Wage Risk and the Skill Premium

The skill premium has increased significantly in the United States in the last five decades. During the same period, individual wage risk has also increased. This paper proposes a mechanism through which a rise in wage risk increases the skill premium. Intuitively, a rise in uninsured wage risk increases precautionary savings, thereby boosting capital accumulation, which increases the skill premium due to capital-skill complementarity. Using a quantitative macroeconomic model, we find that the rise in wage risk observed between 1967 and 2010 increases the skill premium significantly. This finding is robust across a variety of model specifications.


Introduction
The substantial increase in the wages of college graduates relative to those without college education, the skill premium, is one of the most notable inequality trends observed in the United States in recent decades. Another important finding that has been documented by Gottschalk and Moffitt (1994), Heathcote, Storesletten, and Violante (2010), Gottschalk and Moffitt (2012) and Hong, Seok, and You (2017), among others, is that U.S. workers face a considerably higher level of individual wage risk now than in the past. This paper uncovers a link between the rise in individual wage risk and the rise in the skill premium.
In particular, we propose a mechanism through which a rise in individual wage risk leads to an increase in the skill premium and shows that this mechanism can be quantitatively significant.
The proposed mechanism is straightforward and rests on two notions with longstanding traditions in economics. The first is the precautionary savings motive, which is the idea that an increase in (uninsured) income risk induces people to save more. 1 The second is capital-skill complementarity: the idea that capital is relatively more complementary with skilled labor than it is with unskilled labor. 2 Intuitively, in a world in which insurance markets are imperfect, a rise in individual income risk increases aggregate savings through the precautionary savings motive, which boosts capital accumulation. The rise in the capital stock then increases the skill premium due to capital-skill complementarity.
We lay this mechanism out using a model that embeds capital-skill complementarity into an incomplete markets model a là Aiyagari (1994), which has been a workhorse model for studying the effects of precautionary savings on the macroeconomy. The precautionary savings motive is present in this model because individuals face uninsured idiosyncratic wage risk. Capital-skill complementarity is built in by assuming a production function that features a higher degree of complementarity between equipment capital and skilled labor than between equipment capital and unskilled labor, as documented empirically for the U.S. economy by Krusell, Ohanian, Ríos-Rull, and Violante (2000). We use this model to evaluate the quantitative significance of our mechanism for the U.S. economy by calculating the extent to which the rise in individual wage risk contributed to the rise in the skill premium between 1967 and 2010.
We solve for the stationary competitive equilibrium of this model and calibrate the model parameters to match the 1967 U.S. economy along certain dimensions including the level of the skill premium. Then, we introduce the changes in the structure of individual wage risk observed between 1967 and 2010, as estimated by Hong, Seok, and You (2017), as well as the changes in two key factors that have found significant support in the literature in terms of determining the changes in the skill premium. These factors are technological advancements that favor skilled workers (skill-biased technical change) and the rise in the relative supply of skilled workers. 3 Following Krusell, Ohanian, Ríos-Rull, and Violante (2000), skill-biased technical change is modelled as a decline in the price of equipment. Having introduced changes in the structure of wage risk, technology, and relative supply of skilled workers, we solve for the new steady state, which corresponds to 2010. The model generates a 41 percentage point increase in the skill premium, from 1.51 in 1967 to 1.92 in 2010. In the data, the U.S. skill premium increases by 39 percentage points to 1.90 in 2010. 4 We conclude that the model provides a reliable laboratory to investigate the sources of changes in the U.S. skill premium.
Next, we investigate the quantitative significance of our mechanism by calculating how much the rise in wage risk contributed to the rise in the skill premium during the period of interest. To do so, we compute a counterfactual steady state in which the structure of wage risk is as it was in 2010, but all other factors are at their 1967 levels. We find that the rise in wage risk alone increases the skill premium to 1.70, an increase of 18 percentage points relative to the 1967 steady state. In an alternative exercise, we feed in the changes in technology and the supply of skilled agents that occurred between 1967 and 2010, but keep the structure of wage risk as in the 1967 economy. In this counterfactual exercise, the U.S. skill premium equals 1.80, which is 12 percentage points shy of the model implied skill premium for 2010. 5 A comparison of these numbers to the observed 39 (or the model implied 41) percentage point change in the skill premium suggests that the rise in wage risk has a quantitatively significant effect on the skill premium. This conclusion is further supported by sensitivity analyses with respect to individuals' degree of risk aversion and the tightness of borrowing constraints.
In the baseline model, the United States is modelled as a closed economy and the supply of skilled agents is exogenous. Section 7 provides an extension in which the United States is treated as a large open economy. We find that modelling the United States as a large open economy does not affect the model's success in generating the overall rise in the U.S. skill premium. More importantly, the rise in the U.S. wage risk still contributes significantly to the rise in the U.S. skill premium. Section 8 analyzes a separate extension which shows that the effect of the rise in wage risk on the skill premium remains significant when education decisions (and hence the supply of skilled workers) are endogenous.
