Trade Prelim Notes

Dynamic Heckscher-Ohlin Model

This problem is from Tim Kehoe. It showed up on the trade prelims in 2013, 2015 (fall), 2016 (fall) and 2017(spring).

Setup

Consider a two-sector growth model in which the representative consumer has the utility function $$\sum_{t=0}^\infty \beta^t \log (c_{1t}^{a_1}, c_{2t}^{a_2})$$ The investment good is produced according to $$k_{t+1} = d x_{1t}^{a_1} x_{2t}^{a_2}$$ Feasible consumption/investment plans satisfy the feasibility constraints $$c_{1t}+x_{1t}=\phi_{1}(k_{1t},l_{1t})=k_{1t}$$ $$c_{2t}+x_{2t}=\phi_{2}(k_{2t},l_{2t})=l_{2t}$$ where $$k_{1t}+k_{2t}=k_{t}$$ $$l_{1t}+l_{2t}=1$$ The initial value of $k_t$ is $\bar{k_0}$. All of the variables specified above are in per capita terms. There is a measure $L$ of consumer/workers.

Problem (-): Define Equilibrium for this economy.

Solution:

Problem (a): Setup Social Planner's Problem

Write out a social planner’s problem for this economy. Explain how the solution to this social planner’s problem is related to that of the one-sector social planner’s $$\sum_{t=0}^\infty \beta^t \log c_t $$ $$ \text{s.t.: } c_t + k_{t+1} = dk_t^{a_1} $$ $$c_t, k_t \geq 0$$ $$k_0 = \bar{k_0}$$ [You can write done a proposition or propositions without providing a proof or proofs, but be sure to carefully relate the variables in the two-sector model to the variables in the one-sector model.]

Solution:
The social planner's problem for the two sector economy is to choose $C \equiv\{c_{1t}, c_{2t}, x_{1t}, x_{2t}, k_{1t}, k_{2t} \}_t$ to solve
$$ \max_{C}\sum_{t=0}^{\infty}\beta^{t}\log\left(d c_{1t}^{a_{1}}c_{2t}^{a_{2}}\right) $$ subject to the constraints: \begin{gather} c_{1t}, c_{2t}, x_{1t}, x_{2t} \geq 0, \quad \forall t \tag{NonNeg}\\ c_{1t} + x_{1t} = k_{1t} \quad \forall t \tag{Resource 1} \\ c_{2t} + x_{2t} = 1 \quad \forall t \tag{Resource 2} \\ k_{1,t+1} + k_{2,t+1} = d x_{1t}^{a_1} x_{2t}^{a_2} \tag{Capital} \\ k_{10} \quad given \tag{initial k} \\ \end{gather}

Assume that $a_1 + a_2 = 1$

Proposition 1: If $C \equiv\{c_{1t}, c_{2t}, x_{1t}, x_{2t}, k_{1t}, k_{2t} \}_t$ is a solution to the two-sector social planner's problem, then $\{c_t, k_t\}_t$ is the solution to the one-sector social planner's problem if for each time period $t$: $$c_t = d c_{1t}^{a_1} c_{2t}^{a_2}$$ $$k_{t+1} = k_{1,t+1} = d x_{1t}^{a_1} x_{2t}^{a_2}$$

Proposition 2: If $\{c_t, k_t\}_t$ is a solution to the two-sector social planner's problem, then $C \equiv\{c_{1t}, c_{2t}, x_{1t}, x_{2t}, k_{1t}, k_{2t} \}_t$ is the solution to the one-sector social planner's problem if for each time period $t: $$k_{1,t+1} = k_{t+1}$$ $$k_{2,t+1} = 0$$ $$c_{1t} = \frac{k_t}{d k_t^{a_1}}c_t$$ $$c_{2t} = \frac{1}{d k_t^{a_1}}c_t$$ $$x_{1t} =\frac{k_t}{d k_t^{a_1}} k_{t+1}$$ $$x_{2t} =\frac{1}{d k_t^{a_1}} k_{t+1}$$

Problem (b): Solve Social Planner's Problem

Solve the one-sector social planner’s problem in part b. [Recall that the policy function for investment has the form $k_{t+1} (k_t) = A d k^{a_1}$ where $A$ is a constant that you remember or can determine with a bit of algebra and calculus.]

Solution:

Assuming an interior solution with binding budget constraints, the FOCs will be: $$\beta^t \frac{1}{c_t} = \lambda_t \tag{c}$$ $$\lambda_t = \lambda_{t+1} a_1 d k_{t+1}^{a_1 - 1} \tag{k}$$ $$c_t + k_{t+1} = d k_t^{a_1} \tag{budget}$$ And the transversality condition will be that: $$\lim \beta^t \frac{1}{c_t} k_t = 0$$

Now assume that the policy function has the form $$k_{t+1} (k_t) = K^\prime (k_t) = A d k_t^{a_1}$$ Combine with the budget to get $$c_t = d k_t^{a_1} - k_{t+1} = d k_t^{a_1} - A d k_t^{a_1} = (1-A) d k_t^{a_1}$$ Divide the above and compare to the FOCs: $$\frac{k_{t+1}^{a_1}}{k_t^{a_1}} = \frac{c_{t+1}}{c_t} = \beta \frac{\lambda_{t+1}}{\lambda_t} = \beta a_1 d k_{t+1}^{a_1 - 1} $$ $$k_{t+1} = \beta a_1 d k_{t}^{a_1 } $$ Therefore, comparing to the policy function, this means that $A = \beta a_1$ and $c_t = (1-\beta a_1) d k_t^{a_1}$.

Then applying this policy function recursively, we can find that $$k_t = (a_1 \beta d)^{\frac{a_1^t - 1}{a_1 - 1}} k_0^{a_1^t}$$ $$c_t = (1-\beta a_1) d k_t^{a_1}$$

Problem (c): Define International Equilibrium

Suppose now that there is a world made up of $n$ different countries, all with the same technologies and preferences, but with different constant populations, $L^i$, and with different initial capital-labor ratios $\bar{b_\theta^i}$. Suppose that goods $1$ and $2$ can be freely traded across countries, but that the investment good cannot be traded. Suppose too that there is no international borrowing. Define an equilibrium for the world economy.

