There are 5 Questions in this assignment. Deadline is Saturday 10 pm
1. (18 marks) Consider general models and implicit functions: a. State the three conditions of the Implicit Function Theorem for n functions of the form FI (1, ..., Yn; X1, ., Xm) Where y; is endogenous for all j € {1,2, ...,n} and x; is exogenous for all i € {1,2,..., m}. That s, for a point (y19, ..., Yno; X10, ---» Xmo), State the three conditions by which there exists a neighbourhood around this point for which y; = f4(xy, ..., Xm) is an implicitly defined and continuously differentiable function, and where F/ (yy, ..., Yu; X1, -.., Xm) = 0 is an identity, for all j € {1,2, ...,n}. b. Assuming the conditions referred to in Part (a) hold, take the total differential of FI (yy, cee) Ys X41, oe, Xp) = O for each j € {1,2, ...,n} to derive the Implicit Function Derivative Rule a _ lil ax; for allj € {1,2, ...,n} where J is the Jacobian matrix [Ft OF] y1 Yn I=]: oF aF™ 9y1 Yn and J; is the matrix J wherein the Jj" column is replaced with the negative of the arm” radient vector [2 8! oxi’ ox c. Apply the concepts involved in Parts (a) and (b) to the basic national income model Y=C+1Io+Go C=a+pY-T) T=y+d6Y to verify the comparative static ac _ pa-9 a 1-pa-9 °° where 8,6 € (0,1), a,y > 0 and I; and G,, are exogenous. You may assume without verification that the conditions of the Implicit Function Theorem are satisfied for the equilibrium point of the model. 2. (12 marks) Find the critical and inflection points of f(x) = 2sin® x + 3sinx for x € [0,7], making sure to classify the critical points. 3 45 a7 x9 gl 3. (4 marks) Apply Taylor's Theorem to prove that (i) sinx = x — 5 + = - =z + = - o> + at as as sr 7 1 ii STE de and (i) InG+ 1) =x — 24 ZZ FT _2 aud diy iia tA —/ 4A . (4 marks) Let the exponential E(x) be the inverse of the logarithm L(x). Use the properties of logarithms to prove that (i) E(a + b) = E(a)E (b) and (ii) E’ (x) = E(x)/L'(1). (12 marks) Suppose a developer owns a parcel of land worth V(t) = Vok2VE (measured in dollars) when developed at its highest and best use at time t = 0 (measured in years from today), where V, > 0 is the value of the parcel today and K > 1 is a parameter. Assume the only cost to the developer of owning the parcel is foregone interest, at an annual rate of r > 0, on the capital tied up in the parcel. Letting A(t) be the present value of the parcel, the developer seeks to maximize A(t) by choosing the time at which the parcel is developed at its highest and best use. a. Derive the optimal time t* at which to develop the parcel at its highest and best use, and then check the second-order condition to confirm that t* indeed provides for a relative maximum. b. Use the result of Part (a) to evaluate t* for the case of K = fe and r = 5%, where e is Euler's number.