In our model, the rise in wage risk between 1967 and 2010 increases aggregate savings through the precautionary savings channel. At a first glance, this may seem to be in contrast with the fact that the U.S. national savings rate declined over the same period.
This conclusion is not necessarily correct, however. Arguably, the main reason for the decline in the U.S. savings rate during this period is the rise in capital inflows from abroad and the resulting decline in interest rates. If we modelled this rise in capital inflows between 1967 and 2010 taking into account transitional dynamics, our model could replicate the decline in the savings rate even in the presence of the rise in precautionary savings. The fact that we do not match the decline in U.S. savings does not mean that the rise in wage risk has been quantitatively unimportant for aggregate savings. 5 The fact that the two counterfactuals attribute different magnitudes to the importance of wage risk (18 vs. 12 percentage points) alludes to the significant non-linearities present in the model. We would also like to stress that this paper does not aim to propose a competing theory of what determines changes in the skill premium. To the contrary, our model confirms the findings in the existing literature that skill-biased technical change and changes in the supply of skilled workers have been the most important determinants of changes in the skill premium. Our analysis merely puts forth the idea that changes in wage risk can have a quantitatively significant effect on the skill premium.
Related Literature. This paper relates to a large literature that aims to explain the rise of the skill premium in the United States in recent decades. Goldin and Katz (2008) is a monumental work analyzing the evolution of the U.S. wage structure in general and the skill premium in particular through the lens of Tinbergen's (1974) model of the race between education and technology. Krusell, Ohanian, Ríos-Rull, and Violante (2000) estimate a production function with equipment and structure capital and skilled and unskilled labor, and use this production function along with the observed changes in inputs to explain the evolution of the skill premium between 1965 and 1992. Buera, Kaboski, and Rogerson (2015) analyze the role of structural change on the skill premium between 1977 and 2005. The authors find that structural change has also been skillbiased and has contributed significantly to the rise of the skill premium during this period. Burstein and Vogel (forthcoming) build a Ricardian model of international trade in which there are skill intensity differences across firms and sectors and skill abundance differences across countries. Using this model, the authors show that reductions in trade costs can generate significant increases in the skill premium in almost all 60 countries in their sample. 6 We add to this literature by uncovering a novel factor that has contributed to the observed rise in the skill premium; the increase in wage risk.
Ex-ante idiosyncratic wage risk implies ex-post within-group inequality in our model. In this sense, one possible interpretation of this paper is that it proposes a mechanism that links within-group inequality and between-group inequality (the skill premium). Gu-6 Other contributions to the study of the determinants of the skill premium include Hornstein and Krusell (1996), Greenwood and Yorukoglu (1997), Galor and Tsiddon (1997), Heckman, Lochner, and Taber (1998), Caselli (1999), Violante (2002), Domeij and Ljungqvist (2006), He and Liu (2008), Guvenen and Kuruscu (2010) and Burstein, Morales, and Vogel (2015), among others. See also Larrain (2015) for a contribution related to the open economy extension analyzed in this paper. venen and Kuruscu (2012) build a model of the labor market where, for all workers, both skilled and unskilled, labor consists of two parts: raw labor and human capital. Using this model, the authors show that (exogenous) skill-biased technical change can explain the evolution of both within-group and between-group inequalities. Acemoglu (1998) proposes a directed technical change model in which an increase in the relative supply of the skilled workers encourages firms to develop technologies that are more complementary to these workers, creating an endogenous skill bias in technological progress. In an extension of this model, the author shows that his framework has the potential to explain the joint evolution of the between-and within-group inequalities.
This paper is also related to a large incomplete markets literature in the Bewley (1986), Imrohoroglu (1989), Huggett (1993) and Aiyagari (1994) tradition. The two papers that are most closely related to ours in this literature are Heathcote, Storesletten, and Violante (2010) and Hong, Seok, and You (2017). Heathcote, Storesletten, and Violante (2010) estimate the changes in wage risk over time and analyze the macroeconomic implications of the rise in wage risk, also taking into account changes in the skill premium and the gender gap. Hong, Seok, and You (2017) estimate changes in wage risk separately for the skilled and unskilled and analyze the implications of the rise in wage risk for the labor supply of skilled and unskilled workers. 7 The rest of this paper is structured as follows. Section 2 describes the model in detail while Section 3 briefly discusses the mechanism and provides a layout of its quantitative assessment. Section 4 explains how the model is calibrated to the 1967 U.S. economy.
Section 5 summarizes the changes in skill premium, wage risk and other factors between 1967 and 2010. Section 6 discusses our main quantitative findings. In Sections 7 and 8, we consider two extensions to the basic framework: An open economy extension with international trade in goods and capital, and an extension to the basic framework in which people's skill supply is endogeneous. Section 9 concludes. The Appendix contains the details of our empirical calculations.

