Problem 1 Suppose there is one input and one output. Let the production function f : R+ > R+ be differentiable with f(0) = 0 and f' > 0. SHOW: f exhibits decreasing returns to scale The average product f(z)/z is decreasing in z > 0. (b) SHOW: f is strictly concave f exhibits decreasing returns to scale. Now consider the special function Z f(z) = 2 + 1+2z+ sin z for z > 0 and its average product g(z) = f(z)/z for z > 0. (c) SHOW: f is not concave. HINT. Use f'(z) = 1+h(z) with h(z) alternating in sign. (d) SHOW: The average product g(z) is decreasing in z > 0. HINT. Use g(z) = 1+1/k(z) with k(z) > 0 and k(z) > 0. ==