A. Heterogeneity in Goods and Production

Using Input-Output accounts data from NIPA provided by the BEA, we first construct three different metrics for the standard 15 commodities classification⁴: (i) the share of each commodity in US domestic demand (or absorption) (ii) the tradability of each commodity and (iii) the labor intensity of each commodity.

In practice, we compute the share of private (and public) consumption and investment of commodity i as a share of total US absorption using the 2015 “use table” after redefinitions and evaluated at producer prices.⁵ Using the same table, we approximate the “tradability” of each commodity by computing the ratio of exports and imports of commodity i to the final US demand of commodity i. Finally, we assess the labor intensity of each commodity using the network-adjusted labor share of each commodity (thereafter NALI). Traditionally, measures of sectoral labor intensity are proxied at the industry level by taking the share of output that is devoted to employees’ compensation. Although spending a dollar on commodity i does stimulate the industry i, such spending also stimulates other industries along the supply chain to a significant extent. Thus, using the standard industry-level labor intensity of industry i as a proxy for the labor intensity of commodity i would be misleading. Intuitively, the NALI accounts for that fact and captures the labor compensation along the whole supply chain. Formally, the NALI of each commodity i is computed as follows:

Where L is the vector of labor compensation as a fraction of output by industry, and w is the industry-by-commodity domestic requirement matrix, which reports the domestic inputs by industry required (directly and indirectly) to deliver one dollar of commodity i output to final users.⁶

Two stylized facts emerge from Figure 1, which reports the three statistics for the 15 commodities. First, the heterogeneity in labor intensity across commodities is significant. Primary products (i.e. agriculture, mining and utilities) and manufactured goods are generally produced with much less labor than services, even after accounting for interlinkages.⁷ On average, out of a dollar spent on such goods in 2015, only 35 cents ended up going to labor, against roughly 60 cents for a dollar spent on services. Second, except for transportation and warehousing services, primary and manufactured goods are much more traded than services.

Tradability, Labor Intensity, and Share of U.S. Final Demand for Commodities

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Tradability, Labor Intensity, and Share of U.S. Final Demand for Commodities

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Tradability, Labor Intensity, and Share of U.S. Final Demand for Commodities

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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A third key aspect we document is the heterogeneity in the type of labor - skilled vs. unskilled - used across commodities, especially across different types of services. To proxy for the skill level by industry, we compute the distribution of educational attainment within each sector using employment data from the latest IPUMS-CPS vintage, distinguishing between (i) high school diploma or less (ii) college or some college and (iii) above college.

Figure 2 shows clearly that although some services display similar aggregate labor intensities per the NALI criteria (e.g. professional and business services vs. recreation, accommodation and food services), they use very different skill levels. For instance, more than half of the labor force employed in construction, recreation, accommodation and food services has only reached a high school diploma (or less). At the other extreme, education, health care or finance sectors employ mostly workers with at least some college background.

Distribution of Educational Attainment, by Sector

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Distribution of Educational Attainment, by Sector

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Distribution of Educational Attainment, by Sector

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Putting these different findings together, this paper argues that the US economy is well represented by three “aggregate” commodities: (i) a highly tradable manufactured good, produced mostly with capital (which incorporates manufacturing, mining and agriculture); (ii) a non-tradable low-skill service (including retail trade, accommodation and food services etc..) and; (iii) a partly-tradable high skill service (including finance, healthcare, education). Table 1 in Appendix A provides detail on the aggregation method and shows the characteristics of the three aggregated sectors in terms of their share in total consumption and production; total employment or their NALI. All these metrics are used later to calibrate the model (Section III)

Table 1.

Preferences Parameters

Table 1.

Preferences Parameters

Parameter	Value	Target
β	0.96	$\frac{K}{Y}=3$
σ	2	Fixed: in the range of common values used for this parameter.
ρ	0.01	Elasticity of substitution between different types of consumptions being 1. $\frac{1}{1-\rho }=1$
γ	0.17	$\begin{array}{c}\frac{p^{L}L+(p^{S}S-m^{M,S}-p^{S}S{m}^{M,S})}{GDP}\\ =0.8\end{array}$
γN	0.45	$\frac{p^{L}L}{GDP}=0.8$
γT	0.4	$\frac{p^{*}{c}^{*}}{p^{c}c}=0.1$
$\overline{c^N}$	0.025	$\begin{array}{c}{\left(\frac{c^M}{p^{c}c}\right)}_{1stQuintile}=0.42\\ {\left(\frac{c^M}{p^{c}c}\right)}_{5thQuintile}=0.33\end{array}$
ω	12	Average hours $\frac{1}{3}$
θ	3	Elasticity of labor demand approx. $\frac{1}{3}$

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Table 1.

