(36 points) Consider a T period problem where consumers have preferences over co
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Question
(36 points) Consider a T period problem where consumers have preferences over consumption and leisure. Consumers can work in competitive rms which hires labor and capital. The government needs to nance a stream of government expenditure {gt}T t=0 ={g0,g1,....gT}. In Economy 1, the government uses only consumption taxes at the rate {Tct}T t=0 In Economy 2, the government only taxes capital income at the rate {Tkt}T t=0
i) (8 points) For Economy 1, dene the Arrow Debreu competitive equilibrium.
ii) (8 points) For Economy 1, set up the Lagrangian, take rst order conditions and nd the Euler equation for consumption.
iii) (8 points) For Economy 2, dene the Arrow Debreu competitive equilibrium.
iv) (8 points) For Economy 2, set up the Lagrangian, take rst order conditions and nd the Euler equation for consumption.
v) (4 points) Find a relationship between the tax rates in the two economies such that the allocations are identical. (This involves nding a condition such that the Euler Equation for the two economies are identical.)
Explanation / Answer
Arrow Debreu competitive equilibrium:
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.
The model is central to the theory of general (economic) equilibrium and it is often used as a general reference for other microeconomic models.
The Arrow–Debreu model specifies the conditions of perfectly competitive markets.
In financial economics the term Arrow–Debreu is most commonly used with reference to an Arrow–Debreu security. A canonical Arrow–Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (a so-called "state price"). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow–Debreu securities.
Euler equation for consumption:
This equation is called Euler equation: it describes how consumption is optimally related across time. - If r(t) = 1 marginal utility of consumption is constant over time. - If r(t) < (>)1 marginal utility of consumption is increasing (decreasing) over time.
The relationship between the tax rates in the two economies:
A consumption tax (also known as a cash-flow tax, expenditure tax, or consumed income tax) is levied on goods and services that are consumed. While an income tax is based upon income earned from labor or capital, a consumption tax is solely based upon consumption. This may sound similar to a sales tax, but in its purest form a consumption tax will not become regressive as is the case with a pure sales tax.
A consumption tax can be designed to be progressive if it includes the following features:
Consumption vs. Income:
Consumption Tax:
Pure tax economists argue that a consumption tax is superior because it comes closest to attaining “temporal neutrality”. Although impossible to attain in reality, a tax would be considered to have attained temporal neutrality if it did not alter spending habits, change behavior patterns, or affect the natural allocation of resources. Because a consumption tax only taxes consumption, the good or service being consumed is largely irrelevant in reference to the allocation of resources.
Income Tax:
An income tax, however, creates a barrier between the value of a person’s labor (how much they earn from working) and what they actually receive (money after taxes). This is a negative force on the economy because it causes people to work less and pursue more leisure activities than would otherwise be the case if income taxes did not exist. In other words, if there were no income taxes people would immediately see a real increase in purchasing power for each additional unit of time they spent working, and thus would be theoretically be more inclined to work.
The barrier created by income taxes also produces
To think of it another way, income taxes will actually cause greater consumption in the present while reducing future savings and future consumption.
A well designed consumption tax is more neutral and does not affect the allocation of resources as dramatically as an income tax. Taxes are only assessed on any income that is consumed (spent on goods, services, etc.) while not taxing savings. This eliminates any barrier to savings and actually would encourage people to save more, increase available capital, and ultimately produce a more solid, robust economy.
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