1.1 Household Behavior

Each year, every household chooses a basket with three types of final consumption goods: food, services and manufactured goods. We denote the basket by the vector array c = (c_f,c_sv,c_m). The annual utility that a given household derives from a particular consumption basket is denoted u(c). The cost of the consumption basket is paid from the household’s income net of taxes plus available financial assets. Households then save any remaining resources for the next year. These savings are denoted a’.

At the begining of each year, the household observes its existing stock of savings a and its productivity z. Then, the household chooses s, the share of time spent supplying formal labor. Once all annual income is accrued, the household chooses c, the basket of annual consumption, and; a’, its savings for next year. ⁴ Function y(z,l,s,w_l) gives the formal labor income of a household with productivity z in location l formally working a share s of its available time, when local wages are w_l. Function y*(z,l, 1 – s,w_l) gives the informal labor income of such household. Note that the disposable time not spent in formal work, 1 – s, is devoted to informal work in its entirety.⁵ Function T(l,c,y,p) gives the total taxes paid by the household. Taxes can depend on location l, consumption basket c and its price p, and on formal income y. The taxes are net of any government transfers. Functions y, y* and T are specified below.

We can describe the dynamic utility maximization problem faced by each and every household in either location l = {rural, urban} using the basic language of dynamic programming that says that households choices lead to the optimal value of expected discounted utility:

subject to the time use constraint:

0 ≤ s ≤ 1.

and the budget constraint:

Equation 1 is the optimal value function for location l. For a household with initial assets a and productivity z, Equation 1 gives the optimized expected value of utility measured at the beginning of the year. Variables s,c,a’ are chosen by the household with the goal of maximizing the sum of current-year utility u(c) plus all future expected utility E[V_l(a’, z’)|z], discounted at rate β < 1.⁶

The choice of s, c, a’ cannot violate the time use constraint or the budget constraint. The time use constraint says that the share of time used to work in the formal labor market has to be between zero and one. The budget constraint says that the total sources of wealth have to be less than or equal to the total uses of wealth. Finally, we define p · c = p_ac_a + P_svc_sv + p_mc_m to denote total consumption expenditures before taxes, i.e. the sum of the element-wise product of vectors p and c.

We note that the functions describing income and taxes (y, y*, T) are explicitly defined below, while function V_l(a, z) is endogenously determined by the solution of the functional equation in Equation 1, via the principle of dynamic programming. Intuitively, the V_l on the left hand side of the functional equation can be thought of the infinite sum of optimized future utility flows. The max operator determines how decisions (s, c, a’) must be made in the current year, taking into account their expected impact on future years, assuming that the household will continue to make optimal decisions in future years.

1.2 Determination of Income and Taxes

Taxes paid by urban and rural households consist of a flat tax on consumption (VAT) and a flat labor income tax (PIT) which are are identical across regions. For each type of tax, there is a tax gap such that effective taxes collected from households are:

Equation 2 simply states that total taxes are the sum of the PIT and VAT levied on the final consumption goods produced in the formal sector, i.e. formal consumption (denoted c^formal). The third term g denotes any government transfers.

While find the optimal decision rules c and a! numerically using the computer, while we can find an analytical solution for time share at formal work, s as a function of z and w_l. We first define formal income as the product of effective labor supply z x s and the wage rate so that

y(l,z,s,w_l) = w_l zs,,

and we define informal income as a concave function of effective informal labor z(1 — s) times the unit price of informal goods so that

Net labor income is thus given by

where $\hat{\tau }$ is shorthand notation for the effective income tax rate τ_income(1 — gap_income) and p_l is defined to represent the price of the good that is produced by the informal sector in region l.

Because s is an intratemporal decision, it can be chosen to maximize household income, in the following auxiliary problem:

subject to the time use constraint:

0 ≤ s ≤ 1.

The auxiliary problem allocates a share s of discretionary time to market work and a share 1 — s to informal work. Marginal informal income from putting a unit of time in the job market is fixed at zw_l times one minus the effective tax rate. Informal marginal income from a unit of time in informal work is strictly decreasing and equal to αεz^α(1 — s)^α-1.⁷ The solution, as a function of the household’s state, taking wages, prices and tax rates as parameters, is:

Equation 3 implies that a corner solution with zero formal work is possible. We can see that formal labor time s is increasing in wages and in the households skill z, and decreasing in the tax rate and the price of the informally supplied good. This simple static decision problem will be at the heart of our model’s reaction to changes in relative prices, regional wages, and shocks.