Preferences Parameters

Parameter	Value	Target
β	0.96	$\frac{K}{Y}=3$
σ	2	Fixed: in the range of common values used for this parameter.
ρ	0.01	Elasticity of substitution between different types of consumptions being 1. $\frac{1}{1-\rho }=1$
γ	0.17	$\begin{array}{c}\frac{p^{L}L+(p^{S}S-m^{M,S}-p^{S}S{m}^{M,S})}{GDP}\\ =0.8\end{array}$
γN	0.45	$\frac{p^{L}L}{GDP}=0.8$
γT	0.4	$\frac{p^{*}{c}^{*}}{p^{c}c}=0.1$
$\overline{c^N}$	0.025	$\begin{array}{c}{\left(\frac{c^M}{p^{c}c}\right)}_{1stQuintile}=0.42\\ {\left(\frac{c^M}{p^{c}c}\right)}_{5thQuintile}=0.33\end{array}$
ω	12	Average hours $\frac{1}{3}$
θ	3	Elasticity of labor demand approx. $\frac{1}{3}$

B. Heterogeneity in Consumption

How do consumers spend their income across such commodities? After aggregating commodities, we find that roughly 20% of US consumption (PCE) was devoted to what we define as primary and manufactured goods, 30% to low-skill services and 50% to high-skill services (see Appendix). This aggregate view, however, conceals an important heterogeneity across income levels.

To show this, we collect annual expenditures on goods and services, broken down by quintiles of income before taxes, from the 2015 Consumer Expenditure Survey (henceforth CEX). Although expenditure categories and accounting methodologies vary between the CEX and national accounts, we approximate the annual spending made by household in different quintiles on what we define as the “manufactured good”. In practice, we sum, for each quintile of income before tax, the average annual expenditure made on (i) food at home (ii) alcoholic beverages (iii) tobacco products and smoking supplies (iv) utilities (minus telephone services) (v) housekeeping supplies and household equipment (vi) apparel (vii) gasoline and motor oil (viii) car maintenance and repair (ix) drugs and medical supplies (x) audio and visual equipment (xi) toys, hobbies and playground equipment. The result is plotted in Figure 3. Overall, we find that the share of spending dedicated to primary and manufactured goods decreases rapidly as income increases, from roughly 35% at the bottom of the income distribution to 25% at the top.

Average Expenditure Share on Manufactured Goods

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Average Expenditure Share on Manufactured Goods

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Average Expenditure Share on Manufactured Goods

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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A. Production and Firms

We assume an economy that produces three goods: (i) a manufactured good, called M (ii) a low-skilled service, called L and (iii) a high-skilled service, called S. Although the manufactured good M is tradable (i.e. its price is determined exogenously on international markets and the country is a price-taker), we assume both types of services to be non-tradable. Production technologies can be summarized by constant returns to scale production functions satisfying standard assumptions. Markets are assumed to be competitive so that firms and individuals act as price takers and each factor commands its after tax marginal product. Different types of labor (h, x, l) will be assumed to deliver different effective levels of average productivity ηⁱ > 0, i = h, x, l.

B. Households

Household’s heterogeneity is driven by permanent and transitory differences in labor productivity: permanent differences result from households being born with different levels of skills -they are either low-skill (l), medium skill (x) or high skill (h)-, while transitory differences result from idiosyncratic shocks to household’s average deterministic productivity.⁹

Households of skill level i = l, x, h maximize their life time utility from consumption cⁱ and leisure oⁱ, accumulate assets aⁱ that give them a market return r, receive a market wage wⁱ for each unit of labor time 1 – oⁱ that they supply to the market, face a transitory idiosyncratic shocks to their labor productivity ζⁱ ∈ ℵⁱ, pay consumption (τ^s, τ^L, τ^M, τ*) and income taxes T(ζⁱwⁱ(1 – oⁱ), (1 + r)aⁱ), and are subject to an exogenous borrowing constraint given by –κⁱ with κⁱ > 0.

s.t

Preferences are identical across individuals and are represented by expected utility, where u is the Bernoulli utility function which satisfies standard assumptions (i.e., u_c > 0, u_o > 0, u_cc < 0, u_oo < 0, u_co > 0). The discount factor is 0< β < 1 and E₀ denotes the expectations operator conditional on information at period 0. Consumption cⁱ denotes a vector of 4 consumption goods: c^s,i is consumption of services with high-skill content, c^L,i is consumption of services with low-skill content, c^*,i is consumption of imported goods, and c^M,i is consumption of tradable goods (manufactured goods). The numeraire of the economy are manufactured goods M, and their price is normalized to 1.

