Economics 5120 spring 2017 assignment write the lagrangian

Question

Economics Assignment -

Q1. Consider the Solow model, with Cobb-Douglas production (i.e. y = k^α). Along the balanced growth path, write expressions for the following in terms of exogenous parameters and (if necessary) the initial levels A(0) or L(0). You need not re-derive the expression for k* that we derived in class, but express it using the notation from Romer (i.e. g ≡ A^·/A and n ≡ L^·/L).

(a) k*

(b) K(t)

(c) y*

(d) Y(t)

(e) c*

(f) C(t)

(g) Derive the k* that maximizes c*. What savings rate yields that value of k*?

Q2. This question details some basic concepts in the Ramsey model.

(a) What rate is an exogenous parameter in the Solow model, but is endogenously determined in the Ramsey model?

(b) In the Ramsey model, but not the Solow model, we model transactions between firms and households. Which two variables do we derive that reflect equilibrium prices in the factor markets where households sell to firms?

(c) A numeraire is a good that has a price of 1, often implicitly. What has a price of 1 in the Ramsey model?

Q3. This question helps you to understand the distinct meaning of ρ and θ in the Ramsey model. Consider the simplified two-period inter-temporal choice problem where consumers choose current consumption C₀ and future consumption C₁ to maximize

U = C₀^1-θ/1-θ + (1/1+ρ)C₁^1-θ/1-θ

subject to the budget constraint

C_o + (1/1+r) C₁ ≤ I,

where I, r, ρ and θ are all exogenous.

(a) Write the Lagrangian for the maximization problem. Use λ as the multiplier.

(b) Derive first-order conditions for C₀ and C₁.

(c) Derive an expression for ln(C₀/C₁) that does not depend on λ.

(d) What is -d ln(C₀/C₁)/d ln(1/1+r)? What is the common name for this expression?

(e) Explain the economic interpretation for how changes in r, ρ and θ affect ln(C₀/C₁). Your answer should give a sense of what each of the mathematical expressions represents and why there is a relationship between them.

(f) How do changes in I affect C₀, C₁, and ln(C₀/C₁) at the optimum? (Describing whether they increase, decrease, or stay the same is sufficient.) Again, provide some economic interpretation.

Q4. This question is about the firm's problem in the Ramsey model.

(a) We assume F(K, AL) has constant returns to scale, k ≡ K/AL, and f ≡ F/AL. From f'(k) = r, derive dF/dK. Is the correct answer r or rAL?

(b) Romer derives expressions for wage per unit of labor W(t) and wage per unit of effective labor w(t). Notice that we never use our expression for w in our dynamic analysis of c and k. Why does the Euler equation for c^·/c not depend on w(t)? You answer should relate to part of your answer to Question 3.

Q5. This question is about the points of the Ramsey diagram that are not on the saddle path.

(a) We use the phase diagram to depict possible paths of c and k that satisfy the Euler equation and the capital accumulation equation. Which additional equation needs to be satisfied to ensure that any trajectory drawn is feasible? (Any name for it is fine)

(b) Consider points where initial consumption is greater than the level that achieves the saddle path. What happens to c and k at such points? Why can these points not be on an equilibrium path? (Note: Romer gives a long explanation. You should give a simple economic explanation for what is happening along this path.)

(c) Same as part (b), but instead consider initial consumption that is less than the level that achieves the saddle path.

6. Now we consider comparative statics for the Ramsey model. Suppose in the Ramsey model that there is a permanent decrease in the population growth rate n. Answer the questions using diagrams with an explanation.

(a) What happens to the k^· = 0 curve?

(b) What happens to the c^· = 0 curve?

(c) Suppose that the economy was initially on the balanced growth path before the change in n. After the change, what happens to c and k instantaneously? What happens to c and k afterward?

(d) Suppose instead that the initial k was less than k* (before the change in n). After the change in n, what happens to c and k instantaneously? What happens to c and k afterward?

• Q7. This question is about the national income and product accounts. If you want a text supplement lecture, you may find it helpful to look over the first 3 sections of Karl Whalen's guide to Chain-Aggregated NIPA Data that I will link to on Blackboard. Suppose in the current quarter that the BEA reported the following:
• the real GDP growth rate was 2.5%
• the real growth in consumption was 3.0%
• the real growth in investment was 5.0%
• the real growth in government expenditure was 2.0%
• the growth contribution of investment to real GDP was 1.0%
• the growth contribution of consumption to real GDP was 2.0%.
• the nominal share of government expenditure in GDP is 0.2 (i.e. the ratio of nominal G to nominal GDP is 1/5)

(a) What is the nominal share of consumption in GDP?

(b) What is the nominal share of investment in GDP?

(c) What is the growth contribution of government expenditure to real GDP?

(d) What is the growth contribution of net exports to real GDP?

(e) If the BEA were to construct a real chain-weighted series for the aggregate of consumption, investment, and government expenditure, what would its growth rate have been?

Q8. Suppose the following:

• nominal GDP was 18 billion in 2015
• nominal GDP growth for 2016 was 3 percent
• real GDP growth for 2016 was 2 percent

(a) What is the 2016 level of nominal GDP (in 2016 dollars)?

(b) What is the 2015 level of real GDP (in 2015 chain-weighted dollars)?

(c) What is the 2016 level of real GDP (in 2015 chain-weighted dollars)?

(d) What is the 2016 level of real GDP (in 2016 chain-weighted dollars)?

(e) What is the 2015 level of real GDP (in 2016 chain-weighted dollars)?