Model
This section develops an infinite horizon closed economy growth model with two types of capital (structure and equipment capital), two types of labor (skilled and unskilled), consumers, a firm, and a government.
Demographics. The total population size is assumed to be unity. We adopt a version of the Yaari (1965) perpetual youth model in which agents are born at age zero and survive from age h to age h + 1 with constant probability δ < 1. A new generation with mass (1 − δ) enters the economy at each date t with zero asset holdings. The assets of deceased people are distributed among the survivors proportional to the survivors' wealth. This assumption is equivalent to assuming that people can buy actuarially fair life insurance policy. Life before labor market entry is not modelled.
Skill Heterogeneity and Wage Risk. Ex-ante, agents differ in their skill levels: they are born either skilled or unskilled, i ∈ {s, u}, and remain so until the end of their lives. Skilled agents can only work in the skilled labor sector and unskilled agents only in the unskilled labor sector. Agents of skill type i receive a wage rate w i for each unit of effective labor they supply. The total masses of skilled and unskilled agents are denoted by π s and π u , respectively. In the quantitative analysis, skill types correspond to educational attainment at the time of labor market entry. Agents with college education or above are classified as skilled agents and the rest of the agents are classified as unskilled agents.
Section 5 describes the increase in the relative supply of skilled workers, π s , observed between 1967 and 2010.
In addition to heterogeneity between skill groups, there is ex-post heterogeneity within each skill group because agents face idiosyncratic labor productivity shocks every period.
The productivity shock z i denotes how many units of effective labor per unit of time an agent is able to provide. As a result, an agent's wage rate per unit of time is w i · z i , where w i is the marginal product of effective labor of skill type i. The stochastic processes that govern this wage risk are allowed to be different for the two skill groups. The logarithm of z i is modelled as a sum of two orthogonal components: a persistent autoregressive shock and a transitory shock. More precisely, (1) where ε i,t and κ i,t are independently and identically distributed across agents and over time according to a normal distribution with mean zero and variances σ i,ε and σ i,κ . ξ i controls the degree of persistence of the persistent component. Agents draw the initial value of the persistent component of their labor productivity at age h = 0 from a normal distribution with mean zero and variance σ i,θ . For notational simplicity, we define the vector of idiosyncratic productivity components by The change in idiosyncratic wage risk is modelled by allowing the variances σ i,ε and σ i,κ to change over time. Section 5 discusses the observed changes in these variances between 1967 and 2010. We normalize the mean levels of the idiosyncratic labor productivity shocks to one, i.e., set E[z i ] = 1 in both 1967 and 2010 for both skill types. This normalization ensures that changes in the stochastic processes for z i s over time are purely changes in risk. As a result, the skill premium in the model economy is given by the ratio of the marginal products of (effective) labor w s /w u .
Preferences. Preferences over sequences of consumption and labor, (c i,h , l i,h ) ∞ h=0 , are defined using a time-separable utility function where β ∈ (0, 1) is the time discount factor. The function u(·) is strictly increasing and concave in consumption and strictly decreasing and convex in labor. The unconditional expectation, E i is taken with respect to the stochastic process governing the idiosyncratic wage risk for an agent of skill type i. There are no aggregate shocks. Modelling elastic labor supply is especially important since this margin gives agents an additional tool to insure themselves against income shocks.
Technology. There is a constant returns to scale production function: Y = F (K s , K e , L s , L u ), in which K s and K e refer to aggregate structure capital and equipment capital and L s and L u refer to aggregate effective skilled and unskilled labor, respectively. δ s and δ e denote the depreciation rates of structure and equipment capital, respectively.
The key feature of the technology we use in our quantitative analysis is equipment-skill complementarity, which means that the degree of complementarity between equipment capital and skilled labor is higher than that between equipment capital and unskilled labor. This implies that an increase in the stock of equipment capital decreases the ratio of the marginal product of unskilled labor to the marginal product of skilled labor. In a world with competitive factor markets, this in turn implies that the skill premium, defined as the ratio of skilled to unskilled wages, is increasing in equipment capital. Structure capital, on the other hand, is assumed to be neutral in terms of its complementarity with skilled and unskilled labor. These assumptions on technology are consistent with the estimation results of Krusell, Ohanian, Ríos-Rull, and Violante (2000).
Since the two types of labor are not perfect substitutes, the production function implies that an increase in the skilled labor supply, which makes skilled labor less scarce, leads to a decrease in the skill premium. An increase in the unskilled labor supply has the opposite effect.
Finally, at time t, one unit of the general consumption good can be converted into one unit of structure or into 1 qt units of equipment capital. This means that the relative prices of structure and equipment capital in terms of the general consumption good are 1 and q t , respectively. Following the literature, we model skill-biased technical change as a decline in q over time. Section 5 discusses the observed change in q between 1967 and

