[0:00]What's up guys? Econ John here, in this video, we're going to talk about the almost ideal demand system. Let's go. So, what is the almost ideal demand system? The almost ideal demand system is a model which was developed by Angus Deaton and John Muellbauer in 1980, which serves as an approximation for any true demand system whether based on theory or not. Some of the unique properties of the almost ideal demand system are that it satisfies the axioms of choice exactly. It aggregates perfectly over consumers without the need to assume that all goods are normal, uh which means we don't require that all the angle curves have to be the same slope. Right? It does not require non-linear estimation methods, can be estimated with sure. Uh restrictions such as homogeneity, symmetry and budget balancedness, which is also referred to as the adding up condition, can be tested or observed within the framework of this model. Many of these properties can be seen in alternative demand systems like the translog demand system and Rotterdam demand system. However, neither of these two alternatives possess all of these properties together. So the almost ideal demand system is a derivation of the price independent generalized logarithmic (PIGLOG) cost function which is defined as: log e(u,p) = (1-u) log(a(p)) + u log{b(p)}, u ", where
[1:25]log of a, right, that is a homothetic in you, uh cost function, right? Translog cost function and log of BP, right, that is our translog cost function by A plus this term over here, right, which is really a derivation from the linear expenditure system. I'm I'm not going to show how we derive it directly, but, you know, we do some rearranging when we get into the budget shares. So anyways, uh, since U lies between zero and one, we can regard A and B as the costs of subsistence and bliss respectively. Um, based on this specification the almost ideal demand system is defined as the following.
[2:06]So since we're dealing with a logged expenditure function, it's often easier to go talk about budget shares as opposed to demands directly. Because we're able to go and really get proportions of income more easily by just deriving with respect to the natural log of these functions. So, the budget shares are defined as the partial derivative of our logged expenditure function with respect to the log of the price of good I. That's equal to our budget share of good I, right? This yields our equation to be W_i is equal to alpha_i plus the summation of gamma_kj times log of P_k plus b_i times b_0 which is times the product of P_k raised to the power of k of beta_k, right? Where k goes from, I guess, one to N, right, times this u term. By rearranging the almost ideal demand system, we can identify this ambiguous term as log of M over P, right? M being our expenditure, right? Function all over P where P is just our A P term, right, or logged A P term, right? This is often referred to as the Stone price index. It should be noted that often in applied work some arbitrary price index is used instead, so that's kind of a little bit cheating. And it follows that our budget share equation can be modified to be alpha_i plus the summation of all gamma_kj times log of P_j, right, plus beta_i times this indexed, uh, term over here, M over P. So, though the almost ideal demand system is very flexible, we have some restrictions on our parameters. These being, mostly due to practicality or mathematical construction. Um, so these are our conditions. Let's just go over what they are. So P1 to P3 is referred to as the adding up condition or budget balancedness condition. Uh this means that our budget shares will sum to one. P2 refers to the fact that our demands are homogeneous of degree zero in prices. P4 is that our Slutsky symmetry condition, right? This means that our Slutsky matrix or matrix of second derivatives is symmetric. P5 is just a general property of about all cost functions.
[4:29]This being that our Slutsky matrix is, uh, negative semi-definite. Um, we can be more meticulous about these conditions by looking at our coefficients directly. However, P1 to P3 can't really be tested in unison due to the construction of the almost ideal demand system. Instead, we should look at the budget share equations and just to see if these results hold. Uh, since the almost ideal demand system is a flexible form, we should have to test for P number five. That's really the only thing that we should test for using the eigenvalues of our substitution matrix. Uh this is the same if we were testing any other type of cost function or expenditure system. Uh this will be covered in another video. So, for this last slide, I just wanted to go and show the uncompensated price elasticity for good I, the compensated price elasticity for good I and the income elasticity of demand, uh, for good I in all these cases. Um, I'm not going to really go into much detail of this because, you know, most of the variables in them, in this are pretty recognizable, and, you know, we've gone over them in the other slides. Uh just the new variable that we have over here is our delta I J which is the the Kronecker delta. So, that was the first video of the almost ideal demand system up on YouTube. Uh, let me know what you thought. Be well.
