We'll apply previously proven proposition to analyze so-called income consumption curve over consumer. What is meant by that? Let's suppose we have utility maximization problem. We're maximizing utility, subject to constraint and quite often, based on the monotonicity or the utility function, we know that the constraint is binding. Moreover, let's suppose that X1 and X2 are both positive. Now, when we solve this maximization problem, we find a so-called optimal bundle or consumer. That's a pair of X, X1-star, X2-star. Now clearly, both values which demands on the first good and the second good, respectively, they're functions of prices and income. Now, we fix P1, P2, and change I. For instance, let I be growing income and we follow them. This is a parametric curve within the first quadrant of the goods plane and we follow this path. This curve we get after fixing the prices and changing the income is called income consumption curve. Now, under the assumption that all utility function is homogeneous, we get the conclusion that this I-C curve is a straight line. So once again, let U of X1, X2 be a homogeneous function. Then, I-C curve is a straight line. Let's check. Let's check how to check that. The necessary conditions based on Lagrange method gives us, so this problem can be reduced to marginal rate of substitution. First good substitutes for the second, this is a function of X1, X2, should equal the ratio of prices, and secondly, the budget constraint should hold. What are the properties of the marginal rate of substitution? So, marginal rate of substitution is simply the quotient of two partial derivatives. Now, we can apply a proposition number one which tells us that given a homogeneous function or some degree, let degree be M, both partial derivatives are also homogeneous of the degree M minus one. When we divide two homogeneous functions of the same degree of homogeneity, we get also homogeneous function but this time, this homogeneous function has degree zero. So, meaning that actually, marginal rate of substitution is a function of the ratio of two goods. These can be considered as equation in terms of the quotient. So, we clearly understand that X2-star is, I'll write how to include the relationship of the prices ratio. So, let it be some phi function. I'm not interested in the specific form of this function but it exists. Since prices are fixed, this is just a number that relates both optimal goods to the quantities. So, that means that they are related in a linear fashion. So, if we change I, then we get this income consumption curve and the angle between this I-C curve and X is X1 is fixed, unless we change the ratio of the prices. If we change the prices in the same amount, I mean we are scaling the prices so both prices are multiplied by the same number, let's say they're increased by factor two, then nothing happens. With the growth of income, the consumption grows along this line. While it's easily seen from the formula itself, but also, we can substitute into the budget constraint. We have P1, X1-star plus P2, phi, P1 over P2, X1-star, and this is I. Clearly, we have X1-star is a function of. So this is a linear independence with the fixed prices, dependence on I, and the same is true for the X2 quantity. That concludes this statement. There is a similar statement for the production. So for instance, if we deal with the production function which is homogeneous function of some degree of homogeneity, then marginal rate of technical substitution is also depends only on the ratio or the factors of production, if the prices or factors are fixed. How to see it? If we consider a similar problem for production, we get marginal rate of technical substitution should equal, well, let's consider only two factors, labor and capital alone. So, the wage rate, the price of labor with the price of capital. Here, we'll have production function f depending on labor capital should equal let it be q the output. Now, if we fix the factor prices and we just increase the output, so for fixed, we get a so-called firms expansion path and it is also a straight line. Also can be seen because from this equation, since the production function is homogeneous and marginal rate of substitution is the ratio of the marginal products of labor and capital, then these marginal products are homogeneous functions themselves of less one degree of homogeneity. So, when we calculate this marginal rate of technical substitution, this function of the factors of production becomes also homogeneous of a zero degree of homogeneity. So, that means that once again, the firm's expansion path looks like that, capital over labor is fixed. This is a function of the factor prices. So once again, this is a straight line, a very similar result to the consumer theory.