Suppose a firm is operating in the short run, holding input 2 fixed at x2. Its production function is given by the Cobb-Douglas one: f(x1,x2) =( x1)^a(x2)^b. The firm wants to find out its least cost combination of input for any given output level, y. The firm therefore solves the following optimisation problem:
c(w1, w2, y) = Min {w1x1 +w2x2(bar)} such that f(x1, x2(bar)) = y
the bar means x2 is constant b/c it is SR
Solve for the optimal conditional input demand for factor 1, x1 (w1,w2,y)
work:
w1 =a(x1)^(a-1)x2bar^b, so solving for x1* becomes (w1/[ax2bar^b])^(1/a-1)
substituting y/(x1^a) = x2bar^b, x1 = (w1x1^a/[ay])^(1/[a-1]), so x1^(a-1) = (w1x1^a)/(ay)
x1^(a-1)/x1^a = w1/(ay)
x1^-1 = w1/(ay) so x1 =(ay)/(w1)
please just tell me if this is correct!! i need to know if substituting y/(x1^a) for x2bar^b is the correct way to have demand for x1 in terms of w1,w2, and y.
THANK YOU!!!
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