Solution:
TODO: DEFINITION

Problem (d): Whole Buncha Theorems

State and prove versions of the factor price equalization theorem, the Stolper- Samuelson theorem, the Rybczynski theorem, and the Heckscher-Ohlin theorem for this particular world economy.

Factor Price Equalization Theorem:
Statement:
This theorem states that if factor of production is used to produce goods in multiple countries, then the prices of that factor should be the same across each of those countries. That is, if $y_{1t}^i, y_{1t}^j$, then $r_t^i = r_t^j$. And if $y_{2t}^i, y_{2t}^j$, then $w_t^i = w_t^j$
Proof:
Stolper-Samuelson theorem
Statement:
If the relative price of a good increases, then the factor most used in producing that good will also increase in relative price. That is, let $\left\{ \hat{p_{1t}},\hat{p_{2t}},\hat{r_{t}^{i}},\hat{w_{t}^{i}}\right\} $ and $\left\{ \widetilde{p_{1t}},\widetilde{p_{2t}},\widetilde{r_{t}^{i}},\widetilde{w_{t}^{i}}\right\} $ represent the prices is two different equilibriums. If production of each good is nonneagtive in both equilibriums then $\frac{\hat{p_{1t}}}{\hat{p_{2t}}}>\frac{\widetilde{p_{1t}}}{\widetilde{p_{2t}}}\implies\frac{\hat{r_{t}^{i}}}{\hat{w_{t}^{i}}}>\frac{\widetilde{r_{t}^{i}}}{\widetilde{w_{t}^{i}}}$
Proof:
Rybczynski Theorem:
Statement:
The Rybczynski Theorem states that an increase in the endowment of one factor of production will cause an increase in the production of the output which uses that good more extensively. For this economy, let there be two different equilibriums as defined above. If $\hat{k_{1t}^{i}}>\widetilde{k_{1t}^{i}} $ and $\hat{L^i} = \widetilde{L^i}$ then $\hat{y_{1t}^{i}}>\widetilde{y_{1t}^{i}}$ but $\hat{y_{2t}^{i}}=\widetilde{y_{2t}^{i}}$
Proof:
Heckscher-Ohlin Theorem:
Statement:
The Heckscher-Ohlin states that the capital abundant country will export the capital intensive good while the labor abundant country exports the labor intensive good. In this economy, this means that. TODO
Proof:
TODO: Rerrange as GENERAL CONCEPT, SPECIFIC STATEMENT, PROOF. WRITE PROOFS

Problem (e): Further Maths

Let $S_t = c_t / y_t$ where $y_t = p_{1t} k_t + p_{2t} = dk_t^{a_1}$ is world GDP per capita. Explain how you could show that $$\frac{y_{t}^{i}-y_{t}}{y_{t}}=\frac{s_{t}}{s_{t-1}}\left(\frac{y_{t-1}^{i}-y_{t-1}}{y_{t-1}}\right)=\frac{s_{t}}{s_{0}}\left(\frac{y_{0}^{i}-y_{0}}{y_{0}}\right)$$ That is, explain the logic of the argument. You do not need to go into details.

Solution:

Problem (f): Finish and Discuss

Use the solution to the one-sector social planner’s problem in part b to solve for $s_t$. Discuss the economic significance of the result that you obtain