2010.
Production. There is a representative firm which, in each period, hires the two types of labor and rents the two types of capital to maximize profits. In any period t, the maximization problem of the firm reads: max Ks,t,Ke,t,Ls,t,Lu,t F (K s,t , K e,t , L s,t , L u,t ) − r s,t K s,t − r e,t K e,t − w s,t L s,t − w u,t L u,t , where r s,t and r e,t are the rental rates of structure and equipment capital, and w u,t and w s,t are wage rates paid to unskilled and skilled effective labor in period t.
Government. The government applies linear taxes to capital income net of depreciation. The tax rates on the two types of capital can, in general, be different. Let {τ s,t } ∞ t=0 and {τ e,t } ∞ t=0 be the sequences of tax rates on structure and equipment capital.
It is irrelevant for our analysis whether capital income is taxed at the consumer or at the corporate level. We assume without loss of generality that all capital income taxes are paid at the consumer level. The government taxes labor income using a sequence of possibly non-linear functions {T t (y)} ∞ t=0 , where y is labor income and T t (y) are the taxes paid by the consumer. This function makes it possible to model the progressivity of the U.S. labor income tax code, which is important because progressive taxes provide (partial) insurance against wage risk. The government uses taxes to finance a stream of Asset Market Structure. Government debt is the only financial asset. It has a one period maturity and return R t . Consumers can also save through the two types of capital. In the absence of aggregate shocks, the returns to savings in the form of the two capital types are certain, as is the return on government bonds. Therefore, all three assets must yield the same after-tax return in equilibrium, i.e., R t = 1 + (r s,t − δ s )(1 − τ s,t ) = qt+(re,t−qt·δe)(1−τe,t) qt . As a result, one does not need to distinguish between savings via different types of assets in the consumer's problem. Consumers' (total) asset holdings will be denoted by a and A = [0, ∞) denotes the set of possible asset levels that agents can hold. Section 6.6 analyzes an extension in which agents are allowed to borrow. Our assumptions imply that, in every period, the total savings of consumers must be equal to the total borrowing of the government plus the total capital stock in the economy.
Our quantitative analysis focuses on the comparison of stationary equilibria where one stationary equilibrium corresponds to 1967 and another one to 2010. For that reason, instead of giving a general definition of competitive equilibrium, we only define stationary recursive competitive equilibria. In order to define a stationary equilibrium, government policies (expenditure, debt and taxes) are assumed not to change over time.
SRCE is two value functions V u , V s , policy functions c u , c s , l u , l s , a u , a s , the firm's decision rules K s , K e , L s , L u , government policies τ s , τ e , T (·), D, G, two distributions over productivity-asset types λ u , λ s and prices w u , w s , r s , r e , R such that 1. The value functions and the policy functions solve the consumer problem given prices and government policies, i.e., for all i ∈ {u, s}: is the after-tax asset return.
2. The firm solves: max Ks,Ke,Ls,Lu 3. The distribution λ i is stationary for each type, i.e. ∀i = {s, u}, is computed according to the initial unconditional distribution of entrants over the persistent component θ i .

