z c term in (8). SZ describe it as a "term increasing in the idiosyncratic productivity draw ζ jc " (p.2591). More specifically, it is the conditional expectation of a monotonic transformation of ζ jc , with the conditioning event occurring with probability E c . This probability depends on common fundamentals of city-level attractiveness as captured in v c . The conditioning therefore implies a dependence of z c with respect to v c .

It is quite intuitive to see why z c and v c are related. Consider a very attractive city c (v c → ∞). The probability that c is chosen is close to 1. Accordingly, almost all firms, independently of their draw ζ jc will be located in c . In that setting, z c , which is a conditional expectation, will be very close to the unconditional expectation, as the conditioning event has a probability close to one. On the contrary, an unattractive city c with a low value of v c will only attract firms with fairly high realization of the random term ζ jc . In that setting, one would expect z c to be very high.

The relationship between the probability of the conditioning event E c and the conditional expectation z c in the case of a vector of iid Type I extreme value random variables was studied by Hanemann (1984).7 Applying Hanneman's result to the computation of z c , we obtain:

z c ≡ E ζ exp -ε PD -1 ζ jc | c = Γ 1 + (ε PD + 1)σ F E (1+ε PD )σ F c , ( 9
)
where the probability that c is the best location, E c , is given in equation ( 4), and Γ() is the gamma function. We will refer to changes in z c following changes in fundamentals in area c as a the "compositional margin" as it captures changes in the composition (in terms of productivity) of the pool of firms choosing to locate in c.

Differences with SZ

We now turn to detailing the two main differences with the original SZ paper. Those are expressed in terms of missing terms in two critical elasticities of the model. The first one relates to how the cost of capital impacts the response of location choice to the tax rate. The second (and most important) one is the omission of the compositional margin in the elasticity of aggregate labor demand in c with respect to local wages.

Cost of capital and location choice.

Based on the definition of E c in equation ( 4) and the expression for the value for firm j to locate in c (v c ) in equation ( 5), we can derive the elasticity of the location choice probability with respect to the net-of-tax rate:

∂ ln E c ∂ ln(1 -τ b c ) = δ σ F - 1 (ε PD + 1)σ F . ( 10
)
The analogous equation in SZ (first equation of page 2592), which the authors refer to as one of their key objects of interest writes as:

(9 SZ) ∂ ln E c ∂ ln(1 -τ b c ) = - 1 (ε PD + 1)σ F .
Equation (9'SZ) does not account for the fact that, because the business tax increases the relative cost of capital in location c, the entry of new firms following a tax cut will be more pronounced the more capital intensive the technology is-where capital intensity is captured by the capital output elasticity δ.

Compositional margin and the macro elasticity of local labor demand. Taking the partial derivative of ( 8) with respect to w c yields the macro elasticity of labor demand with respect to local wage (denoted ε LD in SZ):

∂ ln L D c ∂ ln w c = ∂ ln E c ∂ ln w c extensive margin + ∂ ln l i,c ∂ ln w c initial intensive margin + ∂ ln z c ∂ ln w c compositional margin = - γ σ F extensive margin + γ(1 + ε PD ) scale effect -1 substitution effect initial intensive margin + - γ σ F (1 + ε PD )σ F compositional margin = - γ σ F + 1 . ( 11
)
As the first line of equation ( 11) makes clear, the wage elasticity of total local labor demand is equal to the sum of three terms: the elasticity of the extensive margin (location choice), the intensive margin and the compositional margin. We express these three terms as a function of parameters in the second line. We further decompose the intensive margin into two components. The first subcomponent of the intensive margin is a scale effect, which captures the fact that firms cut down labor demand because of the reduced sales induced by the rise in labor cost. This depends on the price elasticity of demand on the product market as well as on the labor intensity of the technology. The second subcomponent is a substitution effect which is equal to -1 due to the Cobb-Douglas production function. In the third line, we simplify the expression using the fact that the scale effect and the composition effect cancel each other out. This cancellation of the intensive margin parameters is closely related to theoretical derivations of the gravity equation in trade models featuring country-level or firm-level heterogeneity in productive efficiency. It is the same mechanism that explains that the response of aggregate trade flows to changes in variable trade costs does not depend on the product demand elasticity in Eaton and Kortum (2002) or Chaney (2008) for instance.8

Instead SZ's equivalent equation for the macro elasticity of labor demand writes as:

(9SZ) ∂ ln L D c ∂ ln w c = - γ σ F + 1 -γ(1 + ε PD
) . The difference between equations ( 11) and ( 9SZ) is taking into account the fact that the compositional margin changes with respect to w c :

∂ ln z c ∂ ln w c = -γ(1 + ε PD ) > 0. ( 12
)
Under the maintained assumption that ε PD < -1 (required by monopolistic competition), we have ∂ ln z c ∂ ln w c > 0. Therefore, under the distributional assumptions made by SZ (in common with most of the literature combining firm location choice with worker/consumer mobility), changes in z c due to a change in wages perfectly offset the scale effect part of the intensive margin, leaving only the extensive margin parameters and the substitution effect to enter the global response. The fact that Suárez Serrato and Zidar (2016) do not account for this effect is the reason why (9SZ) should be replaced by ( 11), which is the most important point of our comment, with substantial quantitative implications that we detail in the next section.

Implications for the effect of business tax change on local labor demand. Both of our points (cost of capital and composition margin) also affect what SZ refer to as the "effect of business tax change on local labor demand":

∂ ln L D c ∂ ln(1-τ b c
) which can be written as:

∂ ln L D c ∂ ln(1 -τ b c ) = ∂ ln E c ∂ ln(1 -τ b c ) -1 = δ σ F - 1 (ε PD + 1)σ F -1, ( 13
)
where we used the definition of ρ c = ρ/(1τ b c ), when differentiating equation ( 8). The analogous equation in SZ writes as:

(9 SZ) ∂ ln L D c ∂ ln(1 -τ b c ) = ∂ ln E c ∂ ln(1 -τ b c ) = - 1 (ε PD + 1)σ F .
There are two main differences between (9"SZ) and ( 13). First, the direct effect of 1τ b c on the cost of capital ρ c implies that the expression for the sensitivity of location choice depends on capital intensity ( δ σ F , a point we made above).9 Second, because z c is itself a function of 1τ b c , the impact of the net-of-tax-rate on labor demand is not the same as its impact on the extensive margin (thus the -1 term).

Incidence on workers and business-owners

The approach developed by SZ to calculate the incidence of corporate taxation on firm owners, workers, and landowners respectively consists of three steps:

1. Establish, using their spatial equilibrium model, that the incidence on each of these three groups can be expressed as a function of the reduced-form effects of local corporate tax on four observables.

