A. Households and Self-Employment Production

Following the literature, there is a representative household with a large number of family members and perfect risk-pooling across household members. We denote variables associated with the salaried sector with a subscript S and variables associated with the self-employment sector with a subscript SE. Households choose consumption c_t, the desired measure of self-employment n_SE,t+1, capital search expenditures v_SE,t, domestic debt d_t, and foreign debt $b_t^{*}$ to maximize $E_0{\mathrm{\Sigma }}_{t=0}^{\infty }{\beta }^{t}u\left(c_t\right)$ subject to the budget constraint

and the household’s perceived law of motion for self-employment⁸

The term κ(v_SE,t) denotes total capital search expenditures, which we interpret broadly as the startup costs of self-employed firms. The government collects lump-sum taxes T_t. The amount (1 – ϕ_v)(1 – ϕ)V_S,t (defined below) is spent by the household when salaried firms (BGG entrepreneurs) exit the market, which happens with probability (1 – ϕ), and transfer a share (1 – ϕ_v) of their net worth to the households, where 0 ≤ ϕ_v ≤ 1. The interest rate on foreign debt is given by $R_t^{*}=R^{*}\left[\mathrm{\Theta }(b_t^{*}-b^{*})\right]$ where $\mathrm{\Theta }(b_t^{*}-b^{*})$ is an adjustment cost function that induces stationarity in debt holdings (Schmitt-Grohé and Uribe, 2003) and b^* represents steady-state foreign debt holdings. The interest rate on domestic debt is R_t.

Total production by self-employed firms is given by y_SE,t = z_SE,tn_SE,t where z_SE,t is a sectoral productivity shock. Each self-employed individual pays r_SE,t to rent capital in frictional markets (determined via Nash bargaining). Thus, individual self-employment earnings are given by (z_SE,t – r_SE,t). Households receive labor income w_S,tn_S,t from workers in salaried firms and bu_t in unemployment benefits, where u_t is unemployment. Households own the matching firms, whose profits are Π_M,t. Note that, while households can hold debt, unemployed individuals still require matching with a capital supplier in order to move into self-employment and produce.⁹ Capital relationships end with exogenous probability ρ^SE (the self-employment separation rate). f(θ_SE,t) is the household’s probability of finding a capital supplier, given by f(θ_SE,t) = m(k_M,t, v_SE,t)/v_SE,t, where m(k_M,t, v_SE,t) is a constant-returns-to-scale matching function that brings together household resources spent on capital search, v_SE,t, and capital from matching firms, k_M,t, and θ_SE,t ≡ v_SE,t/k_M,t is capital market tightness.

The solution to the household’s problem yields an optimal decision to enter self-employment:

This condition equates the expected marginal cost of searching for capital to the expected marginal benefit of doing so. The latter is given by individual self-employment earnings if the match survives next period, (z_SE,t+1 – r_SE,t+1), as well as the continuation value from the capital relationship, κ′ (v_SE,t+1)/p(θ_SE,t+1) (Finkelstein Shapiro, 2014). We also obtain standard Euler equations for domestic and foreign debt: u_c(c_t) = E_tβ [R_tu_c(c_t+1)] and $u_c\left(c_t\right)=E_t\beta \left[R_t^{*}{u}_c\left(c_{t+1}\right)\right]$. For future reference, we define unemployment as u_t = 1 – n_S,t – n_SE,t where n_S,t and n_SE,t denote the equilibrium measures of salaried employment and self-employment, respectively. Also, Ξ_t+1|_t = β u_c (c_t+1)/u_c (c_t) is the household’s stochastic discount factor.

B. Salaried Production

Our description of salaried production follows closely the exposition of financial frictions in Gertler, Gilchrist, and Natalucci (2007) (or Gilchrist and Zakrajšek, 2011). For expositional purposes, we abstract from firm-specific indices.

