Essays on Human Capital and Inequality

This thesis conducts positive and normative analysis of inequality based on human capital theory. In Chapter 2, we document important differences in early child investments by family income and study four leading mechanisms thought to explain these gaps: intergenerational ability correlation, consumption value of investment, information frictions, and credit constraints. We evaluate whether these mechanisms are consistent with other stylized facts related to the marginal returns on investments and the effects of parental income on child investments and skills. In Chapter 3, I study optimal higher education subsidies when parents’ willingness to pay for their children\u27s education differs due to heterogeneity in altruism. I first document substantial heterogeneity in the fraction of college expenditure paid by parents across families and provide evidence that this heterogeneity can be explained by parental altruism. Then I analytically characterize optimal education subsidies when the social planner minimizes distortions generated by borrowing constraints and can observe neither the amount of parental transfers nor parental altruism. I show that redistributing towards constrained students of low altruism parents is socially beneficial, but it involves substantial deadweight loss. The calibrated model suggests that the deadweight loss due to unobservable heterogeneity in parental altruism can be quantitatively large and therefore limit redistribution towards students with low parental transfers. In Chapter 4, we study the role of returns to unobserved skills in the rising residual earnings inequality for the past few decades in the U.S. We identify and estimate a general model of earnings residuals that incorporates (i) changing returns to unobserved skills, (ii) changing distribution of unobserved skills, and (iii) changing volatility of earnings that is not related to skills. Using data from the PSID, we find that the returns to unobserved skills went down since the mid-1980s despite the steady increase in the residual inequality. Using a simple demand and supply framework, we show that both demand and supply factors contributed to the downward trend in the returns to skills over time

N-Body Oscillator Interactions of Higher-Order Coupling Functions

We introduce a method to identify phase equations that include $N$-body interactions for general coupled oscillators valid far beyond the weak coupling approximation. This strategy is an extension of the theory from [Park and Wilson, SIADS 20.3 (2021)] and yields coupling functions for $N\geq2$ oscillators for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions enable the study of oscillator networks in terms of phase-locked states, whose stability can be determined using straightforward linear stability arguments. We demonstrate the utility of our approach with two examples. First, we use $N=3$ diffusively coupled complex Ginzburg-Landau (CGL) model and show that the loss of stability in its splay state occurs through a Hopf bifurcation \yp{as a function of non-weak diffusive coupling. Our reduction also captures asymptotic limit-cycle dynamics in the phase differences}. Second, we use $N=3$ realistic conductance-based thalamic neuron models and show that our method correctly predicts a loss in stability of a splay state for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak and recent non-weak coupling theories can not.Comment: 29 pages, 6 figure

Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation

We study a deterministic version of a one- and two-dimensional attractor neural network model of hippocampal activity first studied by Itskov et al 2011. We analyze the dynamics of the system on the ring and torus domain with an even periodized weight matrix, assum- ing weak and slow spike frequency adaptation and a weak stationary input current. On these domains, we find transitions from spatially localized stationary solutions ("bumps") to (periodically modulated) solutions ("sloshers"), as well as constant and non-constant velocity traveling bumps depending on the relative strength of external input current and adaptation. The weak and slow adaptation allows for a reduction of the system from a distributed partial integro-differential equation to a system of scalar Volterra integro-differential equations describing the movement of the centroid of the bump solution. Using this reduction, we show that on both domains, sloshing solutions arise through an Andronov-Hopf bifurcation and derive a normal form for the Hopf bifurcation on the ring. We also show existence and stability of constant velocity solutions on both domains using Evans functions. In contrast to existing studies, we assume a general weight matrix of Mexican-hat type in addition to a smooth firing rate function.Comment: 60 pages, 22 figure