Markets clear:
5. The government budget constraint is satisfied:

The Mechanism and its Quantitative Assessment
The main purpose of this paper is to propose a mechanism through which a rise in individual wage risk leads to an increase in the skill premium. This section first discusses how this mechanism works in the model laid out in Section 2. Second, we describe how the mechanism is evaluated quantitatively in the rest of the paper.
In the model outlined in Section 2, a rise in wage risk increases people's (precautionary) savings because insurance markets are incomplete. The resulting increase in aggregate savings implies a corresponding increase in aggregate capital stock, which implies an increase in the levels of equipment and structure capital. Recall that, by construction, productivity shocks are mean one in the model. As a result, the skill premium, which is defined as the average wage rate of the skilled, divided by the average wage rate of the unskilled, equals w s /w u , which satisfies The assumption of equipment-skill complementarity means that the ratio on the right-hand side of equation (3) is increasing in the level of equipment capital. Under the assumption that structure capital is neutral, this implies that the skill premium is increasing in aggregate capital stock.
In order to evaluate this mechanism quantitatively, we first solve for

Calibrating the Model to 1967
One period in the model corresponds to one year. The structural parameters of the model are set assuming that the SRCE of our model economy under the 1967 technology, relative supply of skilled workers, residual wage risk, and government policies coincides with the U.S. economy of 1967. We first fix a number of parameters to values from the data or from the literature. These parameters are summarized in Table 1. We then calibrate the remaining parameters so that the SRCE matches the U.S. data in 1967 along selected dimensions. The internal calibration procedure is summarized in Table 2. 1967 is chosen as the starting year because the earliest available estimates for individual labor income risk, from the Panel Study of Income Dynamics (PSID), are from 1967. For data availability reasons, we focus on working age males, when we compare the model with data. This concerns the skill premium and educational attainment as well as the idiosyncratic productivity processes. The details of our data work are included in the Appendix.
Technology. The production function takes the same form as in Krusell, Ohanian, Ríos-Rull, and Violante (2000): In this formula, ρ controls the degree of complementarity between equipment capital and skilled labor while η controls the degree of complementarity between equipment capital and unskilled labor. The parameter α gives the income share of structure capital. Krusell, Ohanian, Ríos-Rull, and Violante (2000) estimate α, ρ, η, and we use their estimates.
Their estimates of ρ and η imply that equipment capital is more complementary with skilled than unskilled labor. The other two parameters in this production function, ν and ω, jointly control the income shares of equipment capital, skilled labor and unskilled labor. These parameters are calibrated internally, as explained in detail later. The price of equipment capital is normalized to one, q = 1, for the benchmark 1967 calibration.
Preferences and Demography. The period utility function takes the Balanced Growth Path compatible form: Here, σ equals the coefficient of relative risk aversion in consumption. The parameter φ together with σ controls the average labor supply and the Frisch elasticity of labor supply.
In the benchmark case, we use σ = 2 and calibrate φ to match the average labor supply. All parameter values regarding the idiosyncratic wage risk are reported in the 'Residual Wage Risk' panel of Table 1. Numerically, these processes are approximated by finite number Markov chains using the Rouwenhorst method described in Kopecky and Suen (2010).
Government. The government consumption-to-output ratio is set to 16%, which approximately equals the average ratio in the United States during the period 1970-2012, as reported in the National Income and Product Accounts (NIPA) data.
As for labor income taxes, modelling the progressivity of the U.S. tax system is important for measuring the importance of changes in risk. This is because progressive tax systems provide partial insurance against labor income risk. A higher degree of progressivity can decrease the after-tax income risk agents face and thereby decrease the need for precautionary savings. To approximate the progressive U.S. labor tax code, we follow Heathcote, Storesletten, and Violante (2017) and assume that tax liability given labor income y is defined as: whereȳ is the mean labor income in the economy, 1 − χ is the average tax rate of a mean income individual, and τ l controls the progressivity of the tax code. Using PSID data Auerbach (1983) documents that the effective tax rates on structure capital and equipment capital have historically differed at the firm level. Specifically, he computes the effective corporate tax rate on structure capital and equipment capital from 1953 to 1983.
According to his estimates, in the 1960s, the average tax rate on equipment capital was approximately 41%, while the average tax on structures was approximately 49% at the firm level. 11 We further assume that the capital income tax rate at the consumer level is 15%, which approximates the U.S. tax code. This implies an overall tax on structure capital of τ s = 1 − 0.85 · (1 − 0.49) = 56.65% and an overall tax on equipment capital of 10 Importantly, the tax liability measure of Heathcote, Storesletten, and Violante (2017) takes into account public cash transfers (AFDC/TANF, SSI and other welfare receipts, unemployment benefits, workers' compensation, and veterans' pensions). As a result, taxes at the bottom of the income distribution are in fact negative, providing insurance against the risk of low income. 11 The annual tax rate estimates of Auerbach (1983) are very volatile due to changing inflation rates across years. For this reason, we use the averages over the 1960s instead of the point estimates for 1967 to represent the tax rates at the initial steady state.  agencies. Because the benchmark model economy is a closed economy, federal debt held by foreign and international investors is also excluded when computing the government debt-to-GDP ratio.
Internal Calibration. There are still five parameter values that are left to be determined: the two production function parameters, ω and ν, which (given the income share of structure capital) jointly govern the income shares of equipment capital, skilled labor and unskilled labor, the utility parameter φ, the discount factor β, and the parameter governing the overall level of taxes in the tax function, χ. We calibrate ω and ν to ensure that the model matches the 1967 U.S. economy regarding the following two moments: the first moment is the share of labor income in total income. The labor share is computed from NIPA using the methodology described in Ríos-Rull and Santaeulàlia-Llopis The preference parameter φ is set so that the aggregate labor supply in steady state equals 1/3 as commonly assumed in the literature. The discount rate β is calibrated to match the U.S. capital-to-output ratio, which is calculated using the NIPA and Fixed Asset Tables. Housing is excluded from both capital stock and output calculation to be consistent with Krusell, Ohanian, Ríos-Rull, and Violante (2000) whose elasticity estimates we use. The resulting annual capital-output ratios vary between 1.8 and 2.4 during 1967-2010, mostly due to short-term fluctuations in output. To abstract away from these fluctuations, we take the average of annual capital-output ratios over this period, which gives a capital-to-output ratio of 2 for the U.S. economy. Finally, χ is chosen to clear the government budget constraint in equilibrium. Table 2

Changes in Wage Risk and the Skill Premium
The skill premium in the United States rose by a significant margin between 1967 and 2010. Heathcote, Perri, and Violante (2010)  During the same time period, the U.S. economy also experienced a significant increase in residual wage risk. Table 3 below reports the rise in the estimates of the variances of both persistent and transitory shocks provided by Hong, Seok, and You (2017) between   In the calculations provided by both Cummins and Violante (2002) and DiCecio (2009), the price of structure capital relative to consumption remains virtually constant during this period. For this reason, we keep the price of structures at its normalized 1967 price of 1.

Changes in Other Factors
Supply of Skilled Workers. We compute the fraction of skilled workers for 2010 following the same procedure we use to compute it for 1967. As before, we consider only males who are 25 years and older and who have earnings. We find that the fraction of We keep government consumption as a fraction of GDP constant between 1967 and 2010, because it is fairly constant in the data. We also assume that the progressivity of the labor tax code did not change between 1967 and 2010. We do so because existing estimates of progressivity rely on the TAXSIM program, which does not include state taxes prior to 1978 and, hence, labor tax progressivity for 1967 cannot be properly measured. 12

Quantitative Results
The main purpose of this section is to evaluate the quantitative implications of the rise in residual wage risk for the skill premium using our model. Before doing that, Section 6.1 analyzes to what extent the model can account for the change in the skill premium 12 Kaymak and Poschke (2015) try to overcome the issue, and estimate that τ l = 0.08 in 1967 and τ l = 0.17 in 2010. We conducted an exercise in which we changed the labor tax progressivity over time using their estimates and found that this does not have a significant effect on our main quantitative results.  it makes more sense to use the national savings rate rather than the household savings rate. The net national savings rate in the data declined between 1967 and 2010 whereas it stays constant in the model at 0%. This is because in both steady states investment simply replaces depreciated capital.