2. Estimate the empirical counterparts of the reduced-form effects using changes in state-level corporate taxation.

3. Plug in these estimates, along with calibrated values for two structural parameters and proceed to compute incidence.

We show in this section that the differences highlighted in the previous sections have implications for step (1), with consequences for the conclusions derived in step (3), even if the empirical estimates obtained in step (2) remain unchanged.

In this section, we use the same notation as SZ, and denote the total elasticity of any local variable x c with respect to changes in (1

-τ b c ) as ẋc ≡ d ln x c d ln(1-τ b c )
. SZ refer to ẋc as the "reduced-form effect" of 1τ b c on x c : it represents the total impact of an exogenous shift in the tax rate through both its direct effect and implied changes in other endogenous variables of the model.

The total wage effect of changes in business tax

We start by deriving the expression for the incidence falling on wages, which is central in the computation of incidence more generally, and also for highlighting the implications of the two omissions in SZ.

The equilibrium change in wages that follows a change in local taxes comes from the labor market clearing condition N c = L D c . Denoting the labor supply elasticity as ε LS ≡ ∂ ln N c ∂ ln w c , market clearing implies that Ṅc = ε LS ẇc = LD c . Combining with the expression for labor demand in c, L D c = E c l i,c z c , we obtain :

ε LS ẇc = Ėc + li,c + żc , with (14)
Ėc = ∂ ln E c ∂ ln(1 -τ b c ) + ∂ ln E c ∂ ln w c ẇc = 1 σ F δ + 1 -(1 + ε PD ) - γ σ F ẇc , ( 15
) li,c = (γ(1 + ε PD ) -1) ẇc -δ(1 + ε PD ), and (16)
żc = (1 + ε PD )σ F Ėc = δ(1 + ε PD ) -1 -γ(1 + ε PD ) ẇc , ( 17
)
where the computation of li,c comes from equation (6).

Equations ( 14) to ( 17) allow to solve for the equilibrium value of ẇc :

10 ẇc = δ σ F -1 - 1 (ε PD +1)σ F ε LS + γ σ F + 1 = δ σ F -1 + µ-1 σ F ε LS + γ σ F + 1 , ( 19
)
with µ used by SZ as a notation for the markup over marginal cost, that is µ = -ε PD -ε PD -1 . SZ do not find the same formula for ẇc . There are two reasons for the discrepancy, that are related to the differences (regarding partial effects of taxes) highlighted in section 3.2. First, the numerator is different due to the role of the cost of capital. As can be seen from equation (9'SZ, page 2592 of their paper) reproduced above, the authors omitted the δ σ F term in ∂ ln E c ∂ ln(1-τ b c ) . Moreover, as revealed by the comparison of equations (9"SZ) and ( 13), the term -1 (reflecting substitution from labor to capital due to lower capital cost) is missing from the numerator. Second, and more important, SZ's derivations of labor demand (implicitly) imply that żc = 0 in the decomposition 10 Another (related) way to derive the same result is to write the total elasticity of labor demand (which depends on both the tax rate and the local wages) as LD

c = ∂ ln L D c ∂ ln(1-τ b c ) + ∂ ln L D c ∂ ln w c ẇc . Combining with Ṅc = ε LS ẇc = LD c , one can then solve for ẇc as ẇc = ∂ ln L D c ∂ ln(1-τ b c ) ε LS -∂ ln L D c ∂ ln w c , ( 18
)
which is the traditional incidence formula. Replacing ( 14), which alters the term in the denominator. Their equilibrium 11 value for ẇc is displayed in (10SZ), that we reproduce here: 12

(10SZ) ẇc = ∂ ln L D c ∂ ln(1-τ b c ) ε LS -ε LD = (µ-1) σ F ε LS -γ ε PD + 1 -1 σ F + 1 .
As a final remark regarding how taxes affect wages in this model, we can investigate the parameter values' restrictions for the effect to be positive (consistent with empirical reduced-form results of SZ). Note first that the denominator in equation ( 19) is positive for any acceptable value of the parameters. For the numerator to be positive, we need some additional condition regarding the link between elasticity of demand ε PD , the dispersion of productivity σ F and the output elasticity of capital δ:

δ σ F -1 - 1 (ε PD + 1)σ F > 0 ⇒ σ F < b ≡ δ - 1 ε PD + 1
The baseline choice of SZ of ε PD = 2.5, γ/δ = 0.9 and γ = 0.15 implies a upper a bound on σ F of b ≈ 0.82. 13 A product elasticity of -4 to -5 still allows firms to be fairly heterogeneous in their valuation of locations.

Identification of the incidence on profit

The incidence of changes in business taxes on the three types of agents in the SZ model depends on the impact on real wages, housing rental rates, and profits. The 11 SZ therefore have the following (incomplete) formulation for the labor market clearing:

ε LS ẇc = - 1 (ε PD + 1)σ F - γ σ F ẇc Ėc + (γ(ε PD + 1) -1) ẇc li,c = - 1 (ε PD + 1)σ F + ẇc × γ(ε PD + 1) -1 - γ σ F .
12 In order to recover exactly their equation, one should use the fact that ε LS = 1+η c -α σ W (1+η c )+α . 13 Reassuringly, this is way in excess of the estimates displayed in SZ's Table 6, although one must keep in mind that the interpretation of the results from the structural implementation might be impacted by the compositional margin as well as the role of capital intensity δ in the extensive margin.

change in welfare for each agent, written as a vector I, is:

Workers: Landlords: Business owners:

= I ≡    ẇ -α ṙ ṙ π    . ( 20
)
The authors obtain empirical estimates regarding the reduced-form effects on four endogenous variables: β W (wage growth), β N (population growth), β R (rental cost growth), and β E (establishment growth):14

β Business Tax =       β W β N β R β E       =       ẇ Ṅ ṙ Ė       . ( 21
)
The authors then proceed to express changes in welfare contained in equation ( 20) as functions of the elements of β Business Tax combined with calibrated parameters (α, δ/γ). The presence of the compositional margin has direct implications for this last step, that is for the identification of local incidence (presented in Table 1 of SZ), as well as for some of the structural parameters. We now proceed to show what are the implied changes.

Incidence on land owners and workers. Backing out changes in the welfare of landowners and workers from reduced-form effects of business taxes on wage and rental rate is fairly direct: The impact on landowners is equal to ṙc and can therefore be directly retrieved from the reduced-form effects of the local impact of corporate tax cuts on rents (denoted as β R ). The change in the welfare of workers will depend on wage and rental rate as well as the weight of housing in the utility function (α) and writes as ẇcα ṙc . Assigning a value to α, one can therefore deduct the change in workers' welfare from β Wαβ R .

