For each of the production functions below, calculate the least cost of producin
ID: 1225954 • Letter: F
Question
For each of the production functions below, calculate the least cost of producing one unit of output for the given input prices. (a) q = x_1 + 1/2x_2; (w_1, w_2) = (2, 3) $ Explain your reasoning. One unit of output can be produced either with 1 unit of input 1 costing $, or two units of input 2, costing $. (b) q = min{2x_1 + x_2, x_3}; (w_1, w_2, w_3) = (2, 4, 4) $ Explain your reasoning. Define a composite input, z = 2x_1 + x_2. Then q = min{z, x_3} and one unit of output can be produced with one unit of input z and one unit of x_3. One unit of z can be produced with half a unit of x_1 costing $1, or one unit of x_2 costing $4. One unit of x_3 costs $. Therefore, the least cost of producing one unit of output is $. (c) q = min{x_1, x_2} + x_3; (w_1, w_2, w_3) = (4, 3, 6) $ Explain your reasoning. Define a composite input z = min{x_1, x_2}. Then q = z + x_3, so one unit of output can be produced with one unit of z or one unit of x_3. The former which requires one unit of x_1 and one of x_2 costs $; the latter costs $ .Explanation / Answer
a. Least Cost = $2.
Reasoning - 1 unit of input 1 cost $2 , Two units of input 2 cost $6.
b. Least cost = $5
Reasoning - $4, Thus, least cost = 1 + 4 = $5. Since, X1 and X2 are substitutes, so the one with low price should be chosen.
c. Least cost = $6
The former cost = 4+ 3 = $7, The latter cost = $6.So, x3 should be used to produce the output at minimum cost.
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