The household sector consists of a continuum of households
\(h\in \left[ 0,1\right]\). A share
\((1-\epsilon )\) of these households are not liquidity constrained and indexed by
\(i\in \left[ 0,1-\epsilon \right]\). They have access to financial markets where they can buy and sell domestic and foreign assets (government bonds), accumulate physical capital which they rent out to the intermediate sector, and they also buy the patents
of designs produced by the R&D
sector and license them to the intermediate goods producing firms.
4 The remaining share
\(\epsilon\) of households is liquidity constrained and indexed by
\(k\in \left[ 1-\epsilon ,1\right]\). These households cannot trade in financial and physical assets and consume their disposable income each period. The members of both types of households offer low-, medium- and high-skilled labour services indexed by
\(s\in \{L,M,H\}\). For each skill group, we assume that both types of households supply differentiated labour services to unions which act as wage setters in monopolistically competitive labour markets. The unions pool wage income and distribute it in equal proportions among their members. Nominal rigidity in wage setting is introduced by assuming that households face adjustment costs for changing wages.
Non-liquidity constrained households maximise an intertemporal utility function in consumption and leisure subject to a budget constraint. These households make decisions about consumption
\(C_{i,t}\), labour supply
\(L_{i,t}\), purchases of investment
good
\(J_{i,t}\) and government bonds
\(B_{i,t}\), the renting of physical capital stock
\(K_{i,t}\), the purchases of new patents
from the R&D
sector
\(J_{A,i,t}\), and the licensing of existing patents
\(A_{i,t}\), and receives wage income
\(W_{s,t}\), unemployment benefits
\(bW_{s,t}\), transfer income from the government
\(TR_{i,t}\), and interest income,
\(i_t,i_{K,t}\) and
\(i_{A,t}\).
5 Hence, non-liquidity constrained households face the following Lagrangian
$$\begin{aligned} \max _{\Bigg \{{\begin{array}{c} C_{i,t},L_{i,s,t},B_{i,t},J_{i,t},\\ K_{i,t},J_{A,i,t},A_{i,t} \end{array}}\Bigg \}_{t=0}^{\infty }} V_{i,0} & = E_0\sum _{t=0}^{\infty } \Bigg (U(C_{i,t})+\sum _{s\in \{L,M,H\}} V(1-L_{i,s,t})\Bigg ) \nonumber \\ & -E_0\sum _{t=0}^{\infty }\lambda _{i,t}\frac{\beta ^{t}}{P_{t}}\Bigg ((1+t_{C,t})P_{C,t}C_{i,t}+B_{i,t} \\ & +P_{i,t}\Big (J_{i,t}+\Gamma _{J}(J_{i,t})\Big )P_{A,t}J_{A,i,t} \nonumber \\ & -(1+i_{t-1})B_{i,t-1} \nonumber \\ & -\sum _s \big ((1-t_{w,s,t})W_{s,t}L_{i,s,t} \\ & +bW_{s,t}(1-NPART_{i,s,t}-L_{i,s,t})\big ) \nonumber \\ & -(1-t_{K})\big (i_{K,t-1}-rp_{K}\big )P_{I,t-1}K_{i,t-1} \\ & -t_{K}\delta _{K} P_{I,t-1} K_{i,t-1} -\tau _{K} P_{I,t} J_{i,t}\nonumber \\ & -(1-t_{K})\big (i_{A,t-1}-rp_{A}\big )P_{A,t-1}A_{i,t-1} \\ & -t_{K}\delta _{K} P_{A,t-1} K_{i,t-1} -\tau _{K} P_{A,t} J_{A,i,t}\nonumber \\ & -TR_{i,t}-\int _{0}^{N}PR_{fin,j,i,t} \textit{d} j\\ & -\int _{0}^{A_{t}}PR_{int,m,i,t} \textit{d} m\Bigg ) \nonumber \\ & -E_0 \sum _{t=0}^{\infty }\lambda _{i,t} \xi _{i,t}\beta ^t\Big (K_{i,t}-J_{i,t}\\ &-(1-\delta _{K})K_{i,t-1}\Big ) \nonumber \\ & -E_0 \sum _{t=0}^{\infty }\lambda _{i,t} \psi _{i,t},\beta ^t\Big (A_{i,t}-J_{A,i,t}\\ &-(1-\delta _{A})A_{i,t-1}\Big ) \end{aligned}$$
(6.1)
where
s is the index for the corresponding low- (
L), medium- (
M) and high-skilled (
H) labour type respectively (
\(s\in \{L,M,H\}\)). The budget constraints are written in real terms with prices for consumption, investment
and patents
(
\(P_{C,t}\),
\(P_{I,t}\),
\(P_{A,t}\)) and wages (
\(W_{s,t}\)) divided by the GDP deflator (
\(P_t\)). All firms of the economy are owned by non-liquidity constrained households who share the total profit of the final and intermediate sector firms,
\(\int _{0}^{N}PR_{fin,j,i,t} \textit{d} j\) and
\(\int _{0}^{A_{t}}PR_{int,m,i,t} \textit{d} m\), where
N and
\(A_t\) denote the number of firms in the final and intermediate sector, respectively. As shown by the budget constraints, all households pay wage income taxes (
\(t_{w,s,t}\)), consumption taxes (
\(t_{C,t}\)) and
\(t_{K}\) capital income taxes less tax credits
(
\(\tau _{K}\) and
\(\tau _{A}\)) and depreciation allowances (
\(t_{K}\delta _{K}\) and
\(t_{K}\delta {A}\)) after their earnings on physical capital and patents
. When investing into tangible and intangible capital, households demand risk premia
\(rp_{K}\) and
\(rp_{A}\) in order to cover the risk inherent to the return related to these assets.