Incidence on firm owners. Firm owners' change in welfare-as measured by the change in the non-stochastic component of profits denoted πc -involves several structural parameters on top the total effect on local wages ẇc . Using the expression for equilibrium profits (3), πc can be written as:

πc = 1 + γ ε PD + 1 ( ẇc -δ/γ) (22) = 1 mechanical + γ(1 + ε PD ) ẇ higher labor cost -δ(1 + ε PD ) lower cost of capital . ( 23
)
Equation ( 23) provides intuition regarding the determinants of incidence on firm owners by decomposing the impact of corporate taxes on profits into 3 components. The first component is mechanical: a lower corporate tax rate, in the absence of any changes in equilibrium variables, increases after tax profits with an elasticity of 1 (with respect to the net-of-tax rate). The second component relates to the equilibrium impact of taxes on wages and how in turn this impacts profits. Quite intuitively one can see that higher wage will erode profits more when demand is price elastic (very negative ε PD ) or when technology is labor intensive (high γ). Finally, cuts in corporate taxes lower the effective cost of capital ρ c = ρ/(1τ b c ) which benefits business owners all the more that technology is capital intensive (high δ) and demand is elastic.

Identifying the incidence on firm-owners from reduced-form effects. As can be seen from equation ( 22), even if one assigns a value to the ratio δ/γ, information on (1 + ε PD )γ is still required in order to deduct πc from ẇc . SZ show that they can retrieve (1 + ε PD )γ without making any further assumption (notably on σ F ).

Their argument starts from the theoretical expression for ẇ in equation (10SZ) and expresses it as a function of the reduced-form effects in β Business Tax . To do so, it uses the expression of Ėc in their own version of equation ( 15)-which omits δ σ F -and the fact that Ṅ = ε LS ẇ ⇒ ε LS = β N /β W . This results in equation (18SZ) which writes as: 15 (18SZ)

β W = β E + γ σ F β W β N β W + γ σ F + 1 -γ ε PD + 1 -ε LD from (9SZ)
, and enables to recover γ(ε PD + 1), which is needed to compute the incidence of busi- 15 The SZ-equivalent version of equation ( 15) writes as Ėc =

1 -σ F (ε PD +1) -ẇc γ σ F and implies for the numerator of (10SZ) that 1 -σ F (ε PD +1) = µ-1 σ F = β E + γ σ F β W .
Combining this with the fact that ε LS = β N /β W in the denominator of (10SZ) yields (18SZ).

ness tax on business owners. 16 However, when starting from the corrected version of ẇ in (19), and using the corrected version of Ėc obtained from equation ( 15), one ends up with a different version of (18SZ):

β W = β E + γ σ F β W -1 β N β W + γ σ F + 1 -ε LD from (11) , ( 24
)
in which the parameter ε PD does not appear anymore. Consequently, the term γ(1 + ε PD ) cannot be directly identified by inverting this equation. The compositional margin implies that ε LD is not informative about ε PD , and hence that the incidence shares falling on each category of agents cannot be uniquely identified from the vector of β Business Tax . 17

We see two potential solutions to this issue:

The first one is to expand the vector of reduced-form effects β Business Tax in order to obtain an estimate of the intensive margin of labor demand on the set of incumbent firms-in the spirit of Head and Mayer (2014)'s suggestion regarding firm-level empirical analysis of exports. If one is willing to assume that idiosyncratic productivity shocks are constant over time, looking at the evolution of labor demand among a set of incumbent firms-i.e. holding ζ jc fixed in equation ( 6)-reveals the equivalent of li,c . Equation ( 16) may be rewritten as li,c = γ(1 + ε PD )( ẇcδ/γ)ẇc , which makes it clear that an estimate of this intensive margin combined with β W enable to back-out γ(1 + ε PD ) under the same assumption as in SZ-i.e. calibrating the ratio δ/γ. This analysis would require to overcome two challenges: 1) the availability of firm-level data over multiple year, 2) circumventing the potential bias stemming from focusing on firms that decide to stay in the same locality in the face of changing taxes.

A second, more readily implementable solution is to make an assumption (i.e. calibrate) the values of γ and ε PD and proceed to what SZ call the "reduced-form implementation"-see their Table 3. This method is all the more attractive that most of the structural approach of SZ relies on the same set of assumptions. We present results applying this second option in the next section.

16 Manipulating (18SZ) yields γ(ε 17 One can furthermore note that equation ( 24) implies a linear constraint on reduced-form effects, namely:

PD + 1) = 1 + β N -β E β W , which once introduced in the profit incidence equation (22) yields πc = 1 + 1 + β N -β E β W (β W -δ/γ).
β N + β W -β E = -1.
Testing this restriction based on the β estimates presented in Table 4 of SZ leads to rejection in most instances-see Table A1 in the appendix.

New estimates regarding incidence shares

Accounting for the compositional margin implies that backing out incidence estimates on business owners from the reduced-form effects requires the calibration of two additional parameters: γ and ε PD . This change naturally impacts the incidence share estimates for all three types of agents. Table 1 presents the results. Column (1) reproduces the results presented in column 1 of Table 5 in SZ. Column (2) presents the incidence and incidence shares using calibrated values for γ and ε PD in equation ( 22), instead of SZ estimate of γ ε PD + 1 = 1.14. The calibrated values are taken from SZ's structural analysis: γ = 0.15 and ε PD = -2.5 (bottom panel of Table 3 in SZ). This has large implications for the welfare gains of firm owners, which are roughly halved in comparison with column (1). This implies that the share of incidence borne by firm owners goes from 42 to 28%, a 14 percentage points or 33 percent decline. As acknowledged by SZ on page 2065, a value of -2.5 for the product elasticity of demand is somewhat lower than what is usually found by the macro/trade literature, where consensual estimates tend to be closer to -4 or -5 (Head and Mayer, 2014). The results associated with such values for ε PD are displayed in columns (3) and ( 5). We see that it contributes to erode further the share of the tax cuts benefiting firm owners, although the differences remain more modest than the one between columns (1) and (2). Results are very similar when considering a specification which, as in column (5) of SZ's Table 5, further accounts for potential confounders by controlling for Bartik shocks and changes in the net-of-personal tax rate-see the estimates Table A3 of the appendix.

Implications regarding the structural implementation. SZ's structural implementation (Section VI of SZ) leads to estimates of the firm owners share of incidence of about 45% when considering business tax (column 4 of their Table 7). Revisiting this result is beyond the scope of this comment. However, it is worth noting that this approach relies on the expression of wage incidence ẇc as a function of structural parameters based on equation (10SZ) which ignores both the compositional margin and the effect of business tax on location choice through the local cost of capital-see our equation ( 19) for a formula incorporating both aspects. Moreover, the expression for Ėc presented in the last row of (17SZ) also ignores the local cost of capital impact (see the last row of equation ( 21) for the corrected version). As such, both the structural estimation procedure and the expressions for incidence, conditional on having the right structural estimates, are necessarily affected by our result.

