ADVANCED MICROECONOMIC THEORY

ROY’S IDENTITY

A full derivation of Roy’s Identity is placed at the end of this slideshow. Not difficult, just fairly long.

Ultimately, Roy’s Identity just shows that you can derive the Marshallian Demand function for any good ‘L’, directly from the Indirect utility function:

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AGGREGATION

Individuals demand functions are functions of prices and wealth.

Therefore, total (aggregate) demand functions are functions of prices and everybody’s wealth (assume a total of I individuals).

It is convenient to think of the aggregate demand function as variant only on the TOTAL level of wealth in society, but when does this occur?

Such a definition circumvents distributional issues.

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AGGREGATION

Such aggregation occurs when all individuals have STRAIGHT and PARALLEL ENGEL CURVES (diagram slide after next, algebra next slide).

If we change the wealth of individual ‘h’, then the change of aggregate demand is the same as the change in his personal demand.

This is because no-one else’s demand is affected by his change.

Keeping prices constant, through a bit of algebra magic (using the chain rule) we find that the effect of a change in wealth for ‘h’ on h’s demand is the same as the change in aggregate demand to a change in aggregate wealth, which is the same as the change in another individual’s (j) demand to a change in j’s wealth.

The wealth effect at ‘p’ must be the same for any consumer i = 1... I, at any level of wealth for aggregation to occur straight Engel curves.

AGGREGATION

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AGGREGATION

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AGGREGATION

A general result under which the above conditions for Aggregation hold (that is, there are linear and parallel Engel curves for all consumers), is that consumers have indirect utility functions of the GORMAN FORM:

This condition is NECESSARY and SUFFICIENT. We should not attempt to prove necessity, because it’s pretty insane. Sufficiency can be easily proved: we need to use Roy’s Identity (just take derivatives and plug them in into the identity).

Note that ‘ai(p)’ and ‘b(p)’ both have to abide by conditions to allow Vi(.) to satisfy conditions of an indirect utility function (homogeneity, increasing in w, non-increasing in p, quasi-convexity – see Topic 1 slides).

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AGGREGATION

Regarding Direct Utility Functions:

No known general functions that allow aggregation to occur (i.e., no standard direct utility forms always lead to the Gorman form).

However, a necessary and sufficient restriction on the direct utility function is that the utility functions are additive (utility of total consumption is equal to sum of utilities of consuming each individual good):

Another restriction is that utility functions for individual goods (u~) are Hyperbolic Absolute Risk Aversion (HARA).

This gives rise to a broad class of utility functions

if a=0, it is iso-elastic, if gamma=2, it is quadratic, if gamma 0, it is logarithmic, if gamma -∞, it is a negative exponential.

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ROY’S IDENTITY

Intuition of Roy’s Identity (rearranged):

If the price of apples rise (Pl) what will be the effect on my indirect utility function?

The answer is that it will be negative (I don’t like higher prices) and it will be proportional to my demand for apples.

The derivation can seem a bit confusing; there are a couple of places where the next step is triggered by leaps of logic (‘this bit looks like what we would get if we differentiated the budget constraint’ ), but otherwise it’s fairly consistent.

Rule of thumb:

2 STAGES – FIND dV/dP, then dV/dw

FOR FIRST, SOLVE THE UMP FOR THE F.O.C., THEN DIFFERENTIATE THE BUDGET CONSTRAINT W.R.T PRICES.

FOR SECOND, LEGWORK IS DONE, JUST SUBSTITUTE IN THE UMP RESULT AND DIFFERENTIATE THE BUDGE CONSTRAINT W.R.T THE WEALTH.

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