For households providing positive formal work, the behavioral reaction to a small increase in skill z is to formal work time. This implies that, keeping wages fixed, effective labor supply increases when all agents experience a productivity increase (like in the case of our educational reform). Since wages and informal goods prices p_l also affect formal labor supply, the labor supply effect of an economy-wide increase in skills z may well be positive or negative, depending on the equilibrium reaction of wages and sectoral prices p_l, which we cannot characterize analitically. Interestingly we will observe how some of our reforms have opposing effects in the rural and urban areas.

1.3 Farmtrepreneurs

The farmtrepreneurs make up the third type of consumer. These agents have the same preferences for consumption as households; however, farmtepreneurs differ from rural households in two key ways. First, farmtrepreneurs do not face idiosyncratic risk, reflecting their size and our model’s focus on the lower end of the income distribution, rather than the top. Second, farmtrepreneurs can hire labor in rural areas to produce two different types of goods: (i) the same food produced informally by rural households, or (ii) an export commodity. Producing the export commodity requires capital and labor, while producing food requires only labor (and, implicitly, land). We assume farmtrepreneurs cannot borrow in financial markets capital so they gradually accumulate the capital required to produce the export commodity. This restriction highlights the limited depth of financial markets in many LIDCs. The optimization problem of farmtrepreneurs is given by:

subject to the budget constraint:

$R_1^{*}and{R}_2^{*}$ in the budget constraint (Equation 5) are the optimal revenues from food and export commodity sales of the farmtrepeneur and π and π* are the optimal profits from farming and production of agricultural export commodities, respectively. Every year, the farmtrepreneur chooses a consumption basket c and the capital stock for next year, k’. Optimal profits depend on the accumulated stock of capital k and wages w_rural per unit of effective labor. In particular, the profits from food production are given by:

where $R_1=p_f{\epsilon }_1{n}_1^{{\alpha }_1}$ is the sales revenue from food sales by the farmtrepreneur. Also, the profits from export commodities, given capital, are given by:

where $R_2=p*{\epsilon }_2{\left(k^{\zeta }{n}_2^{1-\zeta }\right)}^{{\alpha }_2}$ is the sales revenue from export commodity sales by the farmtrepreneur. This problem includes new parameters, ε₁ and ε₂, which are productivity constants; the two α₂ parameters, controlling returns to scale; and parameter ζ, controlling capital and labor shares. Note that, for simplicity, all of the farmtrepreneur’s capital is assumed to be used for production of export commodities so they cannot save in the financial market. This completes the specification of farmtrepreneurs in our model. The taxes on optimal revenues are specified as follows: $T_f(R_1^{*},R_2^{*})={\hat{\tau }}_f{R}_1^{*}+{\hat{\tau }}_{fx}{R}_2^{*}-g$ where the two τ parameters are effective revenue tax rates on food and export commodities revenues, and g are lump sum government transfers that are also received by farmtrepreneurs.

1.4 Extractive Multinational Sector

Our model of extractive multinationals in Guinea’s mining sector features a single firm that uses foreign capital and local unskilled labor under a Cobb-Douglas technology with decreasing returns to scale. We assume the global market for extractive export goods is competitive in the sense that Guinean output does not affect the world market price.⁸

We follow Schmitz (2005) and Greenwood and Weiss (2018) and assume a smooth technology and a unit elasticity of substitution between labor and capital. We thus also employ a Cobb-Douglas production function.⁹

Our model features decreasing returns to scale.¹⁰ This assumption allows us to match the observed elasticity of supply, which will determine the impact of government policy on the size of the foreign extractive sector, generating a behavioral response consistent with the views of the Guinean authorities that basically says that more taxes imply less business. In particular, the behavioral response generates a Laffer curve by which increased taxation increases the share of the pie obtained by the government, but only at the expense of making the total pie (e.g. mining output) smaller and smaller.