Quantitative Significance of the Changes in Wage Risk
Having checked the success of the model in replicating the observed rise in the skill premium between 1967 and 2010, we now use the model to perform two counterfactual exercises that allow us to quantify the effect of rising wage risk on the skill premium.  Table 5 shows that the increase in wage risk has a quantitatively significant effect 13 The labor share in the model remains roughly constant between 1967 and 2010. This is in contrast to recent empirical work which argues that labor share has declined in the last few decades; see, for instance, Karabarbounis and Neiman (2014). The fact that the production function we use does not capture the recent decline in the labor share is known from Krusell, Ohanian, Ríos-Rull, and Violante (2000).  and 2010 that the model predicts. The change in wage risk affects the skill premium through the following mechanism. An increase in residual wage risk leads to higher precautionary savings. 14 Higher savings then lead to higher levels of equipment capital.
Due to equipment-skill complementarity present in the production function, this leads to an increase in the skill premium. We verify this mechanism by computing the change in the stock of equipment capital that occurs due to the rise in wage risk. In the "Only Risk" exercise, equipment capital stock increases by about 20%. In the "All but Risk" exercise, the level of equipment capital is about 17% lower than in the exercise in which all factors, including risk, change to their 2010 levels.
The significance of our mechanism hinges upon the fact that (precautionary) savings respond strongly to changes in wage risk. The strong precautionary savings response in our model is consistent with Pijoan-Mas (2006) who shows that in a similar incomplete markets model calibrated to the U.S. economy precautionary savings are quantitatively important.
Insurance. The rise in wage risk increases the skill premium through the precautionary saving channel to the extent that wage risk is uninsured. In this sense, it is important that our model generates a reasonable degree of insurance possibilities for agents. In this section, we provide a measure of the degree of insurance available in the model and compare it with the empirical literature. One way economists measure insurance is by computing the degree of pass-through from earnings to consumption. Formally, the passthrough coefficient from earnings to consumption is defined as the regression coefficient b of the following panel regression: where ∆c t denotes change in individual log consumption between t − 1 and t and ∆y t denotes change in individual log earnings. The estimated coefficient b is expected to take values between 0 and 1. A pass-through coefficient of 0 would mean perfect insurance, whereas that of 1 would imply no insurance.
Computing the pass-through coefficient using model simulated data, we find it to be 0.19 in the 1967 steady state. This value is close to the pass-through coefficients implied by Blundell, Pistaferri, and Preston (2008) who use the Panel Study of Income Dynamics and Consumer Expenditure Survey data to analyze the evolution of the extent of partial insurance in the United States in the 1980s and early 1990s. Replicating the simple regression above gives an average value of around 0.2 for the pass-through from consumption to earnings in the 1980s in their work. We also calculate the passthrough coefficient in the 2010 steady state and find it to be 0.17. This decline in pass-through is also in line with Blundell, Pistaferri, and Preston (2008), who find that the consumption pass-through of earnings declined in the 1990s. The fact that our model generates reasonable pass-through values suggests that through endogenous labor supply, progressive taxation and self-insurance the model provides a realistic degree of insurance opportunities.

Two Decompositions
To shed more light at the main quantitative results, this section provides two decomposition exercises. These results are reported in Table 6. The changes in the persistent components of the wage processes of the skilled and unskilled agents account for most of the change in the skill premium arising from changes in risk. This is despite the fact that the increases in the variances of the transitory and persistent components are of similar magnitudes (see Table 3). The fact that the persistent component is more important for the skill premium is expected, since transitory shocks are well insured by precautionary savings in this economy, while persistent shocks are not. The increase in persistent shock variances then implies larger precautionary savings responses.
Wage Risk of Skilled vs. Unskilled Agents. Table 7 reports the results for the second decomposition exercise. The column entitled 'Skilled only' displays that the skill  Table 3, skilled agents' wage risk has increased more (in particular, the volatility of the persistent component, which is more important for the skill premium as discussed above, has risen), and, hence, one expects a larger response in terms of precautionary savings and the skill premium. This turns out to be true despite the fact that there are relatively few skilled people in the 1967 economy, specifically, 13.56%.

Race Between Technology and Education
This paper does not claim that wage risk is the most important individual determinant of the skill premium. We acknowledge that skilled-biased technical change and the rise in the supply of skilled workers have individually been more important for the skill premium.
We view this paper as proposing a novel idea that changes in wage risk can have a quantitatively significant effect on the skill premium.
We perform two additional counterfactual exercises which illustrate the importance of the skill-biased technical change and the rise in the supply of skilled workers for the skill premium in our model. In the first exercise, we solve for a steady state in which we keep all parameters as in 1967 and only change the fraction of skilled workers, π s , from its 1967 value of 13.56% to its 2010 value of 31.69%. The resulting skill premium is 0.58, meaning that skilled workers have, on average, lower wages than unskilled workers. values. The first and third rows report the same set of values for σ = 1 and σ = 3 cases.
In the second exercise, we solve for a steady state in which we only change the price of equipment, q, from its 1967 normalized value of 1 to its 2010 value of 0.1577. The resulting skill premium is 3.75.

Sensitivity to Risk Aversion
In the proposed mechanism, the link between wage risk and the skill premium works through precautionary savings. It is then natural to think that the strength of this mechanism may depend on the degree of risk aversion. The aim of this section is to measure how sensitive our results are to the degree of risk aversion. To this end, this section repeats the main quantitative exercises for relative risk aversion coefficients of σ = 1 and σ = 3, in addition to the benchmark value of σ = 2. We recalibrate the internally calibrated parameters for each σ exercise. The results of the quantitative exercises are presented in Table 8.
Each row in Table 8   The first row of this table report skill premia for the benchmark economy without any borrowing. increases.