Solution:
Taking the solution from the 1SSPP, TODO: type out maths

LaTeX Dump

Consumer utility: \[ \sum_{t=0}^{\infty}\beta^{t}\log\left(dc_{1t}^{\alpha_{1}}c_{2t}^{\alpha_{2}}\right) \] with $\alpha_{1}+\alpha_{2}=1$ both nonneg. The investment good is produced according to \[ k_{t+1}=dx_{1t}^{\alpha_{1}}x_{2t}^{\alpha_{2}} \] Feasibility Constraints: \[ c_{1t}+x_{1t}=\phi(k_{1t},l_{2t})=k_{1t} \] \[ c_{2t}+x_{2t}=\phi\left(k_{2t},l_{2t}\right)=l_{2t} \] where $k_{1t}+k_{2t}=k_{t}$ and $l_{1t}+l_{2t}=1$ $k_{0}$ given. Measure $L$ of consumers/workers. \section*{(a) Define an equilibrium:} An equilibrium is HH allocation: $\left\{ c_{1t},c_{2t},x_{1t},x_{2t},k_{t+1}\right\} _{t}$ Firm allocations $\left\{ k_{1t},l_{1t},y_{1t}\right\} _{t}$$\left\{ k_{2t},l_{2t},y_{2t}\right\} _{t}$ Prices $\left\{ p_{1t},p_{2t,}r_{t},w_{t}\right\} _{t}$ Such that \begin{itemize} \item Given prices, the HH allocation solves \[ \max_{HH}\sum_{t=0}^{\infty}\beta^{t}\log\left(dc_{1t}^{\alpha_{1}}c_{2t}^{\alpha_{2}}\right) \] \begin{align*} \ni & p_{1t}\left(c_{1t}+x_{1t}\right)+p_{2t}\left(c_{2t}+x_{2t}\right)=r_{t}k_{t}+w_{t}\\ & k_{t+1}=dx_{1t}^{\alpha_{1}}x_{2t}^{\alpha_{2}}\\ & k_{0}\text{ given}\\ & c_{1t},c_{2t},x_{1t},x_{2t}\geq0 \end{align*} \item Given prices and firm 2's allocation, firm 1's allocation solves \[ \max_{k_{1t},l_{1t},y_{1t}}p_{1t}y_{1t}-r_{t}k_{1t}-w_{t}l_{1t} \] \begin{align*} \ni & y_{1t}=k_{1t}\\ & 0\leq l_{1t}\leq1-l_{2t}\\ & 0\leq k_{1t}\leq k_{t}-k_{2t} \end{align*} \item Given prices and firm 1's allocation, firm 2's allocation solves \[ \max_{k_{2t},l_{2t},y_{2t}}p_{2t}y_{2t}-r_{t}k_{2t}-w_{t}l_{2t} \] \begin{align*} \ni & y_{12t}=l_{2t}\\ & 0\leq l_{2t}\leq1-l_{2t}\\ & 0\leq k_{2t}\leq k_{t}-k_{2t} \end{align*} \item Market clears: \begin{align*} c_{1t}+x_{1t} & =y_{1t}\\ c_{2t}+x_{2t} & =y_{2t}\\ k_{1t}+k_{2t} & =k_{t}\\ l_{1t}+l_{2t} & =1 \end{align*} \end{itemize} - Note that in the FOCs \[ \beta^{t}\frac{\alpha_{1}}{c_{1t}}=\lambda_{bt}p_{1t} \] \[ \beta^{t}\frac{\alpha_{2}}{c_{2t}}=\lambda_{bt}p_{2t} \] \[ \frac{c_{1t}}{c_{2t}}=\frac{\alpha_{1}p_{2t}}{p_{1t}\alpha_{2}} \] \section*{(b) Write out a social planner's problem. Explain how the solution to this social planner's problem is related to thtat of the one-sector social planner's problem \[ \sum_{t=0}\beta^{t}\log c_{t} \] \[ \ni c_{t}+k_{t+1}=dk_{t}^{\alpha_{1}} \] \[ c_{t},k_{t}\protect\geq0 \] \[ k_{0}=\bar{k_{0}} \] } Because in this economy, an equilibrium is Pareto optimal, the problem can be rewritten as a SPP. The 2 sector social planner's problem for this economy is \[ \max_{\{c_{1t},c_{2t}\}_{t}}\sum_{t=0}^{\infty}\beta^{t}\log\left(dc_{1t}^{\alpha_{1}}c_{2t}^{\alpha_{2}}\right) \] \begin{align*} \ni\forall t\ & c_{1t}+x_{1t}=k_{t}\\ & c_{2t}+x_{2t}=1\\ & k_{t+1}=dx_{1t}^{\alpha_{1}}x_{2t}^{\alpha_{2}}\\ & k_{0}\text{ given}\\ & c_{1t},c_{2t},x_{1t},x_{2t}\geq0 \end{align*} Going from the 1 sector SPP to the 2 sector SPP: \begin{align*} c_{1t} & =\frac{k_{t}}{dk_{t}^{\alpha_{1}}}c_{t}\\ c_{2t} & =\frac{1}{dk_{t}^{\alpha_{1}}}c_{t}\\ x_{1t} & =\frac{k_{t}}{dk_{t}^{\alpha_{1}}}k_{t+1}\\ x_{2t} & =\frac{1}{dk_{t}^{\alpha_{1}}}k_{t+1} \end{align*} Going from the 2 sector SPP to the 1 sector SPP: \begin{align*} c_{t} & =dc_{1t}^{\alpha_{1}}c_{2t}^{\alpha_{2}}\\ k_{t+1} & =dx_{1t}^{\alpha_{1}}x_{2t}^{\alpha_{2}} \end{align*} \textbf{Proof:} In the FOCs for the 2SSPP, \begin{align*} \beta^{t}\frac{\alpha_{1}}{c_{1t}}=\lambda_{1t} & & \beta^{t}\frac{\alpha_{2}}{c_{2t}}=\lambda_{2t}\\ \lambda_{1t}=\lambda_{kt}+d\alpha_{1}x_{1t}^{\alpha_{1}-1}x_{2t}^{\alpha_{2}} & & \lambda_{2t}=\lambda_{kt}+d\alpha_{2}x_{1t}^{\alpha_{1}}x_{2t}^{\alpha_{2}-1} \end{align*} \[ \therefore\beta^{t}\frac{\alpha_{1}}{c_{1t}}=\lambda_{1t}=\lambda_{2t}\frac{\alpha_{1}}{\alpha_{2}}\frac{x_{1}}{x_{2}}=\beta^{t}\frac{\alpha_{2}}{c_{2t}}\frac{\alpha_{1}}{\alpha_{2}}\frac{x_{1}}{x_{2}} \] \[ \frac{c_{2t}}{c_{1t}}=\frac{x_{2t}}{x_{1t}} \] Combined with the feasibility constraints, this means that any optimal