Microeconomics Questions: Production and Game Theory

Solution Preview

It is best if you look at the attached file as it contains graph and tables.

1- Let a firm have production function where are the inputs.
(i) Sketch the isoquants. What is the MRTS (marginal rate of technical substitution).
By definition, isoquant is the set of all combinations of inputs that result in a given level of output. E.g. there is an isoquant for 10 units of output and it is given by the set of solutions to equation f(z1,z2) = 10, which can be re-written as 2(z1^0.5 + z2^0.5) = 10. While for certain functions it may be hard to find a simple analytical answer, the best approach is to substitute a few numbers for one input and solve for the other one. Here's an example of what I mean: let's consider z1 = 1, then to produce 10 units of output we must have 2(1^0.5 + z2^0.5) = 10, which is
(1 + z2^0.5) = 5
z2^0.5 = 4
z2 = 16. So what we found is that on (z1,z2) graph one of the points that corresponds to isoquant of 10 units is (1,16). Can you find other points? (Hint: (16,1) also belongs to the same isoquant. Why? Hint: Symmetry of production function.)
Now keeping the above calculations, let's try to obtain analytical solution (i.e. in functional form). In general, the isoquant is solution to problem of
f(z1,z2) = y, where y is a real number that denotes output. So:
2(z1^0.5 + z2^0.5) = y
Z1^0.5 + z2^0.5 = y/2
Z2^0.5 = y/2 - z1^0.5
Z2 = (y/2 - z1^0.5)^2 = (y^2)/4 - 2* (y/2) * (z1^0.5) + z1
We can plot the above equation on a diagram (see next page) and it will look like a generic isoquant. So we can put in the specific points we've calculated at the beginning of this problem.

Now the remaining question is MRTS. First, definition of MRTS is the rate at which input 2 will need to be substituted for a small change in input 1 to keep output constant. The intuition is if you change amount of input 1 (z1), then to produce the same amount you will need to change amount of input 2 also. Graphically, it is the movement along the isoquant. E.g. if you were to reduce z1 from 16 by a small amount then you would need to add some amount of z2 to stay on the same isoquant. Hence, MRTS is simply the slope of the isoquant.
Mathematically, this means that we need to take the derivative of equation for isoquant. The equation for our function was:
Z2 = (y^2)/4 - 2* (y/2) * (z1^0.5) + z1
So MRTS = d z2 / d z1 = 1 - (y/2) / (z1^0.5). Hint: recall that derivative of a constant is zero and since we are doing this exercise for a fixed level of y, the derivative of (y^2)/4 is zero. Now, derivative of z1 with respect to z1 is simply 1. (recall if f(x) = x, then f'(x) = 1). Now for the middle component, I use the following two rules of differentiation: 1. if f(x) = a * g(x), where g(x) is some other function of x, then f'(x) = a * g'(x). 2. if f(x) = x^0.5, then f'(x) = ...