For instance, SZ structural approach is forced to consider a fairly low product 1). Column (2) takes the original formula for the incidence on firm owners πc = 1 + γ ε PD + 1 ẇc -δ demand elasticity in their calibration in order to obtain macro local labor demand elasticites in line with estimates from the literature (around -1.5, see Hamermesh, 1996; Kline and Moretti, 2014a). Once corrected to account for the compositional margin, and assuming γ = 0.15, we have ε LD = -1 -0.15/σ F ≈ -1.5 for σ F ≈ 0.3 which is close to estimates shown in SZ's Table 6 panel A. Ignoring that margin overstates the influence played by a larger product elasticity on the macro elasticity of labor demand, which occurs only through its impact on the estimated value of σ F . Consequently, using the corrected formula for the macro labor demand elasticity might help to allow for larger response of product demand to price changes without running into counterfactual values regarding the local labor demand elasticity. Everything else equal, allowing for larger product elasticities (in absolute value) will tend to lower the share of the tax cuts accruing to firm owners.18

= 1 + β N -β E β W + 1 β W -δ γ (see SZ Table

Conclusion

In this comment, we revisit Suárez Serrato and Zidar (2016)'s seminal contribution on the local incidence of corporate taxation. We show that comparative statics regarding the impact of changes in the local corporate tax rate computed by SZ ignore two channels implied by their theoretical model: i) the impact of corporate taxes on location choice through the effective cost of capital and, ii) a "compositional margin", i.e. the fact that the average idiosyncratic productivity of firms is affected by changes in taxes, wages and all other local fundamentals. Accounting for the compositional margin is especially important as it changes the expression for the macro elasticity of labor demand (ε LD ) and causes it to be independent from the elasticity of demand on the product market (ε PD ). This change impacts how SZ reduced-form results can be used to identify how much corporate tax cuts benefit business owners-requiring either the estimation of additional reduced-form effects or the calibration of more parameters. Calibrating the parameters based on consensual values-including values used by SZ in part of their analysis-suggests that the share of the tax burden borne by firm owners hovers around 25% rather than 40%.

C Additional results and tables

C.1 Implications for ε PD of reduced-form estimates.

As acknowledged by the authors on page 2612, an additional issue with the use of the original equation (18SZ) for identification is that solving for γ(1 + ε PD ) yields the following equation (the second equation page 2599 of SZ):

γ ε PD + 1 = β N -β E β W + 1 . (A4)
Given the estimates presented in Table 4 of SZ, the ones used in the computation of the incidence in SZ's Table 5, yields a positive number as β N is consistently found to be larger than β E .

As SZ write: "Having determined the incidence on wages, the incidence on profits is straightforward; it combines the mechanical effects of lower corporate taxes and the impact of higher wages on production costs and scale decisions." Given that the mechanical effect of a change in the log of net-of-tax-rate is simply 1, it is natural to expect the sum of the mechanical effect and the impact of higher wage on profit to be lower than 1, i.e. πc < 1, as long as the change in wages ẇc is larger than the output elasticity ratio γ δ . Surprisingly, column (1) of SZ's Table 5 shows that the overall change in profits is higher than the mechanical effect, despite Table 4 showing that β W = 1.45 > γ δ = 0.9. This surprising result stems from using equation (A4) in order to identify γ ε PD + 1 .

The implication that γ ε PD + 1 > 0 is at odds with the assumption that the product demand elasticity is below -1 (see page 2588). The assumption that ε PD < -1 is necessary for monopolistic competition to admit a solution with positive prices. Therefore, when ignoring the compositional margin, interpreting the reduced-form results through the theoretical formula for local labor elasticity leads to an incompatibility. In Table A2, we list the values of structural parameters implied by the reduced-form results based on SZ's formulas (reported in the last row of their Table 1). We see that estimates for parameters pertaining to the labor demand side of the economy (ε PD , σ F ) display the wrong sign. On the contrary, following the baseline calibration of Table 3, ε PD = -2.5 and γ = 0.15, and applying the corrected formula for σ F , we obtain consistently positive values.

Hence, accounting for the compositional margin loses the identification of the term γ ε PD + 1 but also bypasses the resulting incompatibility between the reduced-form results and the theoretical model.

C.2 Structural form of the model

Here we specify the differences with respect to SZ regarding the structural form of the model. Equilibrium changes in wages, population, rents and number of establishments are stacked in vector Y c,t while changes in taxes are stacked in Z c,t .

Y c,t =       ∆ ln w c,t ∆ ln N c,t ∆ ln r c,t ∆ ln E c,t       , Z c,t = [1 -τ b c,t ].
Denoting e c,t a structural error term, we obtain what SZ refer to as the "structural form": AY c,t = BZ c,t + e c,t where :

A =       -1 σ W 1 α σ W 0 1 -1 ε LD 0 0 -1 1+η -1 1+η 1 0 γ σ F 0 0 1       , B =       0 1 ε LD σ F (ε PD +1) 0 δ σ F + 1 -σ F (ε PD +1)       (A5)
We highlight in blue the terms that are different with respect to SZ. ε LD is included in SZ initial derivation but its expression as a function of structural parameters should follow (11) as a opposed to (9SZ). δ σ F was omitted from the expression.

C.3 Tables

  ε PD and γ based on the baseline values chosen by SZ (see SZ Table 3, Panel: Additional parameters for structural implementation). Columns (3) to (4) experiment with higher value of |ε PD |. Calibration of the housing cost share and γ/δ follows SZ baseline choice. *** p < 0.01, ** p < 0.05, * p < 0.1.

Figure A1 :

 A1 Figure A1: Scatter of ln E c against v c

Figure A2 :

 A2 Figure A2: Scatter of ln z c against ln E c

Figure A3 :

 A3 Figure A3: Scatter of ln z c against v c

Table 1 :

 1 Revisiting Estimates of Economic Incidence Using Reduced-Form Effects SZ Table 5 col.1 SZ BL param. ε PD = -4 ε PD = -5 Notes: This table shows the estimates of the economic incidence expressions. Results are produced based on the coefficients from specification (1) displayed in Table 4 in SZ unless otherwise specified. Regressions use population as weights (see SZ Table 5 notes for more details). Standard errors clustered by state are in parentheses. Column (1) reproduces the results from Table 5 column (1) of SZ-which is based on SZ formula for business owners: πc

	(1)	(2)	(3)	(4)

Table A1 :

 A1 Tests of model-based restrictions on reduced-form estimates. Reduced form estimates from: Table 4 SZ column 1 Table 4 SZ column 5 Table 4 SZ column 6 R= β N + β Wβ E + 1

		2.65	2.30	1.55
	χ 2 : R = 0	4.98	3.63	9.89
	p-value: R = 0	0.03	0.06	0.00

Notes: This table shows nonlinear test implied by equation (

24

).