consumption and investment plan will satisfy \[ \frac{c_{1t}}{c_{2t}}=k_{t}=\frac{x_{1t}}{x_{2t}} \] Define $f(x_{1},x_{2})=dx_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}$, $c_{t}\equiv f(c_{1t},c_{2t})$, and suppose the constraints for the 2SSPP hold. Note that \begin{align*} dk_{t}^{\alpha_{1}} & =dk_{t}^{\alpha_{1}}\cdot1=d\left(\frac{c_{1t}}{c_{2t}}\right)^{\alpha_{1}}\left(c_{2t}+x_{2t}\right)\\ & =d\left(\frac{c_{1t}}{c_{2t}}\right)^{\alpha_{1}}c_{2t}+dx_{2t}\left(\frac{c_{1t}}{c_{2t}}\right)^{\alpha_{1}}=d\left(\frac{c_{1t}}{c_{2t}}\right)^{\alpha_{1}}c_{2t}+dx_{2t}\left(\frac{x_{1t}}{x_{2t}}\right)^{\alpha_{1}}\\ & =dc_{1t}^{\alpha_{1}}c_{2t}^{1-\alpha_{1}}+dx_{2t}^{1-\alpha_{1}}x_{1t}^{\alpha_{1}}=dc_{1t}^{\alpha_{1}}c_{2t}^{\alpha_{2}}+dx_{2t}^{\alpha_{2}}x_{1t}^{\alpha_{1}}\\ & =c_{t}+k_{t+1} \end{align*} So the constraints for the 1SSPP hold. (NNC is trivial) Now suppose the constraints for the 1SSPP hold. And let $c_{1t}\equiv\frac{k_{t}}{dk_{t}^{\alpha_{1}}}c_{t}$, $c_{1t}\equiv\frac{1}{dk_{t}^{\alpha_{1}}}c_{t}$, $x_{1t}\equiv\frac{k_{t}}{dk_{t}^{\alpha_{1}}}k_{t+1}$, $x_{2t}\equiv\frac{1}{dk_{t}^{\alpha_{1}}}x_{t+1}$. Then \[ c_{1t}+x_{1t}=\frac{k_{t}}{dk_{t}^{\alpha_{1}}}c_{t}+\frac{k_{t}}{dk_{t}^{\alpha_{1}}}k_{t+1}=\frac{k_{t}}{dk_{t}^{\alpha_{1}}}\left(c_{t}+k_{t+1}\right)=k_{t} \] \[ c_{2t}+x_{2t}=\frac{1}{dk_{t}^{\alpha_{1}}}c_{t}+\frac{1}{dk_{t}^{\alpha_{1}}}k_{t+1}=\frac{1}{dk_{t}^{\alpha_{1}}}\left(c_{t}+k_{t+1}\right)=1 \] \[ dc_{1t}^{\alpha_{1}}c_{2t}^{\alpha_{2}}=d\left(\frac{k_{t}}{dk_{t}^{\alpha_{1}}}c_{t}\right)^{\alpha_{1}}\left(\frac{1}{dk_{t}^{\alpha_{1}}}c_{t}\right)^{\alpha_{2}}=dk_{t}^{\alpha_{1}}\left(\frac{c_{t}}{dk_{t}^{\alpha_{1}}}\right)^{\alpha_{1}+\left(1-\alpha_{1}\right)}=c_{t} \] \[ dx_{1t}^{\alpha_{1}}x_{2t}^{\alpha_{2}}=d\left(\frac{k_{t}}{dk_{t}^{\alpha_{1}}}k_{t+1}\right)^{\alpha_{1}}\left(\frac{1}{dk_{t}^{\alpha_{1}}}k_{t+1}\right)^{\alpha_{2}}=k_{t+1} \] So the constraints for the 2SSPP hold. (NNC is trivial) So each SPP is clearly optimizing the same value and with optimal behavior, the constraints characterize the same set of possibilities. Thus the solutions to the two SPPs must be related by the above. definitions. \section*{(c) Solve the 1SSPP:} Recall the the budget constraint is \[ c_{t}+k_{t+1}=dk_{t}^{\alpha_{1}} \] FOC for 1SSPP: \[ \beta^{t}\frac{1}{c_{t}}=\lambda_{t} \] \[ \lambda_{t}=\lambda_{t+1}\alpha_{1}dk_{t+1}^{\alpha_{1}-1} \] So \[ \frac{c_{t+1}}{c_{t}}=\beta\frac{\lambda_{t}}{\lambda_{t+1}}=\beta\alpha_{1}dk_{t+1}^{\alpha_{1}-1} \] TVC: \[ \lim\frac{\beta^{t}}{c_{t}}k_{t}=0 \] Guess that the policy function for investment has the form \[ k_{t+1}=K'(k_{t})\equiv Adk_{t}^{\alpha_{1}} \] From the budget, this means that \[ c_{t}=C(k_{t})\equiv\left(1-A\right)dk_{t}^{\alpha_{1}} \] \[ \frac{c_{t+1}}{c_{t}}=\frac{k_{t+1}^{\alpha_{1}}}{k_{t}^{\alpha_{1}}}=\beta\alpha_{1}dk_{t+1}^{\alpha_{1}-1} \] \[ k_{t+1}=\beta\alpha_{1}dk_{t}^{\alpha_{1}} \] So $A=\beta\alpha_{1}$, and $c_{t}=\left(1-\beta\alpha_{1}\right)dk_{t}^{\alpha_{1}}$. Then recursively, \begin{align*} k_{1} & =\beta\alpha_{1}dk_{0}^{\alpha_{1}}\\ k_{2} & =\beta\alpha_{1}dk_{1}^{\alpha_{1}}=\beta\alpha_{1}d\left(\beta\alpha_{1}dk_{0}^{\alpha_{1}}\right)^{\alpha_{1}}=\left(\beta\alpha_{1}d\right)^{1+\alpha_{1}}k_{0}^{\alpha_{1}^{2}}\\ k_{3} & =\beta\alpha_{1}dk_{2}^{\alpha_{1}}=\left(\beta\alpha_{1}d\right)^{1+\alpha_{1}+\alpha_{1}^{2}}k_{0}^{\alpha_{1}^{3}}\\ k_{t} & =\left(\beta\alpha_{1}d\right)^{\frac{\alpha_{1}^{t}-1}{a_{1}-1}}k_{0}^{\alpha_{1}^{t}} \end{align*} \[ c_{t}=\left(1-\beta\alpha_{1}\right)dk_{t}^{\alpha_{1}}=\left(1-\beta\alpha_{1}\right)d\left(\beta\alpha_{1}d\right)^{\alpha_{1}\frac{\alpha_{1}^{t}-1}{a_{1}-1}}k_{0}^{\alpha_{1}^{t+1}} \] - Also note that $x_{2t}(k_{t})=\frac{1}{dk_{t}^{\alpha_{1}}}k_{t+1}=\beta\alpha_{1}$ \section*{(d) International Economy. There are n countries, all with the same technology and preferences, but different populations, $L^{i}$. And with different initial capital-labor