Salaried firms (or BGG entrepreneurs) are risk neutral, discount profits at rate β, and produce using a constant-returns-to-scale production function that takes salaried labor n_S,t and effective capital ω_S,tk_S,t as inputs, where ω_S,t is the fraction of capital k_S,t used in salaried production (a choice).

Each salaried firm faces two shocks, a sectoral productivity shock z_S,t and an idiosyncratic i.i.d. shock ζ_t, where E[ζ_t] = 1. Based on the above, salaried output for a given firm is y_S,t = ζ_tz_S,tF(n_S,t, ω_S,tk_S,t) where F(·) is constant-returns-to-scale.

The timing of firms’ decisions is as follows. At the end of period t – 1, firms have chosen the stock of capital k_S,t available for production in period t, where this choice is based on the expected return in t – 1, E_{t – 1}R_S,t since the idiosyncratic shock in period t has not been realized yet. At the beginning of period t the shocks materialize and salaried firms maximize profits by making the following choices. They choose salaried labor demand n_S,t, which is purchased at price p_n,t from matching firms in frictionless markets. Firms decide on the fraction ω_S,t of previously acquired capital k_S,t that is used in production in the current period, while the remaining capital, (1 – ω_S,t)k_S,t, is sold in frictionless markets to matching firms at price p_k,t.¹⁰ Firms also purchase their desired stock of capital for next period, k_S,t+1, from capital producers at price Q_S,t. To make capital purchases, firms must borrow funds b_S,t+1, which are subject to an external finance premium. In particular, borrowed funds by the firm in t + 1 are given by

where nw_S,t+1 is the firm’s net worth (defined below). Once production has taken place, firms can sell back the effective stock of capital that was used in production (net of depreciation) to capital producers and obtain Q_S,t (1 – δ)ω_S,tk_S,t in revenue.

The solution to the firm’s problem yields an optimal decision to supply purchased capital to matching firms:¹¹

and an optimal demand for salaried employment:

so that the price of each unit of salaried labor purchased from matching firms is equal to the marginal product of salaried labor.¹²

Following Gertler, Gilchrist, and Natalucci (2007), we denote the ex-post gross return on capital by R_S,t so that the marginal return to a unit of capital is characterized by:

where the right-hand-side is the optimality condition with respect to k_S,t+1.

We can express the external finance premium as s (Q_S,tk_S,t+1/nw_S,t+1) = (Q_S,t+1k_S,t+1/nw_S,t+1)^v_S, where v_S > 0 is the elasticity of the external finance premium with respect to leverage, (Q_Sk_S/nw_S).¹³

Net worth nw_S,t evolves according to

where φ is the survival probability of salaried firms, and V_S,t is defined as

In the expression above, z_nw,t is an i.i.d. net worth shock and the term in brackets is the expost cost of borrowing funds to purchase capital (Gertler, Gilchrist, and Natalucci, 2007). The term ϕ_v(1 – ϕ)V_S,t in net worth represents the startup resources available for entering salaried firms, which are left behind by exiting firms. Also, in contrast to existing models with financial frictions, net worth in our model includes the return to selling capital purchased from capital producers to matching firms, p_k,t (1 – ω_S,t)k_S,t.

With the above definitions in mind, the optimal demand for borrowed funds can be expressed as

where the right-hand-side is the ex-post cost of borrowing and R_t is the gross domestic interest rate.

C. Matching Firms

Matching firms supply the capital they purchased from salaried firms to potential self-employed individuals via frictional capital markets. They also post vacancies to find salaried workers, who are then employed in salaried firms (Zhang, 2012). Specifically, matching firms choose salaried vacancies v_S,t, desired salaried employment n_S,t+1, capital demand k_M,t,, and desired self-employment capital supply n_SE,t+1 to

subject to the perceived law of motion for self-employment capital (or self-employment)

