Input z is the only variable input for Joe's manna factory. The supply function of the input is 10 + 2z and the MRP function is 110 - z. Then: A)w^*= 60. B)w^*= 90, z^*= 20. C)w^*= 50, z^*= 20. D)w^*= 100/3.

Question

Quizplus · Accepted Answer

To find the optimal input level, we need to set the MRP equal to the input price (which is the derivative of the cost function with respect to z). So we have:

110 - z = 10 + 2z
3z = 100
z = 100/3

Now we can plug this value of z back into either the supply function or the MRP function to find the optimal output level. Using the supply function, we have:

Q = 10z + z^2 = 10(100/3) + (100/3)^2 = 400/3

So the optimal input level is z = 100/3, and the optimal output level is Q = 400/3. From here, we can use the given formulas to calculate the optimal price and profit:

P = 110 - (100/3) = 230/3

π = PQ - (10z + z^2) = (400/3)(230/3) - (10(100/3) + (100/3)^2) = 12600/9 - 500/3 = 1000/3

Now we can plug this back into the given formulas to see which choice(s) match the optimal values. Only choice C matches the optimal price and input values.

Input Z Is the Only Variable Input for Joe's Manna