C) Determine the equilibrium price of good x (setting the price of good y as 1) that prevails at Home and Foreign under autarky – that is, when they do not trade with each other. Explain why any other price could not be the equilibrium price in autarky." 2 graphs one for home and one for foreign should be "Home and Foreign’s PPF" In autarky, each country has to produce both goods, otherwise, consumers would achieve a utility of 0 if one of the goods they like to consume is not available (plug 0 in one of the goods in the utility function and you get 0 level of utility). From the profit maximization problem in each industry at Home, we know that PxMP LxH = wH = PyMP LyH, At Home, the relative price of x in autarky is given by: PxMP LxH = wH = PyMP LyH Px/Py = A_H = MP LyH / MP LxH = 1/2 Px/Py = MPLyH/MPLxH = (1 unit of good y produced per worker) / (2 units of good x produced per worker) = 1/2 Thus, the relative price of good x in autarky at Home is equal to 1/2. Similarly, at Foreign, the relative price of x in autarky is: PxMP LxF = wF = PyMP LyF Px/Py = A_F = MP LyF / MP LxF = 2 Px/Py = MPLyF/MPLxF = (2 units of good y produced per worker) / (1 unit of good x produced per worker) = 2 Thus, the relative price of good x in autarky in Foreign is equal to 2. Px/Py (Home) = 1/2 Px/Py (Foreign) = 2 In other words, the relative price of x in autarky in a given country must be equal to the opportunity cost of producing one additional unit of good x, expressed in units of good y. This is because the relative price of x in autarky must match the relative productivity of labor in producing good x and good y in each country, ensuring an efficient allocation of labor between the two industries. Any other price would not be the equilibrium price in autarky, as it would lead to an inefficient allocation of labor and imbalanced consumption of goods x and y because the relative price of good x is determined by the opportunity cost of producing one additional unit of good x, which is reflected in the slope of the PPF. (d) Determine the optimal consumption and production at Home and Foreign under autarky. Depict this situation in a graph that includes each country’s PPF and indifference curves for the representative consumer. 2 graphs one for home and one for foreign of "Home and Foreign’s Consumption and Production under Autarakcy" To find the optimal consumption at home and foreign we maximise the representative consumer’s utility subject to the PPF, noting that in autarky, consumption of each good has to be equal to domestic production, i.e. CxH = QxH and CyH QyH. For Home, we want to maximize the utility function subject to the PPF constraint. The utility function is given by: U(CxH, CyH) = CxH^α * CyH^(1-α) Since in autarky CxH = XH and CyH = YH, we can rewrite the utility function using the PPF constraint: U(XH, YH) = XH^α * (10 - 2XH)^(1-α) Take the FOC with respect to XH and set it equal to zero: α * XH^(α-1) * (10 - 2XH)^(1-α) - 2(1-α) * XH^α * (10 - 2XH)^(-α) = 0 Divide by XH^(α-1) * (10 - 2XH)^(1-α) to get: α - 2(1-α) * (XH / (10 - 2XH)) = 0 Solve for XH: XH = 5α Now, we can find YH using the PPF constraint: YH = 10 - 2(5α) = 10(1 - α) So, the optimal consumption and production bundle for Home under autarky is (5α, 10(1 - α)). For Foreign, the utility function is: U(CxF, CyF) = CxF^α * CyF^(1-α) In autarky, CxF = XF and CyF = YF. Using the PPF constraint, we can rewrite the utility function as: U(XF, YF) = XF^α * (5 - 0.5XF)^(1-α) Take the FOC with respect to XF and set it equal to zero: α * XF^(α-1) * (5 - 0.5XF)^(1-α) - 0.5(1-α) * XF^α * (5 - 0.5XF)^(-α) = 0 Divide by XF^(α-1) * (5 - 0.5XF)^(1-α) to get: α - 0.5(1-α) * (XF / (5 - 0.5XF)) = 0 Solve for XF: XF = 10(1-α) Now, we can find YF using the PPF constraint: YF = (10 - 10(1-α))/2 = 5α So, the optimal consumption and production bundle for Foreign under autarky is (10(1 - α), 5α). In summary, the optimal consumption and production bundles under autarky are: Home: (5α, 10(1 - α)) Foreign: (10(1 - α), 5α) The graphical representation should show the tangency of the highest indifference curve with the PPF for each country, indicating the optimal consumption and production points under autarky. The slope of the indifference curve at the tangency point should be equal to the slope of the PPF, which is the equilibrium price of good y under autarky, as derived in part (c). f) 2 graphs one for home and one for foreign should be "Home and Foreign’s Consumption and Production under Free trade"
Here are the questions and answers : Please draw the graphs for each answers and include all axis and details: PLEASE DO NOT SOLVE THE QUESTIONS THEY HAVE ALREADY BEEN SOLVED. Any further information needed is in the image.
