# Mathematics assignments on derivatives.

## Please solve following questions step by step and provide all of the derivations.

ECON 451 Assignment 3
Due May 13th 2021
Please submit the assignment to Canvas
Instruction: Please solve following questions step by step and provide all of the derivations.
Otherwise, your assignment will be considered incomplete.
Question 1 Government budget (30 points)
Let’s consider the governments budget constraint (for time t) equates the total real expen-
diture to total real revenues, as follows
gt + bt􀀀1 = qbt + t; (1)
where gt is real government expenditure, q = 1
1+r , r is real interest rate, bt is real debt
holding, and t is real tax revenue. Assume the transversality condition
lim
t!1
qtbt+1 = 0
prohibiting the government from always borrowing to pay its debt (at some point all gov-
ernment expenditures must be backed by government tax revenues.).
(a) Please solve forward equation (1) for government’s in nite-horizon budget con-
straint [Hint: the initial budget constraint is equation (1)].
for (a) in terms of how government expenditure is nanced.
Question 2 The Ricardian Equivalence (75 points)
Let’s consider following representative consumer problem.
max
fctg1
t=0
E
X1
t=0
tu(ct) (2)
subject to
ct + qbt + t = bt􀀀1 (3)
1
(c) Please solve forward equation (3) for consumer’s in nite-horizon budget constraint
[Hint: the initial consumer’s budget constraint is equation (3)].
(d) Let’s assume that bt􀀀1 = 0 as well. Please merge government’s in nite-horizon
budget constraint in Question 1 (a) with consumer’s in nite-horizon budget constraint
fPor a new budget constraint of consumer. (Hint: the new constraint should be 0 = 1
t=0 qt(ct + g):)
government expenditure a ects consumer’s consumption.
(f) Please solve the representative consumer problem using the new budget constraint
that you get in (d) and nd the rst order condition with respect to ct. Note there is
only one budget constraint now.
(g) Assume (for simplicity) that = q. Please nd ct. Moreover, please show con-
sumption is constant over time and una ected by government taxes, t, used to nance
government spending.
Question 3 Regime-switching model (45 points)
Please apply the regime-switching model or the VAR model for an economics question that is

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