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Instructions:
Microeconomics is a branch of economics that focuses on the behavior and decision-making of individuals, households, and firms in markets. It examines how these economic agents allocate their limited resources to meet their needs and wants, and how they interact with one another in buying and selling goods and services.
Microeconomics also looks at how market forces of supply and demand affect prices and quantities of goods and services, as well as how different market structures, such as perfect competition, monopolistic competition, oligopoly, and monopoly, impact the behavior of firms and consumers.
In essence, microeconomics provides insights into how individuals and businesses make decisions about what to produce, how much to produce, and at what price to sell, and how these decisions affect the allocation of resources in society.
With your answer to the previous question in mind: what do you think of the usual risk sharing regressions? Why might a significant effect of individual income in the consumption regression not be informative about the absence of risk-sharing?
Solution: This model suggests, that individual income depends on total income and all wage rates. In particular: if consumption and leisure are substitutes (i.e. the marginal utility of consumption is high if leisure is low), individual consumption will be positively related to the wage rate – if wages are high, the individual should work and hence receive consumption due to the complementarity. If the wage rate is positively correlated with personal income, we will find in a regression that personal income is correlated with consumption conditional on aggregate income. This however, is an implication of optimal risk sharing.
Now suppose that consumption and leisure are separable in individuals’ preferences. Formally, suppose that $u^{i, h}(c, l)=v^{i, h}(c)+q^{i, h}(l)$. Which variables determine individual leisure and consumption now?
Solution: If preferences are separable, $(7)$ simplifies to
$$
\frac{u_l^{i, h}\left(l^{i, h}, c^{i, h}\right)}{u_c^{i, h}\left(l^{i, h}, c^{i, h}\right)}=\frac{q_l^{i, h}\left(l^{i, h}\right)}{v_c^{i, h}\left(c^{i, h}\right)}=w^{i, h} .
$$
This shows that (5) still applies, i.e. $l^{i, h}=\phi^{i, h}\left(c^{i, h}, w^{i, h}\right)$. All the other steps to derive the consumption allocation rule did not use the separability between consumption and leisure. Hence, individual consumption allocation depend on the same variables as above despite the separability of preferences. But: there is another relation we can use! By looking at (4) right away we get that
$$
\mu^{i, h} v_c^{i, h}\left(c^{i, h}\right)=\mu^{g, h} v_c^{g, h}\left(c^{g, h}\right)
$$
from which we get that for all $i=2, \ldots, H$
$$
c^{i, h}=\chi^{i, h}\left(c^{1, h}\right)
$$
Hence, $C^h \equiv \sum c^{i, h}=\sum \chi^{i, h}\left(c^{1, h}\right)$ so that $c^{1, h}=h\left(C^h\right)$ so that we can still write an equation of the form
$$
c^{i, h}=f^{i, h}\left(C^h\right)
$$
i.e. individual consumption should only depend on aggregate consumption (not aggregate income).
Does individual labor supply of agent $i$ in household $h$ depend on wages and incomes of individuals in the household? Why or why not?
Solution: As clearly seen from above: individual labor supply is given by
$$
T^{i, h}-l^{i, h}=T^{i, h}-\zeta^{1, h}\left(w^{1, h}, w^{2, h}, \ldots, w^{3, h}, \sum_{i=1}^{I^h} w^{i, h} T^{i, h}+X^h\right),
$$
i.e. labor supply depends on wages of other people in the household but not on their non-labor income (conditional on aggregate income $\sum_{i=1}^{I^h} w^{i, h} T^{i, h}+X^h$ ). The intuition that the wages of other individuals determine labor supply is the same as for consumption: with non-separable preferences pareto-optimality requires that marginal utilities of consumption are equalized. As this marginal utility depends on individual leisure, the labor allocations depend on the distribution of wages in the economy.