ratios $\bar{k_{0}^{i}}$. Goods 1 and 2 can be freely traded, but the investment good cannot. There is no international borrowing. Define an equilibrium:} An equilibrium is HH allocation $C^{i}$ for each country $i\in\left\{ 1,...,n\right\} $: $\left\{ c_{1t}^{i},c_{2t}^{i},x_{1t}^{i},x_{2t}^{i},k_{t+1}^{i}\right\} _{t}$ Firm allocations $F_{j}^{i}$ for each country $i$ and goods $j\in\left\{ 1,2\right\} :$ $\left\{ k_{jt}^{i},l_{jt}^{i},y_{jt}^{i}\right\} _{t}$ Prices $\left\{ p_{1t},p_{2t,}\left\{ r_{t}^{i},w_{t}^{i}\right\} _{n}\right\} _{t}$ (Goods prices equalize across countries, but labor and capital can't move freely.) Such that \begin{itemize} \item Given prices, each HH allocation $C^{i}$ solves \[ \max_{C^{i}}\sum_{t=0}^{\infty}\beta^{t}\log\left(d\left(c_{1t}^{i}\right)^{\alpha_{1}}\left(c_{2t}^{i}\right)^{\alpha_{2}}\right) \] \begin{align*} \ni & p_{1t}\left(c_{1t}^{i}+x_{1t}^{i}\right)+p_{2t}\left(c_{2t}^{i}+x_{2t}^{i}\right)=r_{t}^{i}k_{t}^{i}+w_{t}^{i}\\ & k_{t+1}^{i}=d\left(x_{1t}^{i}\right)^{\alpha_{1}}\left(x_{2t}^{i}\right)^{\alpha_{2}}\\ & k_{0}^{i}=\bar{k_{0}^{i}}\text{ given}\\ & c_{1t}^{i},c_{2t}^{i},x_{1t}^{i},x_{2t}^{i},k_{t+1}^{i}\geq0 \end{align*} \item Given prices and other firms' allocations, firm allocation $F_{1}^{i}$ solves \[ \max_{k_{1t},l_{1t},y_{1t}}p_{1t}y_{1t}^{i}-r_{t}k_{1t}^{i}-w_{t}^{i}l_{1t}^{i} \] \begin{align*} \ni & y_{1t}^{i}=k_{1t}^{i}\\ & 0\leq l_{1t}^{i}\\ & 0\leq k_{1t}^{i} \end{align*} \item Given prices and other firms' allocations, firm allocation $F_{2}^{i}$ solves \[ \max_{k_{2t},l_{2t},y_{2t}}p_{2t}y_{2t}^{i}-r_{t}^{i}k_{2t}^{i}-w_{t}^{i}l_{2t}^{i} \] \begin{align*} \ni & y_{2t}^{i}=l_{2t}^{i}\\ & 0\leq l_{2t}^{i}\\ & 0\leq k_{2t}^{i} \end{align*} \item Market clears: \begin{align*} \sum_{i}L^{i}\left(c_{1t}^{i}+x_{1t}^{i}\right) & =\sum_{i}L^{i}y_{1t}^{i}\\ \sum_{i}L^{i}\left(c_{2t}^{i}+x_{2t}^{i}\right) & =\sum_{i}L^{i}y_{2t}^{i}\\ k_{1t}^{i}+k_{2t}^{i} & =k_{t}^{i}\\ l_{1t}^{i}+l_{2t}^{i} & =1 \end{align*} \end{itemize} \subsection*{For HW version} WTS that AVG k = k implies AVG k' = k'. Then the problem will follow from the sequence of capital. Note from the discussion later in proving the HO theorem that if $A_{t}\equiv\frac{\alpha_{1}p_{2t}}{p_{1t}\alpha_{2}}$, then for all $i$, $t$, $x_{1t}^{i}=A_{t}x_{2t}^{i}.$ Also, $\sum_{i}L^{i}x_{1t}^{i}=A_{t}\sum_{i}L^{i}x_{2t}^{i}.$ Also $A_{t}=c_{1t}^{i}/c_{2t}^{i}$ . $\sum_{i}L^{i}c_{1t}^{i}=A_{t}\sum_{i}L^{i}c_{2t}^{i}.$ So \begin{align*} \frac{\sum_{i}L^{i}k_{t+1}^{i}}{\sum_{i}L^{i}} & =\frac{\sum_{i}L^{i}\left(d\left(x_{1t}^{i}\right)^{\alpha_{1}}\left(x_{2t}^{i}\right)^{1-\alpha_{1}}\right)}{\sum_{i}L^{i}}=\frac{\sum_{i}L^{i}\left(dx_{2t}^{i}A^{\alpha_{1}}\right)}{\sum_{i}L^{i}}\\ & =dA^{\alpha_{1}}\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}}=d\left(\frac{\sum_{i}L^{i}\left(x_{1t}^{i}\right)}{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}\right)^{\alpha_{1}}\frac{\sum_{i}L^{i}x_{2t}^{i}}{\sum_{i}L^{i}}\\ & =d\left(\frac{\sum_{i}L^{i}x_{1t}^{i}}{\sum_{i}L^{i}}\right)^{\alpha_{1}}\left(\frac{\sum_{i}L^{i}x_{2t}^{i}}{\sum_{i}L^{i}}\right)^{\alpha_{2}} \end{align*} So \[ k_{t+1}=d\left(x_{1t}\right)^{\alpha_{1}}\left(x_{2t}\right)^{\alpha_{2}} \] Similarly, \[ c_{t}\equiv\frac{\sum_{i}L^{i}\left(d\left(c_{1t}^{i}\right)^{\alpha_{1}}\left(c_{2t}^{i}\right)^{1-\alpha_{1}}\right)}{\sum_{i}L^{i}}=d\left(\frac{\sum_{i}L^{i}c_{1t}^{i}}{\sum_{i}L^{i}}\right)^{\alpha_{1}}\left(\frac{\sum_{i}L^{i}c_{2t}^{i}}{\sum_{i}L^{i}}\right)^{\alpha_{2}} \] And by a process analogous to that from the 1SSPP, we can find that \[ c_{t}+k_{t+1}=k_{t} \] And \[ c_{1t}+x_{1t}=\frac{\sum_{i}L^{i}\left(c_{1t}^{i}\right)}{\sum_{i}L^{i}}+\frac{\sum_{i}L^{i}\left(x_{1t}^{i}\right)}{\sum_{i}L^{i}}=\frac{\sum_{i}L^{i}k_{t}^{i}}{\sum_{i}L^{i}}=\frac{\sum_{i}L^{i}\left(c_{1t}^{i}+x_{1t}^{i}\right)}{\sum_{i}L^{i}\left(c_{2t}^{i}+x_{2t}^{i}\right)}=A_{t}=k_{t} \] \[ c_{2t}+x_{2t}=\frac{\sum_{i}L^{i}\left(c_{2t}^{i}\right)}{\sum_{i}L^{i}}+\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}}=\frac{\sum_{i}L^{i}\left(1\right)}{\sum_{i}L^{i}}=1 \] Note that because \[ \frac{\sum_{i}L^{i}p_{1t}\left(c_{1t}^{i}+x_{1t}^{i}\right)}{\sum_{i}L^{i}p_{2t}\left(c_{2t}^{i}+x_{2t}^{i}\right)}=\frac{\alpha_{1}}{\alpha_{2}} \] the portion of total income spent on good 1 is fixed at $\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}=\alpha_{1}$. Because each consumer has the same preferences, technology, and choice set, their policy functions will be the same as those found above, except that the level of capital stock will differ. As such, we can use the previous result that \[ \frac{c_{2t}^{i}}{x_{2t}^{i}}=\frac{1-\alpha_{1}\beta}{\alpha_{1}\beta} \] \[ 1=\frac{\sum_{i}L^{i}\left(c_{2t}^{i}\right)}{\sum_{i}L^{i}}+\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}}=\frac{\sum_{i}L^{i}\left(\frac{1-\alpha_{1}\beta}{\alpha_{1}\beta}c_{2t}^{i}\right)}{\sum_{i}L^{i}}+\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}}=\left(1+\frac{1-\alpha_{1}\beta}{\alpha_{1}\beta}\right)\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}} \] \[ \alpha_{1}\beta=\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}} \] \[ k_{t+1}=\frac{\sum_{i}L^{i}k_{t+1}^{i}}{\sum_{i}L^{i}}=dA^{\alpha_{1}}\frac{\sum_{i}L^{i}\left(x_{2t}^{i}\right)}{\sum_{i}L^{i}}=\alpha_{1}\beta dk_{t}^{\alpha_{1}} \] \[ k_{t+1}=\frac{\sum_{i}L^{i}\left(\alpha_{1}\beta dk_{t}^{\alpha_{1}}\right)}{\sum_{i}L^{i}}=\alpha_{1}\beta d\frac{\sum_{i}L^{i}\left(k_{t}^{\alpha_{1}}\right)}{\sum_{i}L^{i}}=\alpha_{1}\beta dk_{t}^{\alpha_{1}} \] So with the correct starting capital level, the sequence of average capitals will impose a sequence of average consumption and investement that forms an equilibrium from part a. \section*{(e) For this particular world econmy, state and prove versions of the Factor Price Equalization Theorem, the Stolper-Samuelson Theorem, the Rybcynzki Theorem, and the Heckscher-Ohlin theorem.} \subsection*{Factor Price Equalization Theorem} If $y_{1t}^{i},y_{1t}^{j}>0$, then $r_{t}^{i}=r_{t}^{j}$. And if $y_{2t}^{i},y_{2t}^{j}>0$, then $w_{t}^{i}=w_{t}^{j}$. In other words, if two countries in this economy are producing the same good, it must be that that good's input prices are the same between the two countries. - Note that technology is CRT, so $p_{1t}\leq r_{t}^{i}$, with equality if $y_{1t}^{i}>0$. Otherwise, the firm's problem cannot be satisfied. Thus for any two countries, with $y_{1t}^{i},y_{1t}^{j}>0$, $r_{t}^{i}=p_{1t}=r_{t}^{j}$. Likewise, $p_{2t}\leq w_{t}^{i}$ with equality if $y_{2t}^{i}>0$. So or any two countries with $y_{2t}^{i},y_{2t}^{j}>0$, then $w_{t}^{i}=p_{2t}=w_{t}^{j}$. \subsection*{Stolper-Samuelson Theorem} If $\left(\hat{p_{1t},}\hat{p_{2t,}}\hat{r_{t}^{i}},\hat{w_{t}^{i}}\right)$ represent prices from an equilibrium with particular endowments, and $\left(\tilde{p_{1t}},\tilde{p_{2t,}}\tilde{r_{t}^{i}},\tilde{w_{t}^{i}}\right)$ represent prices from an equilibrium with different endowments, an if $\hat{y_{1t}^{i}},\hat{y_{2t}^{i}},\tilde{y_{1t}^{i}},\tilde{y_{2t}^{i}}>0$, then $\frac{\hat{p_{1t},}}{\hat{p_{2t,}}}>\frac{\tilde{p_{1t}}}{\tilde{p_{2t,}}}\implies\frac{\hat{r_{t}^{i}}}{\hat{w_{t}^{i}}}>\frac{\tilde{r_{t}^{i}},}{\tilde{w_{t}^{i}}}$. This theorem states that if the relative price of a good rises, then the relative price of the factor used most heavily in that good will also rise. - By the factor price equalization theorem, $\hat{y_{1t}^{i}},\hat{y_{2t}^{i}},\tilde{y_{1t}^{i}},\tilde{y_{2t}^{i}}$ implies that $r_{t}^{i}=p_{1t}$ and $w_{t}^{i}=p_{2t}$, so the proof of the theorem trivially follows. \subsection*{Rybcynzki Theorem} Similar to the SS theorem above, let there be two different equilibriums. The Rybcynzki theorem in this economy states that If $\hat{k_{1t}^{i}}>\tilde{k_{1t}^{i}}$, then $\hat{y_{1t}^{i}}>\tilde{y_{1t}^{i}}$, but $\hat{y_{2t}^{i}}=\tilde{y_{2t}^{i}}$. - From the production technology $y_{1t}^{i}=k_{1t}^{i}$, $y_{2t}^{i}=l_{2t}^{i}$ and the market clearing conditions, we conclude that $y_{1t}^{i}=k_{t}^{i}$, $y_{2t}^{i}=1$. The theorem trivially follows. \subsection*{Heckscher-Ohlin theorem} In equilibrium, \[ \frac{L^{i}k_{t}^{i}}{L^{i}}\geq\frac{\sum_{j}\left(L^{j}k_{t}^{j}\right)}{\sum_{j}L^{j}}\implies\begin{array}{cc} y_{1t}^{i} & >c_{1t}^{i}+x_{1t}^{i}\\ y_{2t}^{i} & \frac{\sum_{j}\left(L^{j}k_{t}^{j}\right)}{\sum_{j}L^{j}}=\frac{\text{total capital}}{\text{total labor}}=\frac{\sum_{j}L^{j}\left(c_{1t}^{j}+x_{1t}^{j}\right)}{\sum_{j}L^{j}\left(c_{2t}^{j}+x_{2t}^{j}\right)}=A_{t}=\frac{c_{1t}^{j}+x_{1t}^{j}}{c_{2t}^{j}+x_{2t}^{j}} \end{equation} Secondly, because $p_{1t}=r_{t}$ , $p_{2t}=w_{t}$ as shown previously, the budget constraint can be rewritten: \[ p_{1t}\left(c_{1t}^{i}+x_{1t}^{i}\right)+p_{2t}\left(c_{2t}^{i}+x_{2t}^{i}\right)=p_{1t}k_{t}^{i}+p_{2t} \] \[ p_{1t}\left(c_{1t}^{i}+x_{1t}^{i}-k_{t}^{i}\right)=-p_{2t}\left(c_{2t}^{i}+x_{2t}^{i}-1\right) \] Prices are nonnegative. Therefore exactly one of the following three statements holds: \[ \begin{array}{ccc} \left(c_{1t}^{i}+x_{1t}^{i}1\right)\\ \left(c_{1t}^{i}+x_{1t}^{i}>k_{t}^{i}\right) & \wedge & \left(c_{2t}^{i}+x_{2t}^{i}<1\right)\\ \left(c_{1t}^{i}+x_{1t}^{i}=k_{t}^{i}\right) & \wedge & \left(c_{2t}^{i}+x_{2t}^{i}=1\right) \end{array} \] But note that the latter two of these statements implies that $\frac{c_{1t}^{j}+x_{1t}^{j}}{c_{2t}^{j}+x_{2t}^{j}}\geq k_{t}^{i}$, which contradicts $(1)$, so the first statement must be true. Note that $y_{1t}^{i}=k_{t}^{i}$, $y_{2t}^{i}=1$, and the proof is complete. \section*{(e) Set $s_{t}=\frac{c_{t}}{y_{t}}$, where $y_{t}=p_{1t}k_{t}+p_{2t}=dk_{t}^{\alpha_{1}}$ is world GDP per capita. Explain how to show \[ \frac{y_{t}^{i}-y_{t}}{y_{t}}=\frac{s_{t}}{s_{t-1}}\left(\frac{y_{t-1}^{i}-y_{t-1}}{y_{t-1}}\right)=\frac{s_{t}}{s_{0}}\left(\frac{y_{0}^{i}-y_{0}}{y_{0}}\right) \] } Aggregate the economy. $f(k,l)=dk^{\alpha_{1}}l^{\alpha_{2}}$, and consumers consume and invest using this aggregate good. $\tilde{p_{t}}$ is the price of this aggregate good. From factor price equalization, we know that the consumer's problem can be rewritten \[ \max_{C^{i}}\sum_{t}\beta^{t}\log(c_{t}^{i}) \] \begin{align*} \ni & c_{t}^{i}+k_{t+1}^{i}\leq p_{1t}k_{t}^{i}+p_{2t}\\ & c_{t}^{i},k_{t+1}^{i}\geq0\\ & k_{0}^{i}\text{ given} \end{align*} Countries vary only in their endowments of capital, and all have the same policy functions. Argue that the average allocations (eg $c_{t}\equiv\frac{\sum L^{i}c_{t}^{i}}{\sum L^{i}}$) also satisfy the conditions for equilbrium and thus also fulfill the SPP above. But the variations in income average out to $dk_{t}^{\alpha_{1}}$ leaving us with a problem equivalent to that from the 1SSPP. Use the first order conditions from each problem and combine to get the desired result. From 1SSPP, \begin{align*} \frac{c_{t+1}}{c_{t}} & =\beta\alpha_{1}dk_{t+1}^{\alpha_{1}-1}\\ c_{t}+k_{t+1} & =dk_{t}^{\alpha_{1}} \end{align*} Plugging in $y_{t}=dk_{t}^{\alpha_{1}}=c_{t}/s_{t}$ or $c_{t}=dk_{t}^{\alpha_{1}}s_{t}$: \begin{align*} \frac{dk_{t}^{\alpha_{1}}s_{t+1}}{dk_{t}^{\alpha_{1}}s_{t}} & =\beta\alpha_{1}dk_{t+1}^{\alpha_{1}-1}\\ \frac{s_{t+1}}{s_{t}} & \frac{\beta\alpha_{1}dk_{t}^{\alpha_{1}}}{k_{t+1}}\\ \\ dk_{t}^{\alpha_{1}}s_{t}+k_{t+1} & =dk_{t}^{\alpha_{1}}\\ s_{t} & =1-\frac{k_{t+1}}{dk_{t}^{\alpha_{1}}s_{t}} \end{align*} - To disaggregate, take FOCs of a country's problem \begin{align*} \beta^{t}\frac{1}{c_{t}^{i}} & =\lambda_{t}^{i}\\ \lambda_{t}^{i} & =p_{1t+1}\lambda_{t+1}^{i} \end{align*} \[ \frac{c_{t+1}^{i}}{c_{t}^{i}}=\beta\frac{\lambda_{t}^{i}}{\lambda_{t+1}^{i}}=\beta p_{1t}=\beta r_{t+1} \] This is the same for all countries, so \[ \frac{c_{t+1}}{c_{t}}=\frac{c_{t+1}^{i}}{c_{t}^{i}}=\beta r_{t+1} \] \[ \frac{c_{t+1}}{r_{t+1}}=\beta c_{t} \] \[ \frac{c_{t+2}}{r_{t+2}r_{t+1}}=\beta\frac{c_{t+1}}{r_{t+1}}=\beta^{2}c_{t} \] \[ \frac{c_{t+n}}{\prod_{i=1}^{n}r_{t+i}}=\beta^{n}c_{t} \] - From the budget, \begin{align*} k_{t} & =\frac{c_{t}}{r_{t}}+\frac{k_{t+1}}{r_{t}}-\frac{w_{t}}{r_{t}}=\beta