Table A2 :

 A2 Implications of reduced-form estimates for structural parameters under SZ formulasReduced form estimates from: Table 4 SZ column 1 Table 4 SZ column 5 Table 4 SZ column 6 Preference Dispersion σ W This table shows the estimates of structural parameters based on the formulas provided in the last row of Table 1 of SZ. The different columns show different values which correspond to different empirical specifications displayed in Table 4 of SZ. PD , the formula used in this table-which come from Table 1 last row of SZ-do not necessarily match the equation (18SZ) in section III.B from which it derives. Equation (18SZ) implies:ε PD = β N +(1-γ)β W -β E γβ W. Instead, Table1last row expresses ε PD as: The formula for σ F in this line is based on the corrected version of the total elasticity of establishment growth to local business tax (see equation 15 ) which implies: σ F = δ-(1+ε) -1 -γβ W PD ) calibrated as in baseline of Table3. *** p < 0.01, ** p < 0.05, * p < 0.1.

		.26	.64	1.12
		(.17)	(.98)	(1.71)
	Productivity Dispersion σ F	-.09*	-.23	-.44
		(.05)	(.21)	(.44)
	Housing Supply η	3.88	.64	1.09
		(5.24)	(1.1)	(1.15)
	Product Demand (ε PD ) a	7.59	5.66	4.8
		(6.25)	(4.76)	(3.15)
	Productivity Dispersion (σ F ) b ,	0.14*	0.33	0.93
	accounting for comp. margin	(0.08)	(0.35)	(0.87)
			β E	with parameters
	(δ/γ, γ, ε

Notes: a Note that regarding ε

β N +β W -β E γβ W which corresponds to ε PD + 1. b

Table A3 :

 A3 Revisiting Estimates of Economic Incidence Using Reduced-Form Effects: Based on Estimates of Specification (5) of Table 5 SZ Table 5 col.5 SZ BL param. ε PD = -4 ε PD = -5 Notes: This table shows the estimates of the economic incidence expressions. Results are produced based on the coefficients from specification (2) displayed in Table 4 in SZ unless otherwise specified. Regressions use population as weights (see SZ Table 5 notes for more details). Standard errors clustered by state are in parentheses. Results are produced based on the coefficients from the specification (5) of Table 4 in SZ unless otherwise specified. Column (1) reproduces the results from Table 5 Column (5) of SZ-which are based on SZ formula πc= 1 + β N -β E β W + 1 β Wδ γ (see SZTable 1). Column (2) takes the original formula for the incidence on firm owners πc = 1 + γ ε PD + 1 ẇc -δ γ and calibrate parameters ε PD and γ based on the baseline values chosen by SZ (see SZ Table 3, Panel: Additional parameters for structural implementation). Columns (3) to (4) experiment with higher value of ε PD . Calibration of the housing cost share and γ/δ follows SZ baseline choice. *** p < 0.01, ** p < 0.05, * p < 0.1.

			(1)	(2)	(3)	(4)
	Panel A. Incidence	Workers	.98	.98	.98	.98
			(.84)	(.84)	(.84)	(.84)
		Landowners	1.86	1.86	1.86	1.86
			(1.56)	(1.56)	(1.56)	(1.56)
		Firm owners	1.54*	.86***	.71	.62
			(.92)	(.25)	(.5)	(.67)
	Panel B. Incidence share	Workers	.22*	.26	.28	.28
			(.12)	(.17)	(.18)	(.2)
		Landowners	.42**	.5**	.52**	.54**
			(.17)	(.2)	(.22)	(.24)
		Firm owners	.35***	.23	.2	.18
			(.09)	(.18)	(.23)	(.26)
	χ 2 : Joint test S W = 1 and S F = 0		76.27	19.33	16.29	14.98
	P-value: Joint test S W = 1 and S F = 0		0.00	0.00	0.00	0.00
		PD		-2.5	-4	-5
		γ		0.15	0.15	0.15
		γ/δ	0.9	0.9	0.9	0.9
		Housing share α	0.3	0.3	0.3	0.3

In addition, ignoring the compositional margin creates inconsistencies between the theoretical framework and the empirical reduced-form results. The formula used by SZ to identify the incidence of business tax changes from reduced-form effects implies values for some structural parameters that are incompatible with a well-defined equilibrium-given the value of SZ's reduced-form point estimates (see appendix C.1).

We do not revisit the structural estimation implemented by SZ (Section VI) in our comment. However, given that this approach relies on a mis-specified formula for the elasticity of labor demand as well as for the partial elasticity of location choice to the net-of-tax rate (see equation (10) and associated comments below), it is likely that results presented in this note have consequences on this front as well. We detail some of these consequences at the end of Section 5.

[START_REF] Carlton | The location and employment choices of new firms: an econometric model with discrete and continuous endogenous variables[END_REF] uses this setup to study the joint decision of location choices and employment of plants in the USA.

For instance equation, (6SZ) will refer to the equation (6) of SZ.

The dependence of the cost of capital on the business net-of-tax rate derives from the tax treatment of the returns to equity holders, which is non-deductible against the business income tax, and the assumption that the marginal source of funds is equity. See SZ footnote 14.

Taking logs of profits π jc and dividing byε PD + 1 does not affect the discrete choice as it is a strictly monotonic transformation, see SZ footnote 16.

See in particular equation (3.15) in[START_REF] Hanemann | Discrete/continuous models of consumer demand[END_REF]. We adapt his derivation to our notations and overall setup in Section A of the appendix.

Recent examples where the demand parameter drops from the trade cost elasticity in firm-level sales equation derived in multinational production models appear in Tintelnot (2017) and[START_REF] Head | Brands in motion: How frictions shape multinational production[END_REF].

The effect on the cost of capital is included here even though we consider a partial derivative. This is because the local cost of capital depends directly on the business tax rate and not through a change in an endogenous variable (which should be only included in a total derivative).

We provide the corrected version of equation (17) of SZ (the vector representation of the solution for the changes in these 4 variables) in equation (A5) in our appendix section C.

For instance, in Panel C of their Table6, SZ present structural estimates for ε PD and find ε PD = -4.704. Taking all estimates from this estimation, the demand elasticity is -2.93 when ignoring the compositional margin, but only -2.38 when using the corrected labor elasticity. Accounting for the compositional margin, using the estimates from Panel C Table6, increases the workers' share of incidence, which goes from 0.25 to 0.296 and reduces that of firm owners which falls from 0.548 to 0.507. Again, accounting for the compositional margin at the estimation stage might impact the value of σ F , resulting in ultimately larger or smaller elasticity of labor demand and incidence shares for firm owners than when the misspecified expressions is been used in the structural estimation-the mispecification occurs through what SZ refer to as m(θ) in their classical minimum estimator.