The stochastic process for the idiosyncratic shock to household’s labor productivity ζⁱ is identical and independent across agents and follows a finite state Markov chain with stationary transitions over time, i.e.

The realization of these idiosyncratic shocks give rise to income and wealth distributions even though a set of households have similar average skills.

C. Government

The government in this economy consumes manufacturing goods G^M, high-skill services G^S, and low skill services G^L, invests in infrastructure I^G, levies taxes (τ^s, τ^L, τ^M, τ*, τ^M,M, τ^S,M, τ^M,S, τ^S,S, T(. ) ) and issues foreign debt B*. Its budget constraint is given by

s.t.

D. Stationary Competitive Equilibrium

In this economy households’ state variables are their skill type i, their asset holdings aⁱ, and their idiosyncratic labor productivity status ζⁱ. The aggregate state of the economy at any point in time is described by a joint measure ℶ_t over those individual states. Let aⁱ ∈ Aⁱ = [–κⁱ, ∞), ${\zeta }^i\in {\aleph }^i=\{{\zeta }_1^i,{\zeta }_2^i,\cdots ,{\zeta }_n^i\}$, A = A^l×A^x×A^h, ℵ = ℵ^l×ℵ^x×ℵ^h and Φ = A×ℵ. Let B(A) be the Borel σ-algebra of A and P(ℵ) the power set of ℵ. Let Ω be the set of all finite measures over the measurable space (Φ, B(A)×P(ℵ)).

Given sequences of the tax rates for final consumption τ^s, τ^L, τ^M, τ*, the tax rates for intermediate demand of inputs τ^M,M, τ^S,M, τ^M,S, τ^S,S, income tax function T(. ), government expenditure G^M, G^S, G^L, and I^G, government’s external borrowing B*, the international interest rate on government’s debt r*, the international prices for consumption goods p* (and manufacturing goods which is normalized to 1), a competitive equilibrium is a sequence of functions for the households {v, c, o, a′: Φ → R}, production plans for the firms in the manufacturing sector {l^M, x^M, h^M, k^M, m^M,M, m^S,M}, the high-skill services sector {x^S, h^S, k^S, m^M,S, m^S,S}, and the low skill services sector {l^L, x^L}, prices [w^l, w^x, w^h, p^L, p^S, r] and measures ϑ with ϑϵΩ such that:

1. Given prices, policies for each period vⁱ solves the functional equation (with cⁱ, oⁱ, a^′,i associated policy functions)

   $${\nu }^i(a^i,{\zeta }^i,i)=\underset{c^i,o^i,a^{\prime ,i}}{\max }\left\{u(c^i,o^i)+\beta {\displaystyle \int {\nu }^{\prime ,i}}\left(a^{\prime },{\zeta }^{\prime ,i},i\right)Q^i\left({\zeta }^i,d{\zeta }^{\prime ,i}\right)\right\}$$
   s.t

   $$\begin{array}{c}{p}^c{c}^i+a^{\prime ,i}={\zeta }^i{w}^i(1-o^i)+(1+r)a^i-T({\zeta }^i{w}^i(1-o^i),(1+r)a^i)\\ p^c{c}^i=(1+{\tau }^s)p^s{c}^{s,i}+(1+{\tau }^L)p^L{c}^{L,i}+(1+{\tau }^M)c^{M,i}+(1+{\tau }^{*})p^{*}{c}^{*,i}\\ c^i=F^c(c^{s,i},c^{L,i},c^{M,i},c^{*,i})\\ \begin{array}{ccc}{a}^{\prime ,i}\ge -{\kappa }^i,& c^i>0,& 0\le o^i\le 1\end{array}\end{array}$$
2. Factor prices {w^l, w^x, w^h, r] satisfy:

   $$\begin{array}{c}{w}^l=p^L{F}_l^L=F_l^M\\ w^x=p^L{F}_x^L=F_x^M=p^S{F}_x^S\\ w^h=F_h^M=p^S{F}_h^S\\ r+\delta =p^s{F}_k^S=F_k^M\end{array}$$
3. Government budget balance:

   $$\begin{array}{l}{\displaystyle \sum _i{\displaystyle \int {\Gamma }^i{Q}^i({\zeta }^i,{\aleph }^i)+\Xi +{\displaystyle \sum _i{\displaystyle \int T({\zeta }^i{w}^i(1-o^i),(1+r)a^i)Q^i({\zeta }^i,{\aleph }^i)+(1+r^{*})B^{*}}}}}\hfill \\ =G^M+G^S+G^L+I^G+B^{*\prime }\hfill \end{array}$$
4. Markets clear:

