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∂ 2 ƒ ( x1 (α ), x2 (α );α )
⋅
+
⋅
+
2
∂ x1 ∂ x2
∂α
∂ x2
∂α
∂ x2 ∂α ≡0 α α This is a set of two linear equations in the two derivatives ∂x1*(α)/∂α and ∂x2*(α)/∂α, and we can
solve for ∂x1*(α)/∂α and ∂x2*(α)/∂α by substitution, or by Cramer’s Rule, or however. Econ 100A 14 Mathematical Handout G. OPTIMIZATION #3: COMPARATIVE STATICS OF EQUILIBRIA Implicit differentiation isn’t restricted to optimization problems. It also allows us to derive how
changes in the parameters affect the equilibrium values in an economic system.
Consider a simple market system, with supply and demand and supply functions
QD = D(P,I) and QS = S(P,w) where P is market price, and the parameters are income I and the wage rate w. Naturally, the
equilibrium price is the value Pe solves the equilibrium condition
D(Pe,I) = S(Pe,w)
It is clear that the equilibrium price function, namely Pe = Pe(I,w), must satisfy the identity
D( Pe(I,w) , I ) I≡ S( Pe(I,w) , w )
,w
So if we want to determine how a rise in income affects equilibrium price, totally differentiate
the above identity with respect to I, to get
∂D ( P e ( I , w), I ) ∂P e ( I , w)
∂D ( P e ( I , w), I )
⋅
+
∂P e
∂I
∂I then solve to get
∂P e ( I , w)
∂I = ≡ w, I ∂S ( P e ( I , w), w ) ∂P e ( I , w)
⋅
∂P e
∂I ∂D ( Pe ( I ,w), I )
∂I
∂S ( Pe ( I ,w),w) ∂D ( Pe ( I ,w), I )
−
∂Pe
∂Pe In class, we’ll analyze this formula to see what it implies about the effect of changes in income
upon equilibrium price in a market. For practice, see if you can derive the formula for the effect
of changes in the wage rate upon the equilibrium price. Summary of How to use Implicit Differentiation to obtain Comparative Statics Results
The approach of implicit differentiation is straightforward, yet robust and powerful. It is used
extensively in economic analysis, and as seen above, consists of the following four steps:
STEP 1: Obtain the first order conditions for the optimization problem, or the equilibrium
conditions of the system.
STEP 2: Substitute the solution functions into these first order conditions, or equilibrium
conditions.
STEP 3: Totally differentiate these equations with respect to the parameter that’s changing.
STEP 4: Solve for the derivatives with respect to that parameter. Econ 100A 15 Mathematical Handout H. OPTIMIZATION #4: COMPARATIVE STATICS OF OPTIMAL VALUES The final question we can ask is how the optimal attainable value of the objective function varies
when we change the parameters. This has a surprising aspect to it. In the unconstrained
maximization problem:
max ƒ( x1 ,..., xn ;α )
x1 ,..., xn *
recall that we get the optimal value function φ(α) by substituting the solutions x1*(α),..., xn (α)
back into the objective function, i.e.:
*
*
φ (α ) ≡ ƒ( x1 (α ),..., xn (α );α )
α Thus, we could simply differentiate with respect to α to get:
= *
*
*
∂ ƒ( x1 (α ),..., xn (α );α ) dx1 (α )
⋅
∂x1
d...

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