We use the x subscript for all variables and parameters related to this sector, for convenience:

The problem of extractive multinationals, given the effective tax rate ${\hat{\tau }}_x^{*}$, the rural area wage rate w_rural and the international price of mining commodities, is to choose the capital and labor to maximize profits:

Parameter r^∗ is the opportunity cost of capital faced by the foreign firms and δ_x is the depreciation rate for mining equipment. The solution to this optimization problem is:

Using this solution we can calculate several statistics that can be helpful for calibrating the parameters of the production function. The share of revenue paid to labor is:

while the mining supply elasticity with respect to prices is:

The share of the labor force and the share of output to GDP can also be directly calculated, as $\frac{p_x{Y}_x}{GDP}$. Finally the tax rate on revenues is given by:

1.5 Food Imports

Guinea is defined as a low-income food-deficit country (LIFD) by the Food and Agriculture Organization of the UN (FAO), meaning that it relies on food imports to stabilize its food supply but remains vulnerable to domestic and external shocks that could affect the nutritional status of its vulnerable populations. We thus follow the empirical literature that estimates the trade elasticity to changes in the relative price. In particular, we adopt the following reduced form equation, which is widely used and estimated in the literature:

Where parameter ε is the trade elasticity, we assume the foreign price of food is $p_f^{*}$, the local price of food is p_f, and the import price of food is m_f. Hillberry and Hummels (2013) review the models that give rise to this type of equation as well as the empirical literature that estimates the parameter ε.

1.6 Manufacturing Sector

A competitive manufacturing firm rents capital from households and foreign financial markets at interest rate r, pays for depreciation, and sells manufacturing goods in the combined domestic and international markets. Profit maximization is thus given by:

Due to the assumption of constant returns to scale, the solution specifies the optimal combination of capital and labor to be employed by the manufacturing firm, given the prices of inputs and outputs. The size of the sector depends on the equilibrium conditions, which dictate that wages be such that this sector has to absorb all the urban formal labor supply not demanded by the government while it rents its desired amount of capital at international rate r.

1.7 Public Service Sector

The public services sector can absorb some of the additional labor supply generated by educational reforms and enhances the supply of services as the economy expands. The provision of public services follows three assumptions. First, the government invests a fixed share of GDP ζ in capital for service enterprises. Second, government services are perfect substitutes with services provided informally by the urban households. Third, the government hires labor competitively in the urban labor market in an efficient proportion to its capital stock. We add the subscript e to all variables related to the public service sector. If all capital is composed of manufactured goods and the depreciation of public enterprise capital is δ_e, the law of motion for public capital is K_e,t+1 = ζGDP/P_m + (1 — δ_e)K_e,t, so the steady-state public service capital is $K_e=\frac{{\zeta }_{e}GDP}{{\delta }_e{P}_m}$. The optimal profits of the public enterprise, given the capital stock, are:

The first order condition for optimality is $w_{urban}=(1-{\alpha }_e)p_{sv}{A}_e{K}_e^{\alpha }{L}_e^{1-{\alpha }_e}$. From this condition, we solve for the public service sector’s demand of labor as a function of GDP and other variables, given by

From this solution we derive two statistics that allow us to calibrate parameters A_e and α_e upon knowledge of investment parameter ζ and depreciation δ_e. First, the ratio of the public service sector revenue to GDP given by

and, second, the public payroll to GDP ratio, given by

1.8 Government

Government surplus is defined as the difference between government revenues and government expenses excluding the government’s consumption of manufacturing. Government revenues equal the sum of all tax revenues (including taxes from mining) and the profits (or losses) of public service enterprises. Government expenses equal the sum expenses in development policies (education, infrastructure and tsocial ransfers), the government’s consumption of services, the government’s investment in capital for public service enterprises and interest payments on existing debt. In our model development policies, government consumption of services and investiments in capital for public services and the exogenous level of existing debt are specified as fixed proportions of nominal GDP.

In the definition of equilibrium, below, the government’s consumption of manufactures is set equal to government surplus, as defined above, to balance the government’s revenues and expenses every year.

1.9 Definition of Equilibrium

Given the external prices of agricultural commodities, food, manufactures and mining goods; given the international interest rate and the investment rates for government service enterprises, government infrastructure, education; given the investment rate for social transfers; given the opportunity cost of capital for mining multinationals; given the effective tax rates on food consumption, manufacturing consumption, labor income, commodity exports, food imports, mining revenue, manufacturing revenue, formal food revenue and commodity exports, an equilibrium is defined as: (i) a set of domestic prices for food and services, wage rates for urban and rural labor and a value of GDP; (ii) optimal decision rules for consumption, saving and formal work for urban and rural households; (iii) optimal value functions for urban and rural households; (iv) a cross sectional joint distributions of individual productivity and asset holdings for urban and rural households; (v) decision rules for capital accumulation, consumption, and production of food and agricultural commodities for farmtrepreneurs and government consumption of manufactures such that the following conditions hold:

1. Optimality.

• Households. Given prices, the value functions are consistent with the optimal decision rules and the value functions solve the Bellman equations (Equation 1) for urban and rural households.