and the perceived law of motion for salaried employment

Matching firms receive p_n,tn_S,t from selling matched labor to salaried firms, pay w_S,tn_S,t to matched salaried workers, and spend ψ_Sv_S,t on vacancy postings to attract salaried workers, where ψ_S is the fixed flow cost of posting a vacancy. They also receive r_SE,tn_SE,t from renting matched capital to the self-employed, but spend p_k,tk_M,t to purchase capital from salaried firms at price p_k,t. Capital from the self-employment sector separates at rate ρ^SE and returns to the matching firms, but matching firms have to cover the depreciation of capital for surviving relationships, δ. This is captured by the term (ρ^SE – δ)n_SE,t. Finally, matching firms can resell purchased capital k_M,t (net of depreciation) to capital producers at price Q_S,t, but a fraction (1 – ρ^SE)q(θ_SE,t) of k_M,t is matched and hence not available for resale.¹⁴ q(θ_SE,t) denotes the probability of matching with a household searching for capital, q(θ_SE,t) = m(k_M,t, v_SE,t)/k_M,t (see Finkelstein Shapiro, 2014). Similarly, the probability of filling a salaried vacancy is q(θ_S,t), where q(θ_S,t) = m(u_t, v_S,t)/v_S,t, m(u_t, v_S,t) is a matching function that brings together unemployed individuals u_t and salaried vacancies v_S,t, and θ_S,t ≡ v_S,t/u_t is labor market tightness.

This problem yields a standard job creation condition:

and the optimal supply of capital to the self-employment sector:

In the above expression, the matching firm equates the expected marginal cost of supplying capital to the expected marginal benefit of doing so. The expected marginal cost consists of three terms. The first one is the price at which matching firms buy capital from salaried firms. The second term represents one component of the opportunity cost of matching a unit of capital, given by the resale value if that unit is sold back to capital producers. The third term is a second component of the opportunity cost of matching a unit of capital, given by the value of an idle unit of matched capital that will become productive next period. All these terms are adjusted by the probability that a match materializes, (1 – ρ^SE)q(θ_SE,t). The right-hand-side of the capital supply condition captures the expected marginal benefit, given by the rental rate on matched capital r_SE,t+1, the benefit of getting back any separated capital net of depreciation, (ρ^SE – δ), and the continuation value of maintaining a capital relationship next period.

Note that in equilibrium, the matching firm’s demand for capital, k_M,t, is equal to the salaried firms’ supply of capital, (1 – ω_S,t)k_S,t, and p_k,t – Q_S,t (1 – δ) = z_S,tF_ωSkS,t. This last fact implies that the optimal capital supply condition is identical to the one in Finkelstein Shapiro (2014).

D. Wage and Rental Rate Determination

We assume bilateral Nash bargaining for wages and the capital rental rate in self-employment. The wage and capital rental rate equations are identical to those in Finkelstein Shapiro (2014) and given by:¹⁵

and

where $v_{SE,t}^u\equiv v_{SE,t}/u_t,{\chi }_S$ is the bargaining power of salaried workers, and χ_SE is the bargaining power of self-employed individuals. The intuition is straightforward: the Nash wage not only depends on the marginal product of salaried labor and salaried labor market conditions, but also on the likelihood of entering self-employment. The latter is embodied in capital market tightness (θ_SE). All else equal, higher capital market tightness implies a lower probability of entering self-employment, which in turn exerts downward pressure on salaried wages. Similarly, the capital rental rate is not only a function of labor market conditions in self-employment (embodied in capital market tightness), but also on salaried labor market conditions (embodied in salaried labor market tightness, θ_S). All else equal, higher salaried labor market tightness improves the self-employed’s outside option and puts downward pressure on the capital rental rate.¹⁶

E. Capital Producers

Capital producers choose investment i_S,t to

subject to the production technology for capital

where Q_S,t is the price that capital producers receive from selling capital to salaried sector firms and Φ (i_S,t/k_S,t) is an investment adjustment cost function. The first-order condition yields a standard expression for the price of capital:

F. Government, Market Clearing, and Definition of Total Output

The government collects lump-sum taxes from households to finance unemployment benefits and government spending, and its government budget constraint is T_t = bu_t + g_t. In turn, the economy’s resource constraint is given by

where the last term on the right-hand side captures the bankruptcy costs of salaried firms. We define total output as y_t = y_S,t + y_SE,t.

