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Solutions to Midterm Exam
Economics 720: Mathematics for Economists
Fall 2002
1 The Kuhn-Tucker Theorem and the Cobb-Douglas
Utility Function
a) Define the Lagrangian as
L(c 1 , c 2 , λ) = ca 1 c^12 − a+ λ(I − p 1 c 1 − p 2 c 2 ). (1.1)
b) According to the Kuhn-Tucker theorem, the values c∗ 1 and c∗ 2 that solve the consumer’s problem and the associated value λ∗^ of the Lagrange multiplier must satisfy the first- order condition for c 1 ,
L 1 (c∗ 1 , c∗ 2 , λ∗) = ac∗ 1 a −^1 c∗ 21 −a− λ∗p 1 = 0, (1.2)
the first-order condition for c∗ 2 ,
L 2 (c∗ 1 , c∗ 2 , λ∗) = (1 − a)c∗ 1 a c∗− 2 a− λ∗p 2 = 0, (1.3)
the constraint, L 3 (c∗ 1 , c∗ 2 , λ∗) = I − p 1 c∗ 1 − p 2 c∗ 2 ≥ 0 , (1.4) the nonnegativity condition, λ∗^ ≥ 0 , (1.5) and the complementary slackness condition,
λ∗(I − p 1 c∗ 1 − p 2 c∗ 2 ) = 0. (1.6)
c) Begin by dividing (1.2) by (1.3) to eliminate λ∗:
ac∗ 2 (1 − a)c∗ 1
p 1 P 2
Now use this result to obtain an expression for c∗ 2 in terms of^ c∗ 1 and the model’s parameters: c∗ 2 =
μ 1 − a a
¶ μ p 1 p 2
c∗ 1.
Substitute this expression for c∗ 2 into the binding constraint to obtain
I = p 1 c∗ 1 +
μ 1 − a a
p 1 c∗ 1 =
μ 1 a
p 1 c∗ 1.
This last result yields the solution for c∗ 1 :
c∗ 1 =
aI p 1
Substitute (1.7) back into the expression for c∗ 2 to obtain the solution for c∗ 2 :
c∗ 2 =
(1 − a)I p 2
d) Equation (1.7) reveals that the consumer spends the fraction
p 1 c∗ 1 I
= a
of his or her income on good 1 and the fraction p 2 c∗ 2 I
= 1 − a
on good 2. A property of the Cobb-Douglas utility function is that it implies that these expenditure shares are constant, and equal to the exponents or weights attached to each good in the utility function itself.
2 The Maximum Principle in Discrete Time
a) Define the Hamiltonian as
H(yt, πt+1; t) = max zt βtF (yt, zt; t) + πt+1Q(yt, zt; t) subject to c ≥ G(yt, zt; t). (2.1)
b) By the Kuhn-Tucker theorem,
H(yt, πt+1; t) = max zt βtF (yt, zt; t) + πt+1Q(yt, zt; t) + λt[c − G(yt, zt; t)],
where zt satisfies the first-order condition
βtFz (yt, zt; t) + πt+1Qz(yt, zt; t) − λtGz (yt, zt; t) = 0. (2.2)
c) According to the Maximum Principle, the solution to the original dynamic optimization problem must satisfy the pair of difference equations
πt+1 −πt = −Hy(yt, πt+1; t) = −[βtFy(yt, zt; t)+πt+1Qy(yt, zt; t)−λtGy(yt, zt; t)] (2.3)
and yt+1 − yt = Hπ(yt, πt+1; t) = Q(yt, zt; t). (2.4)
Now substitute this expression for π˙(t) into (3.6) to obtain
−(ρ/φ)e−ρt[(1 − θ)q−θ^ + θαq−θα−^1 ] = −e−ρt[q^1 −θ^ − αq−θα]
or, more simply,
(ρ/φ)[(1 − θ)q−θ^ + θαq−θα−^1 ] = [q^1 −θ^ − αq−θα]. (3.8)
Equation (3.8) is the equation involving only the steady-state equilibrium price q and the model’s parameters. Although it would take some additional algebra, this equation could be used to solve for q in terms of θ, α, ρ, and φ.
4 Dynamic Programming Under Certainty: Optimal
Monetary Policy
a) The Bellman equation is
v(πt; t) = max yt −π^2 t − αy^2 t + βv(πt + ψyt; t + 1). (4.1)
b) Using the guess for v, the Bellman equation becomes
−γπ^2 t = max yt −π^2 t − αy t^2 − βγ(πt + ψyt)^2.
The first-order condition for yt is
− 2 αyt − 2 βγψ(πt + ψyt) = 0 (4.2)
and the envelope condition for πt is
− 2 γπt = − 2 πt − 2 βγ(πt + ψyt). (4.3)
c) Use the first-order condition to solve for the optimal choice for yt in terms of the inflation rate πt, the unknown constant γ, and the model’s parameters:
αyt = −βγψπt − βγψ^2 yt
(α + βγψ^2 )yt = −βγψπt or yt = −
μ βγψ α + βγψ^2
πt. (4.4)
d) Given that γ, β, α, and ψ are all positive, (4.4) reveals that the optimal value for the output gap yt is negative when inflation is positive; conversely, the optimal yt is positive when inflation is negative. The central bank finds it optimal to ”lean against the wind.”