• Farmtrepreneurs. Given the taxes rates food revenue and commodity export revenue the domestic price of food and the external price of commodity exports, the decisions rules for capital accumulation, production of food and commodity exports and consumption, solve the Bellman equation (Equation 2).

2. Stationarity. The cross sectional joint distributions of individual productivity and asset holdings are jointly generated by the household’s decision rules and the stochastic process for individual productivity. Furthermore, these distributions are time-invariant.

3. Market Clearing:

• Agricultural Commodities. The exports of agricultural commodities equals their production by farmtrepreneurs.

• Mining. The export of mining goods equals its production by foreign multinationals.

• Manufactures. Given the tax rate on manufacturing revenue, domestic manufacturing firms hire labor to equate its marginal product and marginal cost, given the urban wage rate and the international interest rate.

• Absorption. The net exports of manufactures equal domestic production minus domestic absorption. Domestic absorption equals the sum of investment in manufacturing firms, government enterprises and government infrastructure plus private and government consumption.

• Food. Given the effective taxe rates on consumption of food, the tax rates on formal food production and the domestic and foreign prices of food, the total food demand by households and farmtrepreneurs equals the imports of food plus the sum of formal and informal domestic production of food.

• Capital. Given the international interest rate and the GDP value, the inflow of capital services equals the demand of capital services by manufacturing firms minus the supply of savings by households.

• Services. Given the GDP value, the domestic price of services and the urban wage rate, the government’s demand for services plus the private consumption of services equals the informal production of services by urban households plus the production of services by government enterprises.

4. Government Budget:

• Given GDP, any positive government surplus (as defined above) is devoted to consumption of manufacturing so that government revenue and expenditure are balanced every period.

• The GDP value assumed has to equal the sum of the output of farmtrepreneurs, informal production (of food, services and agricultural commodities) by households and farmtrepreneurs, the output of manufacturing firms, the output of the public service enterprises and the output of the foreign mining sector, times their respective prices.

First, we make some restrictions on international trade that simplify our abstraction. We assume that food is not exported while agricultural commodities are exported. Food can be imported. Services cannot be imported or exported. Because of a limited elasticity of import supply, the domestic price of food can differ from the foreign price. Manufactured goods can be exported and imported; in addition, their price always equals the foreign price.

The aggregate supply of formal labor in both the rural and urban area is defined as the sum total of every household’s supply of effective labor zs, which equals the time spent in market work s times the labor productivity of the household, z. The demand for formal labor is also measured in units of effective labor.¹¹ While the savings of urban and rural households can be invested in the formal urban sector or in the foreign financial sector at the interest rate, farmtrepreneurs need to finance their own productive enterprises through savings.

Total government revenues consist of (i) the revenue tax on formal firms (i.e. manufacturing firms, farmtrepreneurs producing food and commodity exports, and foreign extractive firms);¹² (ii) the VAT, whereby consumption expenditures on food and manufactured goods are taxed at flat rates;¹³ and (iii) flat taxes on formal labor income.

Government expenditures consist of (i) interest payments to households holding domestic government debt (for simplicity, we abstract from modeling foreign debt), (ii) transfers to households, (iii) payments to the educational system (iv) investment in public infrastructure (v) investments in capital for the public service sector.