A. Business Cycle Dynamics without Macroprudential Policy

Table 6 presents some basic business cycle statistics for the benchmark (no-policy) economy. The labor and credit market facts for Mexico are shared by other emerging economies (see Bosch and Maloney, 2008; Loayza and Rigolini, 2011; Fernández and Gulan, 2014).

First, a model with productivity shocks as the sole driver of business cycles (as in Fernández and Gulan, 2014) qualitatively replicates the countercyclicality of leverage, interest rates, the current account-output ratio, unemployment, self-employment, as well as the countercyclicality of the transition probability from unemployment into self-employment.²³ Moreover, the model generates high unemployment persistence. As shown in column 4 of Table 6, adding net worth shocks – the main specification – allows us to better capture the cyclicality of interest rates and does not alter the general dynamics of the labor market relative to the model with only productivity shocks. Moreover, adding net worth shocks improves the model’s ability to quantitatively capture the cyclical correlation between output and the transition from unemployment to self-employment, as well as the cyclical correlation between output and the current account-output ratio. However, the model does underestimate the volatility of interest rates and unemployment, where the latter is a well-known limitation of standard search models.

Despite the model’s limited shortcomings, we can capture the behavior of the labor market, credit market, and standard macro variables surprisingly well. The key message to take from Table 6 is that the model is jointly consistent with the stylized facts about credit markets in Fernández and Gulan (2014) as well as important cyclical facts about emerging economy labor markets, including the countercyclicality of self-employment. This is one key contribution. More notably, comparing the benchmark model with a model with endogenous self-employment but no capital search frictions (i.e., no input credit channel) (column 5 of Table 6) shows that the latter model fails to deliver countercyclical self-employment, leverage, and interest rates, a relative volatility of consumption greater than one, a countercyclical current account-output ratio, and the persistence of unemployment in the data.²⁴ This gives additional validity to the search frictions and the inclusion of input credit in the benchmark model.

B. Business Cycle Dynamics with Macroprudential Policy

We follow Unsal (2013) and consider a policy that affects the external finance premium when credit to salaried firms deviates from trend. More specifically, we introduce a countercyclical regulatory Pigovian tax τ_S,t such that the return on capital R_S,t becomes

where τ_S,t responds to deviations in credit to salaried firms from steady state as dollows:

Above, η_S ≥ 0 governs the intensity of macroprudential policy over the business cycle.²⁵ This policy generates a smaller (larger) spread between the domestic interest rate R and the cost of borrowed funds, R_S, during recessions (booms), which is similar to countercyclical reserve requirements on banks. To illustrate the role of policy in the model, we calibrate η_S such that the volatility of credit b_S is halved.

(d) Response to a Negative Aggregate Productivity Shock Figure 1 shows the response of the economy to a negative aggregate productivity shock under two scenarios: the benchmark economy (η_S = 0), and an economy with an active macroprudential policy that halves the volatility of credit (η_s > 0).

An active policy exerts downward pressure on the external finance premium during downturns (and upward pressure during expansions). After a negative productivity shock, total output and consumption are less persistent under the policy (the difference in total output dynamics relative to the economy without an active policy is small quantitatively; the effect of the policy on salaried output is somewhat larger than for total output). Investment exhibits a smaller contraction on impact and returns to trend earlier, and net worth is initially more resilient to the shock due to the smaller contraction in borrowed funds under an active policy.

The response of total output hides important compositional differences. Indeed, while the policy makes the fall in salaried output less persistent (which in part explains the behavior of total output), the contraction in self-employment output becomes substantially more persistent. To understand this result, we turn to the response of the labor market.