   $$\begin{array}{c}k=k^M+k^S={\displaystyle \sum _i{\displaystyle \int a^i{\vartheta }^i(d{a}^i\times d{\zeta }^i)}}\\ l=l^L+l^M={\eta }^l{\displaystyle \int {\zeta }^l{o}^l{\vartheta }^l(d{a}^l\times d{\zeta }^l)}\\ x=x^L+x^M+x^S={\eta }^x{\displaystyle \int {\zeta }^x{o}^x{\vartheta }^x(d{a}^x\times d{\zeta }^x)}\\ h=h^M+h^S={\eta }^h{\displaystyle \int {\zeta }^h{o}^h{\vartheta }^h(d{a}^h\times d{\zeta }^h)}\\ {\displaystyle \sum _i{\displaystyle \int c^{L,i}(a^i,{\zeta }^i,i){\vartheta }^i(d{a}^i\times d{\zeta }^i)+G^L=L}}\\ {\displaystyle \sum _i{\displaystyle \int c^{S,i}(a^i,{\zeta }^i,i){\vartheta }^i(d{a}^i\times d{\zeta }^i)+G^S+m^{S,M}+m^{S,S}=S}}\end{array}$$
5. Law of motion

   $${\vartheta }^{\prime }=H\left(\vartheta \right)$$
   where the function H:Ω → Ω can be written explicitly as:

   $${\vartheta }^{\prime }(A\times \aleph )=P((a,\zeta ,i);A\times \aleph )\vartheta (da\times d\zeta \times di)$$
   where

   $$P((a,\zeta ,i);A\times \aleph )\vartheta (da\times d\zeta \times di)={\displaystyle \sum _{i\in I}{\displaystyle \sum _{{\zeta }^{\prime ,i}\in {\aleph }^i}\{Q^i\left({\zeta }^{\prime ,i}\right|{\zeta }^i)\underset{0}{if}\underset{else}{a^{\prime ,i}}(a^i,{\zeta }^i,i)\in A^i}}$$

   A stationary equilibrium is a competitive equilibrium in which all policy variables, international prices, domestic prices, aggregate variables and all other variables (and functions) are time-invariant.

A. Preferences

We assume preferences over consumption and leisure that can be represented by a period utility of the form

The consumption aggregator is given by

To calibrate our preferences, we fix our coefficient of risk aversion to σ = 2. The discount factor β is chosen so that the equilibrium of the benchmark has the capital-output ratio close to its value in the data. ω and θ are chosen so that average hours worked in the economy by the household correspond to $\frac{1}{3}$ of their time, and the labor elasticity to wages is approximately $\frac{1}{3}$. The elasticity of substitution between consumption goods is assumed to be 1 (so that ρ = ρ^T = ρ^N, and ρ → 0. The setting the parameter $\overline{c^N}>0$ ensures that income elasticity demand for services increases with income, this parameter is then chosen such that consumption of manufacturing and imported consumption goods share of total consumption is approximately eight percentage points larger for the top quintile than for the bottom quintile of income distribution. The share of services on total consumption 1 – γ and low skill services in total services consumption γ^N are calibrated in such way that, given the shares of high-skill intermediate inputs in the production of manufacturing and high-skill services sectors (derived from input-output data), the model matches the total share of low skilled and high skilled services production in GDP (which corresponds respectively to 30 percent and 50 percent of total GDP). The share of domestically produced goods in tradable goods consumption γ^T is calibrated to match a share of consumption of imports in total consumption expenditure of approx. 10 percent.

B. Household’s labor productivity

Household’s labor productivity has a permanent component given by their skill level η_i, and a transitory idiosyncratic stochastic component. The permanent component given by skill levels in the model is proxied by education attainment levels in the US population as reported by the Census Bureau (2016). We classify individuals with high school degree or less as low skill individuals, and individuals with more than bachelor’s degree as high skill individuals. We normalized the average labor productivity of medium skill level households to 1 (i.e., η_x = 1), and consider differences in average years of education as proxies for differences in skill levels, therefore as high school years of education are 66 percent or less than the years required to achieve a bachelors’ degree we set η_l = 0.7. In the other hand, while a Master’s degree takes approximately 16% more years of education to achieve than a Bachelor’s degree, we consider the possibility of decreasing returns to human capital accumulation, and opt to set conservatively the skill level of those households with more than a bachelor’s degree as being only 10 percent higher than the skill level of those classified as medium skill (i.e., η_h = 1.1)

Table 2.

Skills Parameters

Table 2.

