7. What is "Polonius' point"? Set up the intertemporal utility maximization problem for an individual over two periods (i.e., \(t=1,\ 2\)). Find the formula for consumption in the first period as a function of the interest rate, the discount rate, and income over the two periods.
Polonius' point is:
"Neither a borrower nor a lender be,
For loan oft loses both itself and friend,
Intertemporal Utility Maximization Problem
\(U(c_{1},c_{2})=\ln{c_1}+\frac{1}{1+\rho}\ln{c_{2}} \)
\(c_{2}=y_{2}-(1+r)(c_{1}-y_{1})\)
\(U(c_{1},c_{2})=\ln{c_1}+\frac{1}{1+\rho}\ln{[y_{2}-(1+r)(c_{1}-y_{1})]} \)
\(\frac{1}{c_1}+\frac{1}{1+\rho}\frac{1}{y_{2}-(1+r)(c_{1}-y_{1})}\cdot (-(1+r))=0\)
\(c^{*}_{1}=\frac{y_{2}+ry_{1}}{2+\rho}\)
8 What is l'Hopital's Rule? Use the rule to find the following limit: \(\lim_{x\to 0}\frac{e^{x}-1}{x}\)
- L'Hopital's rule is used to determine the limit when \(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{0}{0}\)
\(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{f^{'}(x)}{g^{'}(x)}\)
\(\lim_{x\to0}\frac{e^x-1}{x}=\frac{e^x}{1}=1\)
9 Define convex, concave, strictly convex, and strictly concave curves using the notion of derivative (hint: not just first derivative!)
A function a concave function if the line segment joining any two points on the graph is below the graph or on the graph
- f''(x) less than or equal to 0
A function is said to be a convex function if the line segment joining any two points on the graph is above the graph or on the graph
- F''(x) greater than or equal to zero
A function is said to be a strictly concave function if the segment joining any two points on the graph is below the graph
f′(c)=0 & f′′(c)<0
A function is said to be a strictly convex function if the segment joining any two points on the graph is above the graph
f′(c)=0 & f′′(c)>0