A study of milk production found that y= 2.90x0.015 0.250 0.350 0.408 0.030 X3 'x5' x4 where y is the output of milk, and x, ..., x5 are the quantities of five different input factors. (a) If all the factors of production were doubled, what would happen to y? (b) Write the relation in log-linear form. Problem 2 Calculate all the first-order partial derivatives of the following six functions: (a) f(x, y, z) = x+ y + z 4 (d) f(x, y, z) = x/yz Problem 3 (b) f(x, y, z) = 5x - 3y3+3z4 (c) f(x, y, z) = xyz (e) f(x, y, z) = (x + y +z+)6 (f) f(x, y, z) = exyz Let Y = F(K, L) = 15K1/5 2/5 denote the number of units of output that are produced when K units of capital and L units of labour are used as inputs. (a) Compute F(0,0), F(1, 1), and F(32, 243). (b) Find an expression for F(K + 1, L) - F(K, L), and give an economic interpretation. (c) Compute F(32 + 1,243) lating FK (32, 243). F(32, 243), and compare the result with what you get by calcu- (d) Show that F(tK, tL) = F(K,L) for a constant k. Problem 4 Use the chain rule to find the following derivatives: (a) dz/dt, if z= F(x,y) = x+e, where x = 13 and y = 2t. (b) dy/dt, if Y=F (K,L) = KL, where K = f(t) and L = g(t). (c) g'(r), if g(r) = F(r,1-r, 1/(1 r)). (d) az/t and z/as, if z = F(x,y) where x = f(t) and y = g(t,s). B Problem 5 Use the chain rule to find the following derivatives: (a) z/t when z = = e, x = t + s, y = t +s (b) z/t when z=. z= xh(x, y), x=t+s, y = t+s Problem 6 (a) If F(x,y) = 3xy-3x-4y, find OF/dy. == - (b) If F (Q, L) = QL QL, find F and FL (c) If x(v1, v2) = (v +v2)", find x1 (v1, v2). (d) If F(x, y) = xy+3xy 4, find F2(x, y). (e) If F (K,L) = InK+InL, find Fk (K,L) and FKK (K,L). Problem 7 Let f be defined for all x, y by f(x, y) = x y x. (a) Use (14.8.3) to show that f is concave. (b) Show that-e-f(x,y) is concave. CONCAVITY AND CONVEXITY CONDITIONS USING SECOND DERIVATIVES Suppose that the function z = f(x, y) is defined and C on a set S that is open and convex. Then, f is concave f is convex f0, f0, and f22 - (12) 0 f 0, 0, and - (f) 0 (14.8.3) (14.8.4) For strict concavity or convexity, here are sufficient but not necessary condi- tions: f0 and ff - (F12) > >0 f is strictly concave f">0 and ff (12) > 0 f is strictly convex The inequalities in all four statements are understood to hold at all points (x, y) in S. (14.8.5) (14.8.6)