Basic Mathematics for Economists phần 7

Để dẫn xuất một phần thứ hai được tìm thấy bởi sự khác biệt đầu tiên để dẫn xuất một phần của một hàm. Khi một hàm có hai biến độc lập sẽ có bốn thứ hai để dẫn xuất một phần. Lấy ví dụ, chức năng sản xuất trong đó C là chi tiêu của người tiêu dùng, YD là thu nhập dùng một lần | where C is consumer expenditure Yd is disposable income Y is national income I is investment t is the tax rate and G is government expenditure. What is the marginal propensity to consume out of Y What is the value of the government expenditure multiplier How much does government expenditure need to be increased to achieve a national income of 700 Second-order partial derivatives Second-order partial derivatives are found by differentiating the first-order partial derivatives of a function. When a function has two independent variables there will be four second-order partial derivatives. Take for example the production function Q 25K 0 4L0 3 There are two first-order partial derivatives dQ dQ 0AL 7 These represent the marginal product functions for K and L. Differentiating these functions a second time we get d Q -6KL dK 2 d2Q -4l 1-7 dL2 These second-order partial derivatives represent the rate of change of the marginal product functions. In this example we can see that the slope of MPl . d2Q dL2 will always be negative assuming positive values of K and L and as L increases ceteris paribus the absolute value of this slope diminishes. We can also find the rate of change of dQ dK with respect to changes in L and the rate of change of dQ dL with respect to K. These will be d 2q d 2 Q dKdL dLdK and are known as cross partial derivatives . They show how the rate of change of Q with respect to one input alters when the other input changes. In this example the cross partial derivative d2 Q dLdK tells us that the rate of change of MPl with respect to changes in K will be positive and will fall in value as K increases. You will also have noted in this example that d2Q d2Q dKdL dLdK In fact matched pairs of cross partial derivatives will always be equal to each other. Thus for any continuous two-variable function y f x z there will be four second-order partial derivatives d2y d2y . d2y x d2y i ii iiit iv dx2 dz2 dxdz .