* Malgouyres thanks ANR for financial support through grants ANR-17-EURE-0001 and ANR-19-CE26-0004. The views expressed herein are those of the authors and do not necessarily reflect those of the Banque de France or the Eurosystem.

Online Appendix Clément Malgouyres

Thierry Mayer Clément Mazet-Sonilhac

A Re-stating Hanemann (1984)'s result

In this section, we restate the result by Hanemann (1984) to make it match SZ conceptual framework and notations more directly.

Definition and setup. Consider a discrete choice by agent j involving c = 1, ..., C options. We denote V F c = v c + ζ jc the value of c for agent j where v c is a common value to all agents in the economy-i.e. the nonstochastic component of the value associated with choice c-and ζ jc is an idiosyncratic taste shock. For simplicity, we will omit the subscript j from now on. We define with A c the set of values of the vector ζ such that the option c yields to highest value to the agent, i.e.

E e tζ

Translation in SZ setting. Based on equation (A1), and on the definition of z c provided in equation ( 8) in the main text, we simply set t = -(1 + ε PD ) and denote the scale parameter σ F , to obtain the result presented in equation ( 9) in the body of the text and which we reproduce here:

B Simulation results

In this section, we provide simple simulation results illustrating the finding by Hanemann regarding the link between z c and E c .

We consider a set of location c = 1, ..., C where we set C = 50. We attribute a value v c to each location c which is defined as

There are N sim discrete choices operated overall. For each chooser n = 1, ..., N sim , we draw a vector e n of C values from an Extreme Value Type I distribution with scale parameter σ F . The sum of v c and the idiosyncratic shock e cn determines the value of location c: V cn = v c + e cn .

We collect two objects from each simulation: i) the chosen location based on c max = arg max c V c n ∀c = 1, ..., C}; and ii) the associated draw e c max n .

We compute the sample equivalent to E c and z c across our N sim choices:

(A3) We set the parameters of the simulation are as follows: C = 50, N sim = 100, 000,

We display the result of our simulation graphically. We start by showing that the relationship between ln E sim c and v c features the theoretical slope of 1/σ F as implied by the multinomial logit formula (see Figure A1).

Figure A2 confirms the negative relationship between ln z c and ln E c with a slope virtually identical to its theoretical value given the value of the parameters considered ((1 + ε PD )σ F = -0.30).

Finally, A3 confirms the negative relationship between ln z c and v c with a slope virtually identical to its theoretical value given the value of the parameters considered ((1 + ε PD ) = -1.5).


Y Kozhemyakin

S Rehault-Godbert
email: 4sophie.rehault-godbert@inrae.fr

D Dubuc #

K Grenier
email: 3grenier@laas.fr

Millifluidic Sensor Designed to Perform the Microwave Dielectric Spectroscopy of Biological Liquids

Keywords: microwave spectroscopy, millifluidic channel, coplanar waveguide, transmission coefficient, liquids

This paper presents a microwave sensor composed of one millifluidic channel that operates over a frequency range from 1 GHz to 4 GHz and is used for microwave spectroscopy of liquids with millilitre volumes. The accuracy of the device is first evaluated with BSA solutions of different concentrations and then with complex biological liquids such as hen egg yolk and albumen samples. Low standard deviations of the measured values of modulus and phase contrasts of the transmission coefficient S21 are obtained. These values correspond to 0.02 and 0,18 respectively for BSA solutions, as well as 0.05 and 0.03 for 10 samples of hen egg yolk and white respectively over the selected frequency range. Such accuracies enable the use of such a sensor for various applicative domains, including agronomy for which the investigation of the dielectric properties of eggs may be of interest for quality, freshness or status discrimination purpose.

I. INTRODUCTION

Microwave dielectric spectroscopy is an analyzing technique able to characterize solids, liquids and gaz based on their dielectric properties. This analyzing method presents the main advantage of being non-invasive and quick. It is well used in different domains, including the control of processes [START_REF] Wellock | Applications of microwave spectroscopy in process analysis[END_REF], such as the evaluation of the quality of water, food, animal feed and others [START_REF] Kupfer | Elektromagnetische Aquametrie[END_REF], [START_REF] Zhang | Determination of quality parameters of tomato paste using guided microwave spectroscopy[END_REF], [START_REF] Cora Jofre | Infrared assisted digestion used as a simple green sample preparation method for nutrient analysis of animal feed by microwave induced plasma atomic emission spectrometry[END_REF]. Not so long ago, microwave spectroscopy has broaden its application field with the study of intermolecular interactions [START_REF] Tang | Advances and Applications of Microwave Spectroscopy[END_REF], the properties of biological tissues [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF], including the discrimination of malignant, benign and healthy tissues [START_REF] Lazebnik | A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries[END_REF] and finally at the microscale the investigation of individual cells [START_REF] Artis | Microwaving Biological Cells: Intracellular Analysis with Microwave Dielectric Spectroscopy[END_REF], [START_REF] Grenier | Low Volume and Label-Free Molecules Characterization and Cell Monitoring with Microwave Dielectric Spectroscopy[END_REF]. Therefore, microwave dielectric spectroscopy is of great interest for the quantitative study of the composition of liquids, in various volumes conditions. Developments have already been performed at the macroscale with the use notably of large waveguide configurations, resonators, coaxial probes, as well as at the microscale with miniature microwave sensor configurations. There is however a lack of solution for the intermediate level, which involves millifluidic samples, whereas an increasing demand for the development of microwave spectroscopy devices for such a size exists at an affordable cost. Previously, a microwave sensor designed with two millifluidic channel has been developed [START_REF] Gonzalez | Millifluidic Sensor Dedicated to the Microwave Dielectric Spectroscopy of Liquids[END_REF], based on the Bianco and Parodi method for permittivity extraction. Even if this device is efficient in the characterization of millifluidic liquids, the presence of the two fluidic channel complicates the use of this sensor as the two channels require to be filled. It is also liquid-consuming. This paper therefore considers a simplified microwave device, which consists of a coplanar waveguide with only a single millifluidic channel.

