Consider a firm that uses labor and capital to produce output x using a homothetic production technology that has increasing returns to scale when output lies between 0 and x_A, constant returns to scale when output lies between x_A, and x_B, and decreasing returns to scale when output exceeds x_B (where 0<x_A<x_B).Although the different parts of the question repeatedly refer to the isoquant graph you first draw in (a), you should probably re-draw the graph several times - each time only with the portions you need for the question -- to indicate the different items that are asked for in the remaining parts of the question (rather than indicating all your answers on literally the same graph).
a.On a graph with labor on the horizontal and capital on the vertical axis, draw isoquants for x_A and x_B.For a given set of input prices w and r, indicate the least cost input bundle A=(l^A, k^A) for producing x_A using an isocost line.Label the slope of the isocost line and then label the slope of the isoquant in terms of the marginal product of labor and capital.
b.Indicate where the least cost input bundle B for producing x_B must lie (in light of the homotheticity property of the production technology.) What does the vertical slice along which all cost-minimizing input bundles lie look like (on a graph with "inputs" on the horizontal and x on the vertical)?
c.Indicate all input bundles in your isoquant graph that could be part of a profit maximizing production plan for some output price p>0.
d.Suppose the actual profit maximizing production plan is (l*,k*,x*).What two conditions involving the marginal products of the inputs hold at this - and only this - production plan?
e.Now suppose that a change in tax policy results in an increase of the rental price of capital r.Indicate all possible input bundles in an isoquant graph that might be long-run profit maximizing assuming no change in p or w.(Include the isoquant corresponding the initial profit maximizing output level x* as well as the isoquant that contains B (from (b)) in your graph.) Explain your reasoning.

We Have Worked a Lot with Homothetic Production Technologies

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