By limiting the rise in the external finance premium, the policy reduces the size of the contraction in borrowed funds, which in turn limits the reduction in capital purchases by salaried firms. This pushes salaried firms to keep more of their purchased capital k_S,t in-house relative to the no-policy case – that is, ω_S,t falls by less – as firms react to the fact that the contraction in available capital next period will be less severe under the policy. Thus, the effective amount of capital within the salaried sector, ω_S,tk_S,t, contracts by less. This leads to a smaller fall in the marginal product of salaried labor, which bolsters salaried vacancy postings and leads to a smaller contraction in capital demand (and hence the price of capital) relative to the no-policy case. This explains the smaller contraction in salaried employment in the aftermath of the shock.

However, the fact that salaried firms decide to keep more of their capital within the sector implies that the reallocation of capital towards matching firms – and hence the available supply of capital to potential self-employed individuals – rises by less relative to an economy without policy.²⁶ In turn, this translates into a smaller expansion of self-employment after the shock. Importantly, as aggregate productivity returns to steady state, the initial increase in the availability of capital for the self-employed – which was already more limited under the policy – falls back to steady state earlier. This creates a long-lasting contraction in self-employment over time compared to the no-policy scenario.

In relative terms, the sharper contraction in self-employment in the aftermath of the shock more than offsets the smaller contraction and faster recovery in salaried employment. This explains the larger and more persistent rise in unemployment. Importantly, even though self-employment accounts for a smaller share of total employment relative to salaried employment, the former’s dynamic response to the shock plays an important role in driving unemployment fluctuations.

(e) Response to a Negative Salaried-Firm Net Worth Shock Figure 2 shows the response of the economy to a negative shock to salaried-firm (BGG entrepreneur) net worth. First, consider the economy without policy. Given that b_S,t = Q_S,tk_S,t+1 – nw_S,t+1, a negative net worth shock not only raises the desired amount of borrowed funds but also increases the external finance premium. In turn, the demand and the price of capital by salaried firms fall and generate a contraction in investment.

The fall in capital demand leads to a reduction in capital usage, ω_S,t, salaried employment, and salaried output. This reduction in salaried-sector capital usage leads to a reallocation of available capital towards the self-employment sector, which generates an increase in entry into self-employment and in turn an expansion in the share of self-employment. The expansion in the self-employment sector more than offsets the contraction in salaried employment, so that unemployment initially falls in response to the shock. However, the fall in salaried output ultimately generates a contraction in consumption and total output as salaried output represents more than 70 percent of total output.

An active policy has the unintended effect of putting upward pressure on the external finance premium after a negative net worth shock. This limits the rise in desired borrowed funds by salaried firms and generates sharper contractions in salaried employment and output, investment, and total output. Moreover, this implies a larger reallocation of available capital towards the self-employment sector as salaried firms reduce capital usage by more under the policy. This leads to a larger expansion in self-employment and self-employment output after the shock, and ultimately explains the sharper movements in unemployment under the policy.

Given that the net worth shock only affects the salaried sector, the policy exacerbates the differences between the salaried and self-employment sectors, thereby leading to more pronounced fluctuations relative to the no-policy scenario. Note that in contrast to the response to productivity shocks, a stronger policy under net worth shocks delivers even larger fluctuations. Thus, while macroprudential policy generates sharper unemployment fluctuations regardless of the type of shock, the latter does matter for the effects of policy on other macro aggregates.

An important caveat regarding the response of the economy to net worth shocks: while a negative shock generates a fall in unemployment in the model, this is due to the fact that the efficiency of the search market for capital is not disrupted by the shock. If we assume that negative net worth shocks can worsen the efficiency of the matching process between available capital and potential self-employed individuals, the expansion in self-employment after a negative net worth shock would be more subdued and unemployment would end up increasing after the shock.²⁷ This is indeed the case when we allow capital matching efficiency to be positively correlated with exogenous innovations to net worth.

