# How to derive utility function from expenditure function

c) Verify that these demand functions satisfy the properties listed in Propositions 3.D.2 and 3.D.3. This is straightforward. 3.G.3 Consider the (linear expenditure system) utility function given in Exercise 3.D.6. a) Derive the Hicksian demand and expenditure functions. Check the properties listed in Propositions 3.E.2 and 3.E.3.

(a) Derive the Marshalian demand functions for the following utility function: u(x 1,x 2,x 3) = x 1 + δ ln(x 2) x 1 ≥ 0, x 2 ≥ 0. Does one need to consider the issue of "corner solutions" here? (b) Derive the Hicksian demand functions and the expenditure function for the following utility function:

A reward function can be defined using a utility function, , as . The utility function can be converted to a cost function as . Minimizing the expected cost, as was recommended under Formulations 9.3 and 9.4 with probabilistic uncertainty, now seems justified under the assumption that was constructed correctly to preserve preferences.

However, at the end, there is no change in the total utility as the consumer remains on the same indifference curve. Hence, we can write that, on the same indifference curve: (marginal utility of x)×(change in x)+(marginal utility of y) ×(change in y) = 0 ( marginal utility of x) × ( change in x) + ( marginal utility of y) × ( change in y) = 0 M U xdx+M U ydy =0 M U x d x + M U y d y = 0 M U ydy = −M U xdx M U y d y = − M U x d x dy dx = −M U x M U y d y d x = − M U x M U y Here ...

Problem 3: Suppose, production of a ﬂrm can be described by function: Q(L;K) = K1 =3L2 (a) Derive the total cost function in a general form with respect to any prices of labor and capital w and r, respectively. (Hint: you gave to go through the optimality conditions and get the function of TC in form TC = f(w;r;Q))

Deriving Direct Utility Function from Indirect Utility Function Theorem. Suppose that u(x , y) is quasiconcave and differentiable with strictly positive partial derivatives. Then for all (x , y) , v(p x , p y , I) , the indirect utility function generated by u(x , y) , achieves a minimum in (p x , p y ) and u(x , y) = min v(p x , p y ,1) s.t. p ...

If we restrict the function y= x2 to the positive numbers, as on the right, we have an invertible function. Its inverse is x= √ y, the square root function. Notice that the root symbol √ yrefers only to the positive root. √ 4 = −2 is incorrect, while (−2)2=4 is correct. When we invert the graph of the cost function in Example 3 above ...

In the following, we proceed to derive the welfare formula in second steps. Step 1: Extensive Margin is zero The expenditure function in country jtakes the following form: e j = min xc ij X i J i Z 1 ’ ij p ij(’)xc ij (’)g i(’)d’ (D.1) s.t. " X i J i Z 1 ’ ij h q ij(’)xc ij (’) + x ˙ 1 ˙ x ˙ 1 ˙ i g i(’)d! # ˙ ˙ 1 U j (D.2) The Lagrange function can be written as: e j = X i J i Z 1 ’ ij p

As an example of this type of cost function, consider C(q) = 5q. The marginal cost function is just MC(q)=5 and the average cost function is AC(q)=5. Deriving Cost Functions from Production Functions If you start out with a production function, you can derive the related cost function. That is, if you start with q = f(k,l) you can derive C = C ... We work with a general money-in-the-utility-function framework. As is well known, this framework nests several speciﬂc models of money. An important assumption of our analysis is that money and work eﬁort are complements, so that the demand for money, conditional on the expenditure of goods, weakly increases with the amount of work eﬁort.