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Microeconomics Questions: Production and Game Theory

This problem set is an assignment for a post graduate student in Economics (Msc) , thus, each question demands a detailed explanation not just a quick or brief answer. There are 2 questions on production theory, specifically on profit maximization problem. The third question is on game theory, particularly: Nash equilibrium.

Please see the attached file for the full problem statement and formulas. The text below is contained within the Word file:

1- Let a firm have production function where are the inputs.
(i) Sketch the isoquants. What is the MRTS (marginal rate of technical substitution).
(ii) Is homogenous of some degree What does this tell you about the return to scale under which the firm operates?
(iii) Set up and solve the firm's profit maximization problem given prices ≫0 (here is the price of the output and the vector of factor prices).
(iv) Using your answer to the previous sub-question, find the level of output supplied to the market when
= (1,1,1) ,

2- Define and explain the notion of a (pure strategy) Nash equilibrium. Give an example of a game and find the Nash equilibrium.
3- Let be a homogenous of degree 1 production function for a firm.
(i) Show that the firm faces constant returns to scale in production (include the definition of CRS in your answer).
(ii) Prove that the firm must earn zero profits when it maximizes profits. (Hint: assume that, given prices , the firm could earn strictly positive profits from some combination of the inputs . Then argue if so, the firm would want to use infinitely much of the inputs and could earn infinite profits. Contradictions).
(iii) Using the conclusion from (ii), show that if is profit maximizing given prices , then so is for any (Hint: simply show that profits will also be zero for the input combination . Since zero is the maximal profit, is then profit maximizing).

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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) = ...

Solution Summary

The solution provides detailed explanations to the questions given. Specifically, there are three questions:
1. Production theory
2. Game theory
3. Production theory under constant returns to scale.

The definitions are provided where required and the formulas are beautifully typed using MS Equation editor.

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