(f) Summary of Main Results and Key Mechanisms To summarize, policy interventions that limit credit fluctuations reduce the persistence of consumption and output contractions and decrease the volatility of investment in response to productivity shocks, while they amplify the response of all these variables in response to net worth shocks. However, the beneficial effects from a countercyclical macroprudential policy under productivity shocks are quantitatively small and more importantly, a countercyclical macroprudential policy tends to amplify unemployment fluctuations regardless of of the type of shock when we consider the linkages between formal credit, informal credit, and labor markets that are more prevalent in emerging economies. The link between salaried firms with access to formal financing and the self-employed via (informal) input credit markets plays a key role in explaining the adverse impact of cyclical macroprudential policy on unemployment dynamics. Indeed, the extent to which policy affects the use of resources in the salaried sector that are potentially available for the creation of self-employment ventures has direct implications for self-employment dynamics. Given the contribution of self-employment to total employment, changes in the cyclical behavior of self-employment due to policy can ultimately lead to important changes in unemployment fluctuations.

C. Additional Results and Robustness

(g) Shocks to Self-Employment Productivity In response to a negative productivity shock to the self-employment sector, the salaried sector’s value of supplying capital to the self-employment sector falls. This reduces the supply of capital to the self-employed, raises salaried employment and output, and generates a reduction in self-employment and self-employment output. Thus, salaried and self-employment output co-vary negatively. The fact that salaried firms keep more of their capital reduces the need to invest. This leads to a contraction in the price of capital and, coupled with a fall in the supply of capital to the self-employment sector, to a fall in net worth and to an increase in the external finance premium. Introducing a countercyclical macroprudential policy increases the incentive to provide capital to the self-employment sector relative to the benchmark case with no policy, so that the drop in capital supply is smaller, leading to a smaller contraction in self-employment and output in the sector. The policy puts further downward pressure on the price of capital and leads to a larger contraction in investment. Ultimately, the larger drop in investment due to the policy limits the rise in salaried output (due to the rise in salaried employment), and the net result is higher total output volatility.²⁸ Despite this fact, the policy does reduce the variability of unemployment, which is explained by the smaller fall in self-employment when credit fluctuations are smaller.

(h) Shocks to Efficiency of Financial Intermediation We follow Gilchrist and Zakrajšek (2011) and consider a shock that increases the cost of external financing for a given leverage ratio.²⁹ An active macroprudential policy reduces the contraction in borrowed funds and leads to smaller reductions in salaried output, employment, investment, and consumption. The policy also reduces the reallocation of resources towards self-employment. Thus, similar to the case of aggregate productivity shocks, self-employment becomes less countercyclical and makes unemployment more volatile.

(i) Shocks to the Foreign Interest Rate Introducing foreign interest rate shocks hinders the model’s ability to capture the cyclicality of the labor market in the data and generates counterfactual self-employment and salaried employment dynamics when these shocks are correlated with productivity shocks. Independent foreign interest rate shocks worsen the fit of the model relative to the benchmark calibration and also fail to generate the cyclicality of interest rates in the data. Importantly, this is the case because, as was the case in Fernández and Gulan (2014), the presence of financial frictions are sufficient to capture several features of emerging economy business cycles without resorting to foreign interest rate shocks.

(j) Household Heterogeneity We modify the benchmark model to introduce two different households and two types of salaried employment. The first household decides on domestic and foreign debt holdings, and receives income from both matching-firm profits and labor income solely from the first type of salaried workers. The second household receives income from the second type of salaried workers and self-employed individuals only.³⁰ The model is in line with the stylized facts captured by the benchmark model. However, a countercyclical macroprudential policy generates asymmetric effects on household consumption in response to productivity shocks, with one household enjoying lower consumption volatility and the other facing higher consumption volatility. These effects offset each other and the end result is that aggregate consumption volatility remains virtually unchanged relative to the no-policy scenario, so that the benefits from policy in terms of lower consumption volatility are less clear. The main conclusions of the paper regarding the adverse impact of policy on unemployment dynamics remain the same.³¹