Millicent runs an Etsy shop on which she sells bracelets (y) made from ounces of
ID: 1203985 • Letter: M
Question
Millicent runs an Etsy shop on which she sells bracelets (y) made from ounces of beads (z1) and inches of silver-plated wire (z2). Millicent’s production function is f(z1, z2) = z1^1/4 z2^3/4 . The price of an ounce of beads is w1 = 16, the price of an inch of wire is w2 = 3, and the price of output is p = 8.
What is the equation for the isocost line that goes through this input combination? Solve for the equation
What combination of ounces of beads and inches of wire (z1, z2) should she use to produce these 24 bracelets at lowest possible cost?
Explanation / Answer
(a) Equation of isocost: C = w1 x z1 + w2 x z2
C = 16z1 + 3z2
Production function: q = z10.25z20.75
Marginal product of z1 = MP(z1) = dq / dz1 = 0.25 x (z2 / z1)0.75
Marginal product of z2 = MP(z2) = dq / dz2 = 0.75 x (z1 / z2)0.25
Profit is maximized when MP(z1) / MP(z2) = w1 / w2 = 16 / 3
(1/3) x (z2 / z1) = 16 / 3
z2 = 16z1 .....(1)
Substituting in isocost function,
C = 16z1 + 3z2 = z2 + 3z2 = 4z2
z2 = C / 2
z1 = z2 / 16 = (C / 2) / 16 = C / 32
(b) q = 24 = z10.25z20.75
Substituting z2 = 16z1,
z10.25(16z1)0.75 = 24
z10.25 x z10.75 x 160.75 = 24
z1 x 8 = 24
z1 = 24/8 = 3
z2 = 16 x z1 = 16 x 3 = 48