Topic 1: Closed HOS Model

Question 1

What are the 5 assumptions in the Heckscher-Ohlin-Samuelson model?

Answer 1

1. Two regions (home, abroad), primary factors (L, K) and two goods (X-labour intensive, Y-capital intensive).
2. Fixed primary factor endowments
3. X & Y produced using the same technology
4. Industries are modelled as a single price taking firm with constant returns to scale.
5. Single household with homethetic preferences (composition of consumption doesn’t change with income)

Question 2

What must we assume to assume a single price taking industry for our whole economy?

Answer 2

• Competition is perfect (all firms are indentical in size, technology and are price taking
• Infinite number of potentual entrants to the industry
• Entry to and exit from the industry is free

Question 3

What are the first two fundamental theorems of welfare economics?

Answer 3

1. Any general equilibtrium achieved through private market trading with all agents as price takers, no externalities and costless exchanges is pareto efficient.
2. Any given pareto optimal allocation of the economy’s resources can be achieved through private market trading if all agents are price takers, there are no externalities and trades, or exchagnes are costless, given an initial lump sum transfer.

Question 4

What is the optimisation problem in the closed HOS model? Give the lagrange multipliers.

Answer 4

Maximise:

U = X^1/2Y^1/2

Subject to:

X = A_XL_X^3/4K_X^1/4 (p_X)

Y = A_YL_Y^3/4K_Y^1/4 (P_Y)

L_X + L_Y = L (w)

K_X + K_Y = K (r)

Question 5

What are shadow prices?

Answer 5

In the HOS model, the shadow prices are the lagrange multipliers from the constraints, which are measured in ulitity per unit of X,Y,L & K - the extra utility if constraints were relaxed.

Question 6

What is Walra’s law?

Answer 6

That in the simply optimization problem, demonstrated graphically below.

If both prices simultaneously doubled, there would be exactly no change in consumption. Income would obviously still match expenditure. This demonstrates that only relative prices mater - the absolute values are meaningless, and so can’t be solved for by themselves.

Question 7

What is a numeraire? Why do we use it?

Answer 7

Because prices are all relative, a numeraire is simply a price we set as 1, so that all other prices are measured relative to it. Best to pick a numeraire not directly affected by the analysis to be conducted. Not a good idea to pick goods that are heavily distorted. In more complicated economies, the numeraire is money!

Question 8

How can we approach solving the closed HOS model without an overall optimization calculation?

Answer 8

We can solve the consumer and producer problem seperately, giving us equations that can be used together to solve for all variables.

Question 9

Set up and solve the consumption problem in the closed HOS model

Answer 9

Consumers must choose C_X & C_Y to maximise U = C_X^1/2C_Y^1/2,

Subject to p_XC_X + p_YC_Y = M

We can then optimise to get - MU_X/ MU_Y = -p_X/p_Y,

From which we get p_XC_X = P_YC_Y

Question 10

Derive MU_X & MU_Y from U=C_X^1/2C_Y^1/2 to get the MRS

Answer 10

MU_X = 1/2C_Y^1/2/C_X^1/2

MU_Y= 1/2C_X^1/2/C_Y^1/2

MRS = - MU_X/MU_Y = 1/2/(1/2)=1

Question 11

Formulate and solve the producer problem:

Answer 11

Profit maximisation!

Max Π_X = p_XX - wL_X- rK_X

Subject to: X=A_XL_X^3/4K_X^1/4

substituting…

Max Π_X = p_XA_XL_X^3/4K_X^1/4- wL_X- rK_X

Now we set the partial derivites equal to zero.

3p_XX/4L_X-w = 0

p_XX/4K_X-r = 0

Due to arbitrage, w_X = w_Y, r_X = r_Y

When we set a numeraire, using the consumer equation, we can solve for everything.

Question 12

Show how GDP can be calculated in closed HOS model

Answer 12

Income: Y = rK+wL

Exp: Y = C+I+G+X-M

Y = p_XC_X + P_YC_Y

Value Added: Y= pXCX + PYCY as no intermediate inputs.