Problem 1
Consider the production function \(f\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}^{2} .\) Does this exhibit constant, increasing, or decreasing returns to scale?
Problem 2
Consider the production function \(f\left(x_{1}, x_{2}\right)=4 x_{1}^{\frac{1}{2}} x_{2}^{\frac{1}{3}} .\) Does this exhibit constant, increasing, or decreasing returns to scale?
Problem 3
The Cobb-Douglas production function is given by \(f\left(x_{1}, x_{2}\right)=A x_{1}^{a} x_{2}^{b}\) It turns out that the type of returns to scale of this function will depend on the magnitude of \(a+b .\) Which values of \(a+b\) will be associated with the different kinds of returns to scale?
Problem 4
The technical rate of substitution between factors \(x_{2}\) and \(x_{1}\) is \(-4 .\) If you desire to produce the same amount of output but cut your use of \(x_{1}\) by 3 units, how many more units of \(x_{2}\) will you need?
Problem 5
True or false? If the law of diminishing marginal product did not hold, the world's food supply could be grown in a flowerpot.
Problem 6
In a production process is it possible to have decreasing marginal product in an input and yet increasing returns to scale?