Borrowing Constraints
In the benchmark model, borrowing is ruled out. This section analyzes a version of the model in which borrowing is allowed, but limited by an exogeneous upper bound, i.e.
a ≥ −ā, whereā is the maximum amount an agent can borrow. Heathcote, Storesletten, and Violante (2010) setā so that the fraction of people with negative wealth in their model is 15%, which approximates the corresponding data moment for their period of interest . We follow them and chooseā to ensure that approximately 15% of the agents have negative wealth in 1967. This exogenous limit is held constant in the '2010' and the 'Only Risk' exercises. A comparison of the two rows of Table 9 shows that the main quantitative results are not affected by the presence of borrowing constraints.

Open Economy
The United States is not a closed economy, but, following the literature, earlier sections use that scenario as a useful benchmark. In a closed economy, the rise in aggregate savings

The Rest of the World Economy
The rest of the world economy is intentionally kept simple. In the rest of the world, labor is inelastically supplied with the following preference specification for consumption: whereσ refers to the coefficient of relative risk aversion of the consumers in the rest of the world. Foreign consumers evaluate stochastic sequences of consumption (ĉ t ) ∞ t=0 according to 15 Economists sometimes model the United States as a small open economy with a fixed interest rate. This assumption is not realistic but in many cases it provides computational or analytical convenience without affecting results. In our analysis, this is not the case since the relative size of the United States matters for what fraction of U.S. savings can be absorbed by the rest of the world. In particular, if one assumes that the United States is a small open economy, any change in U.S. savings would be fully absorbed by the rest of the world with no impact on the U.S. skill premium (assuming there are no frictions in international financial markets).
whereβ is the discount factor of the rest of the world andψ is an exogenous savings wedge.ψ should be thought of as an exogenous parameter, the rise of which provides a reduced form way of introducing global imbalances into the model economy. In doing so, we are following the "global savings glut" hypothesis, which argues that global imbalances are caused by a shift in savings behavior in the rest of the world. 16 We calibrate two values forψ, one for the 1967 and another for the 2010 steady states, to ensure that the model matches the observed drop in the U.S. net foreign asset position during this period.
There is only one type of capital and labor in the rest of the world, and production takes place according to a standard Cobb-Douglass production function Agents in the rest of the world face idiosyncratic labor income risk,ẑ, where, as in the U.S. economy, the logarithm ofẑ is the sum of two orthogonal components: a persistent autoregressive shock and a transitory shock. More precisely, whereε t andκ t are drawn from distributions with mean zero and variances σε and σκ. ξ represents the persistence parameter of the AR(1) process that governs the persistent component.
Agents from the two countries (the United States and the rest of the world) can engage in intertemporal bond trading with one another at the world interest rate R. Letting IIP andÎIP denote the net international investment positions of the United States and rest of the world economies respectively, the market clearing for the world bond market is given by There is no government in the rest of the world.  LG, Maddison and NIPA refer to LeBlanc and Georgarakos (2013)

Model Fit
The results of the open economy exercise are reported in the third and fourth columns of Table 11. For comparison purposes we also report the findings from the closed economy Second, as the third row of Table 11 shows, domestic savings in the open economy in 2010 are smaller than domestic savings in the closed economy, because the interest rate is

Quantitative Significance of the Rise in Wage Risk
Next we investigate whether the rise in individual wage risk is still quantitatively important for the skill premium between 1967 and 2010 if the U.S. is modelled as an open economy. The third column of Table 12 reports that the rise in wage risk still contributes substantially to the rise in the skill premium: if the U.S. wage risk is the only factor allowed to change, and all other factors in the United States and the rest of the world remain constant, the skill premium in the United States would still go up by 11 percentage points (see the column "Only Risk"). The effect of the rise in wage risk on the skill premium is smaller than it is in the closed economy, because in an open economy part of the rise in aggregate savings is absorbed by the rest of the world. As a result, the stock of equipment capital increases only by 8% (relative to the 20% in the closed economy exercise). Notice that in the counterfactual exercise above the rest of the world absorbs a large fraction of the increase in U.S. (precautionary) savings because the rest of the world economy is kept as it was in 1967. In particular, the rise in the savings in the rest of the world (the savings glut) is not modelled. As an alternative exercise, we conduct the "Only Risk" exercise in a world in which we simulate the savings glut by increasing the rest of the world economy's preference parameterψ from its calibrated level in 1967 to 2010. The last column of Table 12 shows that the rise in wage risk increases the skill premium by 14 percentage points in this case. The effect of the rise in wage risk on the skill premium is larger in this case since, with the savings glut, the rest of the world absorbs the rise in U.S. savings (coming from the rise in U.S. wage risk) to a lesser degree.