c_{t-1}+\frac{k_{t+1}}{r_{t}}-\frac{w_{t}}{r_{t}}\\ k_{t+n} & =\frac{c_{t+n}}{r_{t+n}}+\frac{k_{t+n+1}}{r_{t+n}}-\frac{w_{t+n}}{r_{t+n}} \end{align*} \begin{align*} c_{t} & =r_{t}k_{t}+w_{t}-k_{t+1}\\ & =r_{t}k_{t}+w_{t}-\left(\frac{c_{t+1}}{r_{+1}}+\frac{k_{t+2}}{r_{t+1}}-\frac{w_{t+1}}{r_{t+1}}\right)\\ c_{t}\left(1+\beta\right) & =r_{t}k_{t}+w_{t}+\frac{w_{t+1}}{r_{t+1}}-\frac{k_{t+2}}{r_{t+1}}=r_{t}k_{t}+w_{t}+\frac{w_{t+1}}{r_{t+1}}-\frac{\frac{c_{t+2}}{r_{t+2}}+\frac{k_{t+3}}{r_{t+2}}-\frac{w_{t+2}}{r_{t+2}}}{r_{t+1}}\\ c_{t}\left(1+\beta+\beta^{2}\right) & =r_{t}k_{t}+w_{t}+\frac{w_{t+1}}{r_{t+1}}+\frac{w_{t+2}}{r_{t+1}r_{t+2}}-\frac{k_{t+3}}{r_{t+1}r_{t+2}}\\ c_{t}\left(\sum_{i=0}^{\infty}\beta^{t}\right) & =r_{t}k_{t}+\sum_{T=0}^{\infty}\left[w_{t+T}\prod_{i=1}^{T}\frac{1}{r_{t+1}}\right]-\lim_{T\to\infty}\left[k_{t+T+1}\prod_{i=1}^{T}\frac{1}{r_{t+i}}\right] \end{align*} By the transversality condition, \[ \lim_{T\to\infty}\left[k_{t+T+1}\prod_{i=1}^{T}\frac{1}{r_{t+i}}\right]=\lim_{T\to\infty}\left[k_{t+T+1}\prod_{i=1}^{T}\frac{\lambda_{t+1+i}^{i}}{\lambda_{t+i}^{i}}\right]=\lim_{T\to\infty}\left[k_{t+T+1}\left(\frac{\lambda_{t+1+T}^{i}}{\lambda_{t+1}^{i}}\right)\right]=0 \] Likewise, for each country, \[ c_{t}^{i}\left(\sum_{i=0}^{\infty}\beta^{t}\right)=r_{t}k_{t}^{i}+\sum_{T=0}^{\infty}\left[w_{t+T}\prod_{i=1}^{T}\frac{1}{r_{t+1}}\right] \] Subtract: \begin{align*} \left(c_{t}^{i}-c_{t}\right)\left(\sum_{i=0}^{\infty}\beta^{t}\right) & =r_{t}\left(k_{t}^{i}-k_{t}\right)\\ \left(c_{t}^{i}-c_{t}\right) & =\left(1-\beta\right)r_{t}\left(k_{t}^{i}-k_{t}\right) \end{align*} - Finally, note that by subtracting the integrated budget constraint from the country bc, \[ \left(c_{t}^{i}+k_{t}^{i}\right)-\left(c_{t}+k_{t}\right)=r_{t}\left(k_{t}^{i}-k_{t}\right)-w_{t}(l_{t}-l_{t}^{i})=r_{t}\left(k_{t}^{i}-k_{t}\right) \] - Summing it up, we have that \begin{align*} \frac{c_{t+1}}{c_{t}} & =\beta r_{t+1}\\ \left(c_{t}^{i}+k_{t+1}^{i}\right)-\left(c_{t}+k_{t+1}\right) & =r_{t}\left(k_{t}^{i}-k_{t}\right) & \implies k_{t+1}^{i}-k_{t+1}=\beta r_{t}\left(k_{t}^{i}-k_{t}\right)=\frac{c_{t}}{c_{t-1}}\left(k_{t}^{i}-k_{t}\right)\\ c_{i}-c_{t} & =\left(1-\beta\right)r_{t}\left(k_{t}^{i}-k_{t}\right) \end{align*} \[ c_{t+1}=\beta r_{t+1}c_{t} \] \[ y_{t}^{i}-y_{t}=r_{t}\left(k_{t}^{i}-k_{t}\right) \] \begin{align*} \frac{y_{t}^{i}-y_{t}}{y_{t}} & =\frac{r_{t}\left(k_{t}^{i}-k_{t}\right)}{y_{t}}=\frac{r_{t}}{y_{t}}\left(\frac{c_{t-1}}{c_{t-2}}\right)\left(k_{t-1}^{i}-k_{t-1}\right)=\frac{r_{t}}{y_{t}}\left(\frac{c_{t-1}}{c_{t-2}}\right)\left(\frac{y_{t-1}^{i}-y_{t-1}}{r_{t-1}}\right)\\ & =\frac{r_{t}}{y_{t}}\left(\frac{c_{t-1}}{c_{t-2}}\right)\left(\frac{y_{t-1}^{i}-y_{t-1}}{r_{t-1}}\right)\frac{\beta}{\beta}\frac{y_{t-1}}{y_{t-1}}=\frac{\beta t_{t}c_{t-1}}{y_{t}}\left(\frac{y_{t-1}}{\beta r_{t-1}c_{t-2}}\right)\left(\frac{y_{t-1}^{i}-y_{t-1}}{y_{t-1}}\right)\\ & =\frac{c_{t}}{y_{t}}\left(\frac{y_{t-1}}{c_{t-1}}\right)\left(\frac{y_{t-1}^{i}-y_{t-1}}{y_{t-1}}\right)=\frac{s_{t}}{s_{t-1}}\left(\frac{y_{t-1}^{i}-y_{t-1}}{y_{t-1}}\right) \end{align*} And \begin{align*} \frac{y_{t}^{i}-y_{t}}{y_{t}} & =\frac{s_{t}}{s_{t-1}}\left(\frac{y_{t-1}^{i}-y_{t-1}}{y_{t-1}}\right)=\frac{s_{t}}{s_{t-1}}\left(\frac{s_{t-1}}{s_{t-2}}\left(\frac{y_{t-2}^{i}-y_{t-2}}{y_{t-2}}\right)\right)=\frac{s_{t}}{s_{t-2}}\left(\frac{y_{t-2}^{i}-y_{t-2}}{y_{t-2}}\right)\\ & =\frac{s_{t}}{s_{t-n}}\left(\frac{y_{t-n}^{i}-y_{t-n}}{y_{t-n}}\right)=\frac{s_{t}}{s_{0}}\left(\frac{y_{0}^{i}-y_{0}}{y_{0}}\right) \end{align*} \section*{(f) Find $s_{t}$and interpret. } From 1SSPP, $s_{t}=\frac{\left(1-\alpha_{1}\beta\right)dk_{t}^{\alpha_{1}}}{dk_{t}^{\alpha_{1}}}=\left(1-\alpha_{1}\beta\right)$, which means $s_{t}$doesn't change over time, and so the portion of aggregate wealth spent on consumption does not vary over time. Also, this means the following is also constant \[ \frac{y_{t}^{i}-y_{t}}{y_{t}}=\left(\frac{y_{0}^{i}-y_{0}}{y_{0}}\right) \] Which means that relative income levels remain constant over time as well.