4.3.1 The Impact of Infrastructure on Sectoral Output

Following the empirical and theoretical literature – which notes that infrastructure consists of a vector of four distinct stocks of capital that represent railroads, roads, communications and energy – we denote the vector of infrastructure capital per person as: E = [E₁,E₂, E₃,E₄]. The stocks of infrastructure capital affect the vector of productivities:

according to a function $A\left(E\right)=R_{+}^4\to R_{+}^3$. Specifically, for each sector we use the elasticities estimated as follows:

Functions of this kind have been estimated in the empirical literature using panel data and time series data. The vector A contains the productivity of the different productive centers of the economy: services (s), informal agriculture (a), farming (f), commodity export (∗) and manufactured goods (m). The accumulation of infrastructure stocks is as follows:

is the law of motion for infrastructure capital, where E’ is next period’s vector of infrastructure stocks. Investment is characterized by a private and a government component:

where g denotes the vector of government investments in infrastructure and λ is a parameter vector measuring the wastefulness in public infrastructure investment (note that × denotes element-by-element multiplication). In a steady state, the stock of each infrastructure component is given by (E_n = I_n/δ). Furthermore, assuming equal prices of 1 for all components of infrastructure, we can easily prove using a Lagrangian optimization that the optimal amount of investment on infrastructure n for sector: j is

where I_j is the ammount of investment exogenously devoted to sector j. This result provides us with the relationship between productivity in a sector and the ammount of total investment in infrastructure:

4.3.2 Data

We combine sectoral real GDP and population data from the World Bank’s World Development Indicators (WDI) with indexes of infrastructure stocks from the Canning (2008) database. Our base sample consists of an unbalanced panel with 158 low, middle and high income countries. The periodicity of the data is annual and the sample covers years 1950 to 2008. Table 6 presents basic sample statistics and variable counts. The list of countries is presented in Appendix 2.

^{Table 6.}

Characteristics of the Dataset

^{Table 6.}

Characteristics of the Dataset

	Initial Year	End Year	Observations
mean	1974.66	1999.86	23.97
median	1973	1999	24
min	1960	1989	7
max	1996	2002	43

View Table

^{Table 6.}

Characteristics of the Dataset

	Initial Year	End Year	Observations
mean	1974.66	1999.86	23.97
median	1973	1999	24
min	1960	1989	7
max	1996	2002	43

4.3.3 Empirical Specification

In order to measure the impact of infrastructure on growth we follow the academic literature that focuses on the long run and short run elasticities of output to elasticity. In particular Shioji (2001) utilizes a convergence equation that holds in many versions of the one sector growth model and state-level data from the United States and Japan, assuming that TFP depends positively on public capital per capita. This convergence equation says that convergence speed toward the steady state is proportional to the distance of GDP per worker from its steady-state level. Shioji (2001) derives this equation from a theoretical one-sector growth model which includes an adjustment cost of investment. In that model, the stock of public capital impacts the steady state level of income per capita, thus increasing the convergence speed (growth rate) compared to regions at similar distance from the steady state but lower stocks of public capital. Shioji (2001) recognizes the difference between short and long run elasticities and accounts for (i) the short-run endogeneity of public capital with respect to output, and (ii) the sluggish adjustment of productivity to increased public capital. Our analysis employs the same technique, but applies it separately to each sector of the economy. Our empirical model rests on two assumptions, which we can verify using numerical simulations of our quantitative model. See Appendix 2 for the formal derivations.

4.3.4 Estimation

Shioji (2001) employs panel data estimators, including Pooled OLS and Pooled OLS with country fixed effects, Arellano and Bond’s (1991) estimation technique (referred to as “Difference GMM” in Blundell and Bond’s (1998)) estimation technique (known as “GMM” in levels), and Kiviet’s (1995) estimation technique (known as the “LSDV-C”) to estimate the convergence equation.²³ The results in Shioji (2001) favor GMM(Diff) as the most stable across all estimations, so we employ this estimator. See Shioji (2001) Table 3 and the explanations in Shioji’s Section 6. We now rearrange and take a first difference of our specification elaborated in Appendix 2 to obtain

The differenced error term is correlated with the lagged dependent variable that multiplies b_1j, so GMM estimation of the b’s is recommended utilizing orthogonality conditions between the disturbance ∆u_ijt and any elements of the vector of lagged log output per worker (∆y_ijt-2, ∆y_ijt-3, ∆y_ijt-4, ..., ∆y_ij0). The idea is to add lags (moment conditions) until the observed error term distribution satisfies the assumption of no serial correlation. We construct the sectoral GDP combining the real GDP in PPP dollars of 2010 with the sectoral shares of value added and divide by total population. The right hand side variables are constructed by taking the stock variable from Canning et al. (2008) dividing by total population from the WDI and taking logarithms. Our specification is similar to Shioji’s (2001) and also to Calderon and Serven’s (2004), but our approach differs in that it applies the methodology separately to each sector of the economy. Our sample includes 158 countries. The characteristics of the panel are described in Table 6.