Regarding its applicative context, which is related to precision farming, there is always a demand for devices that could facilitate a strict quality control of the products produced and delivered to the end consumer. Methods for checking the freshness of fruit, eggs and vegetables using microwave spectrometry have already been shown in [START_REF] Nelson | Dielectric spectroscopy of watermelons for quality sensing[END_REF], [START_REF] Trabelsi | Sensing Egg Quality during Storage by Radiofrequency Complex Permittivity Measurement[END_REF], [START_REF] Ates | The freshness analysis of an apple and a potato using dielectric properties at the microwave frequency region[END_REF]. In addition, methods for determining the moisture content of cereal grains and meat have also been described [START_REF] Nelson | Using cereal grain permittivity for quality sensing by moisture determination[END_REF], [START_REF] Ji | Study of water-injected pork and pork freshness using microwave techniques[END_REF]. These investigations demonstrate the high potential of microwave dielectric spectroscopy for studying the qualitative and even quantitative composition and condition of various materials ranging from chemistry to biology, including product quality controls.

In the present paper the construction of a millifluidic sensor with a single fluidic channel designed for microwave dielectric spectroscopy is first discussed. Several reasons of changing the microwave sensor with two millifluidic channels to the one with one millifluidic channel are given. In a next section, a description of the experiments performed from 1 GHz to 4 GHz for bovine serum albumin (BSA) solutions of different concentrations ranging from 5 ml/mg to 100 ml/mg are evaluated. Next section is dedicated to the investigation of more complex and sticky biological materials, which correspond to hen egg yolk and albumen, carried out with the developed microwave sensor.

II. ARCHITECTURE AND FABRICATION OF THE MILLIFLUIDIC MICROWAVE SENSOR ELABORATED WITH A

SINGLE FLUIDIC CHANNEL In [START_REF] Gonzalez | Millifluidic Sensor Dedicated to the Microwave Dielectric Spectroscopy of Liquids[END_REF], a microwave sensor with two millifluidic channels is considered to characterize the permittivity of several liquids. In the present paper, the simplified microwave sensor is presented with the schematic shown in Fig. 1.

The operating frequency band of the sensor is from 1 GHz to 4 GHz. The microwave structure is fabricated on the Rogers RO2003 substrate with a thickness ℎ = 0.8 mm, dielectric constant 𝜀 ' = 4.3 and dissipation factor 𝑡𝑎𝑛𝛿 = 0.003.

The device consists of a coplanar line with a central conductor width of 𝑤 = 3.2 mm and a slot of 𝑠 = 0.2 mm. The line length L is equal to 56 mm. These dimensions have been defined to match with a 50Ω characteristic impedance in air. On top of the substrate, a drilled Plexiglas block is used to realize the millifluidic channel. It presents a length 𝑙 = 26 mm long and has an approximate volume 𝑉 = 0.8 mL. The Plexiglas block containing the millifluidic channel and two holes allowing the liquid injection on one side and the fluid output on the other side provides a precise positioning of the liquid under consideration on the coplanar line. In addition, the tight fit of the block to the metal substrate prevents any leakage of the liquid under test outside the channel. For this purpose, the Plexiglas is fastened with four lateral plastic screws to a second Plexiglas substrate located below the Rogers one, thereby creating the necessary pressure on the metal substrate and ensuring a tight fit, as shown in Fig. 2.

In operation, the new microwave sensor, which integrates a single millilitre channel, has proven to be practical and easy to use. This fact is the consequence of three main advantages compared to the microwave sensor, which includes two milliliter channels as described in [START_REF] Gonzalez | Millifluidic Sensor Dedicated to the Microwave Dielectric Spectroscopy of Liquids[END_REF]:

• The smaller required volume of the liquid to be tested.

As a result of the fact that the device consists of a single millilitre fluidic channel, the required minimum volume of the liquid to be tested is reduced by more than 35% compared to the device in [START_REF] Gonzalez | Millifluidic Sensor Dedicated to the Microwave Dielectric Spectroscopy of Liquids[END_REF],

• Increased simplicity and ease of operation. The microwave sensor with a single millilitre fluidic channel is a device that relies on proper installation, calibration of the device, as well as smooth and even filling of the channel with the liquid under test. These steps are simplified compared to the microwave sensor with two millifluidic channels, • Cleaning. Insufficient cleaning of the millifluidic channel leads to additional measurement errors, which have a direct effect on the overall result of the experiments. Switching from microwave sensor with two millilitre fluidic channels to microwave sensor with one channel ensures that the probability of experimental error due to insufficient cleaning of working volume of the channel is lower. Experimental analysis of the liquids under test is made with a measuring unit consisting of a measuring and a calculation block, a vector network analyser (VNA) Copper Mountain S5085 and the microwave millifluidic sensor (Fig. 2).

In the next section the results of measurements on BSA solutions of different concentrations from 5 mg/ml up to 100 mg/ml as well as on egg yolks and albumens with the microwave millifluidic sensor will be presented.

III. TWO CASE STUDIES FOR EVALUATING THE CAPABILITIES OF THE MICROWAVE MILLIFLUIDIC SENSOR

A. Microwave measurements of solutions of BSA with different concentrations

The first sensitivity studying of microwave millifluidic sensor is carried out with 4 BSA solutions of different concentrations: 5, 10, 50 and 100 mg/ml. Each solution is measured 5 times, starting with the lowest concentration. Deionized water is also measured between each BSA solutions as a reference.

In the BSA experiments, the modulus and phase of the S21 parameter are measured. Since the liquids present increased viscosity and adhesiveness, a special measurement protocol is developed for their experimental analysis with the microwave millifluidic sensor. Modulus and phase of the transmission coefficient are calculated using the formulae (1) and ( 2):

𝛥|𝑆 78 | = |𝑆 78 | :;<=;> -|𝑆 78 | @ABC' (1)
𝛥𝜑 E FG = 𝜑 E FG :;<=;> -𝜑 E FG @ABC' .

(

) 2
The obtained average values for the differences of modulus and phase contrasts for S21 are presented on fig. 3 and fig. 4 respectively. Fig. 5 shows these values depending on the concentration of the BSA solutions. The 2,7 GHz and 4 GHz frequencies are selected for the modulus and phase contrasts respectively. They have been chosen on the basis of a search for a frequency with a maximum modulus and phase contrasts values over the overall frequency range. The measurements enable to distinguish the Fig. 2. Photographies of the millifluidic and microwave device, seen from the side (in the left) and from above (in the right), with the coplanar waveguide, the millifluidic channel, SMA connecters, Plexiglas block and the inlet and the outlet hole and pipe and an experimental test setup with the microwave millifluidic sensor Fig. 1. Schematic of the simplified millifluidic and microwave sensor with two ports with dimentions. The coplanar waveguide is represente in yellow, whereas a plexiglas fluidic channel is placed on top. Fig. 3. The average values of differences of modulus contrasts of measured S21 for the BSA solutions for different concentrations, with marked standard deviations Fig. 4. The average values of differences of phase contrasts of measured S21 for the BSA solutions for different concentrations, with marked standard deviations Fig. 5. Measured standard deviation and average modulus contrasts of S21 for each BSA solution versus concentration, at 2,7 GHz (in blue) and measured standard deviation and average phase of S21 for each BSA solution versus concentration, at 4 GHz (in orange) solutions of BSA with different concentrations. Calculated average values and the standard deviations σ of measured modulus and phase contrasts of S21 for every BSA solution are shown in Table 1. It is possible to observe the σ -values for the modulus contrasts of S21 of the same order for all the BSA solutions. For phases, the standard deviation is increasing with respect to an increase of the BSA solution concentration, taking the maximum value as 100 mg/ml. This fact may be explained by a pollution of millifluidic channel because of higher viscosity of the liquid at high concentration. Despite a definite dispersion of values of σ for the modulus and the phase contrasts of S21 on examined frequencies, these standard deviations may be considered acceptable and applicable specially in the field of rapid, omnipresent and low-cost analyses.

