solve problem?

Question

A)
the function F is defined for all x and y by F(x,y)=xe^y-3+xy^2-2y. show the point (1,3) lies onthe level curve F(x,y)=4 and find the equation for the tangent line to the curve at the point (1,3).
B)
the NerloveRingstad production function y=y(K,L) is defined implicitly by y^1+elny=AK^mL^n, where A,m,n are positive constants. Find the marginal productivities of y w.r.t. K and L- that is, find dy/dK and dy/dL.
(take the logarith of each side and then differentiate implicitly)

Answer 1

A) F(1;3) = 1e^0 + 1*9 - 2*3 = 4 then the point (1;3) is in the curve F(x;y) = 4

F'_x(x;y) = e^(y - 3) + y²
F'_x(1;3) = e^0 + 3² = 10

F'_y(x;y) = xe^(y - 3) + 2xy - 2
F'_y(1;3) = 1e^0 + 2*1*3 - 2 = 5

The tangent vector is (10 ; 5)

Th equation of the tangent line is y = mx + p

m = 5/10 = 1/2
3 = 1/2 * 1 + p
p = 3-1/2 = 5/2

y = 1/2 x + 5/2

B) y^(1 + eln(y)) = A K^m L^n : is it right ?

So, taking the logarithm of each side :

(1 + elny)lny = lnA + m lnK + n lnL

lny + e (lny)² = lnA + m lnK + n lnL

defferentiating :

[1/y + 2e 1/y lny] dy = m/K dK + n/L dL


(1 + 2e lny) dy/y = m dK/K + n dL/L

... but ... I'm not sure !!