Endogenous Skill Supply
In the baseline environment, the fraction of skilled and unskilled agents are assumed to be fixed since the main focus of this paper is the relative prices of labor given the observed relative supply of skilled agents. In particular, in the counterfactual exercise in which only risk changes, we do not allow for skill supplies to change. This section provides an alternative measure of how much the rise in wage risk affects the skill premium, which takes into account the effect of the rise in risk on people's education decisions.
Education. In this version of the model, agents make education decisions at the beginning of their lives, just before they enter the labor market and before they draw the first wage shocks. They can choose to pursue a college degree, in which case they will be called skilled agents, i = s, or a lower level of education, in which case they will be called unskilled agents, i = u. As before, skilled agents can only work in the skilled labor sector, and unskilled agents only in the unskilled labor sector. As in Heathcote, Storesletten, and Violante (2010), there is a utility cost of attaining a college degree, ψ, which is idiosyncratic and drawn from a distribution F (ψ). This distribution is a reduced form way of capturing the cross-sectional variation in the psychological and pecuniary costs of acquiring a college degree such as variation in scholastic talent, tuition fees, parental resources, access to credit, and government aid programs.
Upon drawing the cost of education, ψ, the agent compares this cost to the benefit of attaining a college degree, which is simply the net present utility gain of receiving the skilled wage rather than the unskilled wage in each date and state after entering the labor Since people choose whether to become skilled or not, the fraction of skilled people in the economy, π s , which was a parameter in the baseline model, becomes an endogeneous variable in this section. The rest of the economic environment is identical to the one developed in the model section and will not be described here.
To conduct quantitative work, the cost distribution for attending college needs to be specified. We assume that the utility cost of attending college is distributed according to an exponential distribution with parameter m, a pdf, f (ψ) = me −mψ , and a cdf, Observe that, for the marginal agent who chooses to go to college, the cost of attending college exactly equals the benefit of doing so,ψ : 0)]. Moreover, the total measure of agents who face an education cost that is at mostψ is equal to π 67 s in 1967. Thus, we calibrate m by setting F (ψ) = π 67 s . 18 The results of the "Only Risk" counterfactual exercise under endogenous skill supply are summarized in the last two columns of Table 13. For comparison purposes, the table also reports the benchmark exogenous skill supply exercise in the first two columns of Table 13. The effect of the rise in wage risk on the skill premium is larger in the endogenous skill supply exercise (34 vs. 18 percentage points). This finding might seem surprising. In particular, we know from the exogenous skill supply analysis that the rise in wage risk increases the skill premium (from 1.51 to 1.70). Other things being equal, an increase in the skill premium makes attaining a college degree more desirable. Therefore, one may expect that, with endogenous skills, a higher fraction of the population would attend college, which would then dampen the rise of the skill premium. However, recall from Table 3 that wage risk has increased more for skilled than for unskilled agents, making college education less attractive. Overall, attending college could become more or less desirable depending on which of the two effects dominates. It turns out that, under our parameterization, the latter effect dominates and fewer people attend college in the 'Only Risk' exercise as the last row of Table 13 shows. This increases the skill premium further.
18 The quantitative results are almost identical if, instead of being exponential, the utility cost distribution F is assumed to be uniform U [0, m] and one calibrates m to the same target, i.e. so that F (ψ) = π 67 s .

Conclusion
This paper proposes a mechanism through which a rise in individual wage risk leads to an increase in the skill premium. Intuitively, a rise in uninsured wage risk increases precautionary savings. The resulting rise in the capital stock increases the skill premium due to capital-skill complementarity. To evaluate the significance of this mechanism, we build a quantitative macroeconomic model with incomplete markets and capital-skill complementarity. The rise in wage risk observed in the United States between 1967 and 2010 increases the skill premium by at least 10 percentage points (25% of the observed change) in all the parameterizations and specifications of the model. We conclude that changes in wage risk can have a quantitatively important effect on the skill premium.
individuals whose hourly wage rate is less than half of the federal minimum wage.

A.2 Closed Economy
Skill biased technical change. We use the declining price of equipment as a measure Fraction of skilled agents. The fraction of skilled agents is calculated using CPS.
There was a change in terms of how CPS reports educational attainment in 1992. We follow Krusell, Ohanian, Ríos-Rull, and Violante (2000) and define the fraction of skilled agents as follows. For 1967, it is the ratio of males aged 25 and older with earnings and 4 years or more of college divided by the total number of males aged 25 and older with earnings in Table P-17. For 2010, it is the ratio of males aged 25 and older with earnings and a bachelor's degree or more divided by the total number of males aged 25 and older with earnings in Table P-16. Government consumption-to-GDP ratio. The government consumption-to-output ratio is recovered from the National Income and Product Accounts (NIPA) data. It is defined as the ratio of nominal government consumption expenditure (line 15 in NIPA Labor share. The labor share is computed from NIPA using the methodology described in Ríos-Rull and Santaeulàlia-Llopis (2010) and for details, we refer the reader to that paper. It offers several alternative ways of calculating the labor share. We use the following: we first calculate what Ríos-Rull and Santaeulàlia-Llopis (2010) call "unambiguously capital income" and "unambiguously labor income." Income which cannot be unambiguously classified as labor or capital income is then divided between capital and labor using the ratio between capital and labor income in unambiguously assigned income. To get the labor share, labor income is then divided by GDP.
Capital-to-output ratio. Housing is excluded from both output and capital when calculating the capital-to-output ratio. For this calculation, output is defined using Table   1.5.5 in NIPA as GDP (line 1) net of Housing and utilities (line 16) and Residential investment (line 41). Capital stock is calculated using the Fixed Asset Tables (FAT),