B. Microwave measurements of hen egg yolk and albumen

This section presents the results of microwave measurements of the yolk and albumen of fertilized chicken eggs. Such a study can be applied in the agro-industrial sector in order to accumulate and process information on the quality and condition of the liquids that constitute the structure of a hen egg.

In order to quantify the degree of repeatability of measurements made with a microwave millifluidic sensor, yolk and albumen samples from two different eggs are examined and measured 10 times each in sequence. The measurements are then made for 9 yolks samples and 9 albumen samples of different hen eggs measured 1 time each to estimate the level of a reproducibility of experiments. This approach allows a qualitative evaluation of the microwave measurements of the tested liquids according to (1) and ( 2).

The choice of frequencies for the presentation of the modulus and the phases contrasts of the S21 coefficient is due to the maximum values of these quantities at the frequencies in question. 3 GHz and 4 GHz are chosen for calculations for the modules and phases respectively. Fig. 6 shows the dependencies of the measured modulus contrasts of the coefficient of transmission on its phase contrasts for 10 consecutive repetitive measurements of yolk and albumen of two fertilized hen eggs at 3 GHz. Fig. 6. Complex graph of S21 modulus on phase contrasts of 10 measurements for the yolk and the albumen of hen eggs with marked standard deviations at 3 GHz Table 2 presents the measured and calculated quantities of the average values and the standard deviations of the modulus and the phase contrasts of S21 for the chosen frequencies. The standard deviations σ for the yolk and the albumen are 0,05 and 0,03 respectively. The difference with the measured values of S21 can be attributed to the contamination of the millifluidic channel during successive measurements. This created a negative effect on the accuracy of the experiment and the values obtained when measuring the same liquid. The egg constituents are indeed particularly sticky compare to Δφ, °F req., GHz 5 mg/ml 10 mg/ml 50 mg/ml 100 mg/ml 0 50 100 BSA samples. Nevertheless, achieved values of σ are of the same order than those obtained in [START_REF] Gonzalez | Millifluidic Sensor Dedicated to the Microwave Dielectric Spectroscopy of Liquids[END_REF].

-0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 -5 0
The fig. 7 shows the results of measurements of S21 in terms of reproducibility of measurements at 3 GHz for 9 yolks and albumens samples from different fertilized hen eggs. Fig. 7. Complex graph of S21 modulus on phase contrasts of 9 measurements for the yolks and the albumens extracted from 9 different fertilized hen eggs with marked standard deviations at 3 GHz In Table 3 the obtained quantities of average values and standard deviations for the modulus and the phase contrasts of S21 at 3 GHz and at 4 GHz respectively are presented. The variation the values of σ presented and those obtained in the previous case may be due, among other things, to differences in the chemical and biological composition of the yolks and albumens extracted from diverse hen eggs. IV. CONCLUSIONS A millifluidic microwave sensor operating in the frequency range from 1 GHz to 4 GHz is presented. The device features a single millifluidic channel to allow the small volume of liquid to be studied to interact with the electromagnetic waves over a wide frequency range.

BSA solutions of different concentrations were measured to determine the sensitivity of the microwave sensor. A small value for the standard deviation of 0.02 and 0.18 for the modulus and the phase contrasts of the transmission coefficient S21 respectively were obtained. The found results of the standard deviation for the modulus contrasts of S21 correspond to that expected in the modeling and development of the considered microwave sensor and present the same order as the values of a standard deviation obtained in [START_REF] Gonzalez | Millifluidic Sensor Dedicated to the Microwave Dielectric Spectroscopy of Liquids[END_REF].

Further work with the device was carried out with samples of yolks and albumens of fertilized hen eggs in the terms of the repetitiveness and the reproducibility of the measurements. The corresponding values of the calculated standard deviations for the modules of the coefficient of transmission S21 were for the yolks 0.05 and 0.07 and for the albumens 0.03 and 0.08 respectively. results were used for the validation of the microwave sensor and methodology of studying complex liquids in millilitre volumes, which can be used in a variety of applications. This opens the door to the development of systems suitable for quality assessment and control of products in the agro-industrial sector.

Table 1 .

 1 Average values and dispersion of the module and phase contrasts of measured and referenced to water ΔS21 for the solutions of BSA with different concentrations at 2,7 GHz for the modulus contrasts and 4 GHz for the phase contrasts calculated for 5 measurements for each solution

	BSA	Mean	σ |ΔS21|	Mean	σ Δφ
	Concentration	|ΔS21|, dB		Δφ, °
	5 mg/ml	-0,06	0,04	1,08	0,18
	10 mg/ml	-0,01	0,04	1,67	0,37
	50 mg/ml	0,22	0,02	7,66	1,6
	100 mg/ml	0,45	0,04	13,4	2,7

Table 2 .

 2 Average values and standard deviations of the modules contrasts at 3 GHz and phase contrasts at 4 GHz referenced to water Δ|S21| for the yolk and the albumen measured 10 times in terms of the repeatability of experiments

	Liquid	Mean	σ |ΔS21|	Mean	σ Δφ
		|ΔS21|, dB		Δφ, °
	Yolk	-0,5	0,05	-12,6	1,8
	Albumen	0,2	0,03	-4,3	1,5

Table 3 .

 3 Average values and standard deviations of the modules contrasts at 3 GHz and phase contrasts at 4 GHz referenced to water ΔS21 for 9 yolks and 9 albumens of different eggs measured 1 time in terms of the reproducibility of experiments

	Liquid	Mean	σ |ΔS21|	Mean	σ Δφ
		|ΔS21|, dB		Δφ, °
	Yolk	0,02	0,07	41,7	2,7
	Albumen	-0,9	0,08	21,7	0,8

ACKNOWLEDGMENTS

These experiments were performed under the framework of the PPILOW project that has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N°816172.