Skills Parameters

Parameter	Value	Target
μl	0.39	Population share with high school degree or less.
μx	0.48	Population with some college, no degree, associate degree or bachelor degree.
μh	0.13	Population share with a Bachelor’s degree or more
ηl	0.7	Approx. 12 years of education.
ηx	1	Approx. 16 years of education.
ηh	1.1	More than 16 years of education.

View Table

Table 2.

Skills Parameters

Parameter	Value	Target
μl	0.39	Population share with high school degree or less.
μx	0.48	Population with some college, no degree, associate degree or bachelor degree.
μh	0.13	Population share with a Bachelor’s degree or more
ηl	0.7	Approx. 12 years of education.
ηx	1	Approx. 16 years of education.
ηh	1.1	More than 16 years of education.

The stochastic idiosyncratic productivity component is calibrated to minimized the distance between the model’s and the data’s Lorenz curve for pre-tax income for the US (See Appendix). Each of the process is governed by a 3 state Markov chain, and can be summarized by a persistence parameter φⁱ and the variance of the process ${\sigma }_{ϵ,i}^2$. The parameters of these three process are shown in Table 3.

Table 3.

Stochastic Productivity Parameters

Table 3.

Stochastic Productivity Parameters

Stochastic Productivity Parameters:		$\begin{array}{cc}{\zeta }_{t-1}^i={\zeta }_{t-1}^i{\phi }^i+{ϵ}_t^i;& {ϵ}_t^i\sim N(0,{\sigma }_{ϵ,i}^2)\end{array}$
Parameter	Value	Target
φl	0.76	Persistence parameter of the stochastic idiosyncratic shock for low skill households.
φx	0.76	Persistence parameter of the stochastic idiosyncratic shock for medium skill households.
φh	0.73	Persistence parameter of the stochastic idiosyncratic shock for high skill households.
${\sigma }_{ϵ,l}^2$	0.01	Variance of the process for the idiosyncratic labor productivity shock for low skill households.
${\sigma }_{ϵ,x}^2$	0.0289	Variance of the process for the idiosyncratic labor productivity shock for medium skill households.
${\sigma }_{ϵ,h}^2$	0.0484	Variance of the process for the idiosyncratic labor productivity shock for high skill households.

View Table

Table 3.

Stochastic Productivity Parameters

Stochastic Productivity Parameters:		$\begin{array}{cc}{\zeta }_{t-1}^i={\zeta }_{t-1}^i{\phi }^i+{ϵ}_t^i;& {ϵ}_t^i\sim N(0,{\sigma }_{ϵ,i}^2)\end{array}$
Parameter	Value	Target
φl	0.76	Persistence parameter of the stochastic idiosyncratic shock for low skill households.
φx	0.76	Persistence parameter of the stochastic idiosyncratic shock for medium skill households.
φh	0.73	Persistence parameter of the stochastic idiosyncratic shock for high skill households.
${\sigma }_{ϵ,l}^2$	0.01	Variance of the process for the idiosyncratic labor productivity shock for low skill households.
${\sigma }_{ϵ,x}^2$	0.0289	Variance of the process for the idiosyncratic labor productivity shock for medium skill households.
${\sigma }_{ϵ,h}^2$	0.0484	Variance of the process for the idiosyncratic labor productivity shock for high skill households.

C. Technology

We assume CES production functions for the three sectors. The parameters that determined the participation of the factors are jointly calibrated such that labor intensities and intermediate output intensities match the data.

D. Government

We set consumption (and intermediate consumption) tax rates to 7.5%, which is the approximate mid-point for the range of state sale taxes in US. For the benchmark calibration government consumption is chosen to be 16% of GDP, as the average in the data for the period 2009-2015 per the World Bank data. Additionally, we assumed that total government consumption is concentrated on manufacturing goods (i.e., G^L = G^S = I^G = 0).¹⁰

To approximate the US income tax code, we used a variation of the function used in Conesa and Kruger (2006) which seems to properly represent the US system. More specifically this function takes the functional form:

where y^j is household j pre-tax income. Following Benabou (2002), Heathcote et al. (2016) and Guner et al. (2016), federal income taxes are parameterized to capture the various effective tax rates paid at different levels of income. The benchmark tax function, which approximates the current US system, is plotted in Figure 4 and reports the effective tax rate as a function of multiples of median income. To account for the presence of an EITC in the current US system, the tax function is negative and very progressive for households earning less than 50% of the median income. At the other extreme, the function is set to peak around 28% for people approaching high multiples of the median income (10 and above).

Baseline Tax Function

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Baseline Tax Function

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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Baseline Tax Function

Citation: IMF Working Papers 2017, 192; 10.5089/9781484316580.001.A001

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