Most of our coefficients have the correct sign and given our inclusive sample, the precision is surprisingly good.

Table 7 indicates that the Arellano-Bond test for autocorrelation of the errors reject serial autocorrelation at the 1% confidence level for all of specifications. The sample includes all countries in the intersection of the Canning (2008) dataset and the WDI that had at least 10 annual observations for all of the variables in the regression. The standard errors are computed using the heteroskedasticity and autocorrelation robust standard errors. The estimates are computed using one step GMM.

^{Table 7.}

Coefficients from Arellano-Bond Estimate of the Convergence Equations

p-values in parentheses ^∗ p < 0.05, ^∗∗ p < 0.01, ^∗∗∗ p < 0.001

^{Table 7.}

Coefficients from Arellano-Bond Estimate of the Convergence Equations

	Aggregate	Agriculture	Manufacturing	Services
Rail	0.0280	0.0753	-0.173	0.00711
Road	0.0271	0.0652	0.165*	0.115***
Telephones	0.0341**	-0.0476	0.0838	0.0765*
Energy	0.0175	0.0499	0.165**	0.0285
N	1811	1421	1290	1286
r2

p-values in parentheses ^∗ p < 0.05, ^∗∗ p < 0.01, ^∗∗∗ p < 0.001

View Table

^{Table 7.}

Coefficients from Arellano-Bond Estimate of the Convergence Equations

	Aggregate	Agriculture	Manufacturing	Services
Rail	0.0280	0.0753	-0.173	0.00711
Road	0.0271	0.0652	0.165*	0.115***
Telephones	0.0341**	-0.0476	0.0838	0.0765*
Energy	0.0175	0.0499	0.165**	0.0285
N	1811	1421	1290	1286
r2

p-values in parentheses ^∗ p < 0.05, ^∗∗ p < 0.01, ^∗∗∗ p < 0.001

For consistency with the theory, we now restrict the coefficients to be positive. We do so in the simplest way possible, which is by omitting the variables that display a negative coefficient from the estimations. Clearly, the negative signs could be caused by multicollinearity issues, but examining the correlation structure of the regressors reveal that the correlation between the differenced regressors is not large.²⁴

We now proceed with the restricted estimation, summarized in Table 8.

^{Table 8.}

Restricted Coefficients from Arellano-Bond Estimate of the Convergence Equations

p-values in parentheses ^∗ p < 0.05, ^∗∗ p < 0.01, ^∗∗∗ p < 0.001

^{Table 8.}

Restricted Coefficients from Arellano-Bond Estimate of the Convergence Equations

	Agriculture	Manufacturing	Services
Rail	0.106* (0.021)		0.00711 (0.881)
Road	0.0428 (0.248)	0.152** (0.003)	0.115*** (0.001)
Telephone		0.104** (0.008)	0.0765* (0.018)
Energy	0.0270 (0.353)	0.0566 (0.219)	0.0285 (0.259)
N	1533	2385	1286
r2

p-values in parentheses ^∗ p < 0.05, ^∗∗ p < 0.01, ^∗∗∗ p < 0.001

View Table

^{Table 8.}

Restricted Coefficients from Arellano-Bond Estimate of the Convergence Equations

	Agriculture	Manufacturing	Services
Rail	0.106* (0.021)		0.00711 (0.881)
Road	0.0428 (0.248)	0.152** (0.003)	0.115*** (0.001)
Telephone		0.104** (0.008)	0.0765* (0.018)
Energy	0.0270 (0.353)	0.0566 (0.219)	0.0285 (0.259)
N	1533	2385	1286
r2

p-values in parentheses ^∗ p < 0.05, ^∗∗ p < 0.01, ^∗∗∗ p < 0.001

The signs of the remaining coefficients are stable and their significance is enhanced by the two restrictions imposed on negative coefficients from Table 7. The measurements from this regression will allow us to control the differential effect of each type of infrastructure across the sectors of the economy in our model. We need to adequately capture the differential impact of infrastructure across sectors in order to credibly model the quantitative response of inequality to a particular infrastructure policy.

Combining our estimates and the derivations for investment, we obtain the equations relating investments to productivity for each of the sectors: