Here are 3 questions in Math for Econ class. Thanks in advance.

(a) Explain what is meant by a homogeneous function of 2 variables of degree h. Show that the partial derivatives of such a function are homogeneous of degree It 1 . For a homogeneous utility 1nction of 2 variables, show that the slope of the indifference curves is constant along the line y = cx where c is a positive constant. Draw a diagram to illustrate this result. (b) Suppose the production function in an economy takes the form Q = F(K, L) where Q, K and L denote aggregate output, capital and labour. Suppose lrther that F (K , L) is homogeneous of degree 1. Show that the production mction can be written in the form Q = L(k) where is a function of a single variable and k = K / L. In the case where F(K, L) = (K2 + L'z)'1/2 show that F (K, L) is homogeneous of degree 1 and nd the function . Consider the following two problems where, in each case, a and b are positive constants: (i) maximise x2+y2 subject to ax+ by =1; (ii) minimise x2+y2 subject to ax+ by =1. Explain by means of a diagram which, if either, has a solution. In a case or cases where a problem has a solution, nd it. Now suppose, in each case, the constraints x 2 0, y 2 0 are imposed. Investigate what happenswhen(a)a = 1,13 = 3, (b)a = 3,b = 2. A rm has a Cobb-Douglas production function Q = KaLb Where Q, K and L denote output, capital and labour respectively and where a and b are positive constants. Determine how the returns to scale depend on a and b and nd the marginal products of capital and labour. Show that each isoquant is negatively sloped, convex and has the axes as asymptotes. Sketch the isoquant diagram. When the prices of capital and labour are r and w respectively, nd the conditional input demand functions and hence show that the minimum cost of producing output level q is (a+b)()aib(a)q