The utility function is additively separable in consumption
\(C_{i,t}\) and leisure
\(1-L_{i,s,t}\). Log-utility for consumption as well as the presence of habit persistence is assumed.
$$\begin{aligned} U(C_{i,t})=(1-habc)\log (C_{i,t}-habcC_{i,t-1}). \end{aligned}$$
(6.2)
CES preferences with common labour supply elasticity are assumed for leisure, but a skill-specific weight
\(\omega _s\) on leisure. This is necessary in order to capture differences in employment levels across skill groups. Thus preferences for leisure are given by
$$\begin{aligned} V(1-L_{i,s,t})=\frac{\omega }{1-\kappa }(1-L_{i,s,t})^{1-\kappa }, \, \quad \text {with} \quad \kappa >0 \, \end{aligned}$$
(6.3)
For the sake of brevity, the following derivations of the optimality equations focus only on the ones related to the R&D
investments made by non-liquidity constrained households. These households buy new patents
of designs produced by the R&D sector
\(I_{A,t}\) and rent their total stock of designs
\(A_t\) at rental rate
\(i_{A,t}\) to intermediate goods producers in period
t. Households pay income tax at a rate
\(t_{K}\) on the period return of intangibles and receive tax subsidies at rate
\(\tau _{A}\).
6 Hence, the first-order conditions with respect to R&D investments
are given by:
$$\begin{aligned} \frac{\partial V_0}{\partial A_{i,t}}:\,\,\,-\lambda _{i,t}\psi _{i,t} + E_t\bigg ( & \lambda _{t+1}^i\psi _{t+1}^i\beta (1-\delta _{A}) \\ & +\lambda _{i,t+1} \beta \frac{P_{A,t}}{P_{t+1}}\big ((1-t_{K})(i_{A,t}-rp_{A})+t_{K}\delta {A}\big )\bigg )=0 \end{aligned}$$
(6.4)
$$\begin{aligned} \frac{\partial V_0}{\partial J_{A,i,t}}:\,\,\,-\frac{P_{A,t}}{P_{t}}(1-\tau _{A})+\psi _{i,t}=0 \end{aligned}$$
(6.5)
Neglecting second-order terms, it can be shown that the rental rate of intangible capital is:
$$\begin{aligned} i_{A,t}\approx E_t\frac{(1-\tau _{A})\big (i_t-\pi _{A,t+1}+\delta _{A}(1+\pi _{A,t+1})\big )-t_{K}\delta _{A}}{1-t_{K}}+rp_{A} \end{aligned}$$
(6.6)
where
\(1+\pi _{A,t+1}=\frac{P_{A,t+1}}{P_{A,t}}\).
Hence, households require a rate of return on intangible capital which is equal to the nominal interest rate minus the rate of change of the value of intangible assets and also covers the cost of economic depreciation plus a risk premium. Governments can affect investment decisions in intangible capital by giving tax incentives in the form of tax credits and depreciation allowances or by lowering the tax on the return from patents.
Within each skill group, a variety of labour services are supplied which are imperfect substitutes to each other. Thus trade unions can charge a wage mark-up
\(\frac{1}{\eta _{s,t}}\) over the reservation wage.
7 The reservation wage is equal to the marginal utility of leisure divided by the corresponding marginal utility of consumption. The relevant net real wage to which the mark-up adjusted reservation wage is equated is the gross wage adjusted for labour taxes, consumption taxes and unemployment benefits, which act as a subsidy to leisure. Thus the wage equation reads
$$\begin{aligned} \frac{U_{1-L,h,st}}{U_{C,h,s,t}}\frac{1}{\eta _{s,t}}=\frac{W_{s,t}(1-t_{w,s,t}-b)}{P_{C,t}(1+t_{C,t})}\qquad \text {for} \qquad h\in \{i,k\}\,\, \text {and}\,\, s\in \{L,M,H\}. \end{aligned}$$
(6.8)
Aggregation
The aggregate of any household-specific variable
\(X_{h,t}\) in per capita terms is given by
$$\begin{aligned} X_t = \int _0^1 X_{h,t}\, dh =(1-\epsilon )X_{i,t} + \epsilon X_{k,t}, \end{aligned}$$
(6.9)
Hence, aggregate consumption and employment is given by
$$\begin{aligned} C_t=(1-\epsilon )C_{i,t} + \epsilon C_{k,t}\,\,\text {and}\,\, L_t=(1-\epsilon )L_{i,t} + \epsilon L_{k,t}. \end{aligned}$$
(6.10)