C) Determine the
2 graphs one for home and one for foreign should be "Home and Foreign’s
In autarky, each country has to produce both goods, otherwise, consumers would achieve a utility of 0 if one of the goods they like to consume is not available (plug 0 in one of the goods in the utility function and you get 0 level of utility). From the profit maximization problem in each industry at Home, we know that PxMP LxH = wH = PyMP LyH,
At Home, the relative price of x in autarky is given by:
PxMP LxH = wH = PyMP LyH
Px/Py = A_H = MP LyH / MP LxH = 1/2
Px/Py = MPLyH/MPLxH = (1 unit of good y produced per worker) / (2 units of good x produced per worker) = 1/2
Thus, the relative price of good x in autarky at Home is equal to 1/2.
Similarly, at Foreign, the relative price of x in autarky is:
PxMP LxF = wF = PyMP LyF
Px/Py = A_F = MP LyF / MP LxF = 2
Px/Py = MPLyF/MPLxF = (2 units of good y produced per worker) / (1 unit of good x produced per worker) = 2
Thus, the relative price of good x in autarky in Foreign is equal to 2.
Px/Py (Home) = 1/2
Px/Py (Foreign) = 2
In other words, the relative price of x in autarky in a given country must be equal to the
(d) Determine the optimal consumption and production at Home and Foreign under autarky. Depict this situation in a graph that includes each country’s PPF and indifference curves for the representative consumer.
2 graphs one for home and one for foreign of "Home and Foreign’s Consumption and Production under Autarakcy"
To find the optimal consumption at home and foreign we maximise the representative consumer’s utility subject to the PPF, noting that in autarky, consumption of each good has to be equal to domestic production, i.e. CxH = QxH and CyH QyH.
For Home, we want to maximize the utility function subject to the PPF constraint. The utility function is given by:
U(CxH, CyH) = CxH^α * CyH^(1-α)
Since in autarky CxH = XH and CyH = YH, we can rewrite the utility function using the PPF constraint:
U(XH, YH) = XH^α * (10 - 2XH)^(1-α)
Take the FOC with respect to XH and set it equal to zero:
α * XH^(α-1) * (10 - 2XH)^(1-α) - 2(1-α) * XH^α * (10 - 2XH)^(-α) = 0
Divide by XH^(α-1) * (10 - 2XH)^(1-α) to get:
α - 2(1-α) * (XH / (10 - 2XH)) = 0
Solve for XH:
XH = 5α
Now, we can find YH using the PPF constraint:
YH = 10 - 2(5α) = 10(1 - α)
So, the optimal consumption and production bundle for Home under autarky is (5α, 10(1 - α)).
For Foreign, the utility function is:
U(CxF, CyF) = CxF^α * CyF^(1-α)
In autarky, CxF = XF and CyF = YF. Using the PPF constraint, we can rewrite the utility function as:
U(XF, YF) = XF^α * (5 - 0.5XF)^(1-α)
Take the FOC with respect to XF and set it equal to zero:
α * XF^(α-1) * (5 - 0.5XF)^(1-α) - 0.5(1-α) * XF^α * (5 - 0.5XF)^(-α) = 0
Divide by XF^(α-1) * (5 - 0.5XF)^(1-α) to get:
α - 0.5(1-α) * (XF / (5 - 0.5XF)) = 0
Solve for XF:
XF = 10(1-α)
Now, we can find YF using the PPF constraint:
YF = (10 - 10(1-α))/2 = 5α
So, the optimal consumption and production bundle for Foreign under autarky is (10(1 - α), 5α). In summary, the optimal consumption and production bundles under autarky are:
Home: (5α, 10(1 - α))
Foreign: (10(1 - α), 5α)
The graphical representation should show the tangency of the highest indifference curve with the PPF for each country, indicating the optimal consumption and production points under autarky. The slope of the indifference curve at the tangency point should be equal to the slope of the PPF, which is the equilibrium price of good y under autarky, as derived in part (c).
f) 2 graphs one for home and one for foreign should be "Home and Foreign’s Consumption and Production under Free trade"
When the equilibrium price of good x is equal to 1 (keeping the price of good y as 1), the terms of trade are 1:1. In this situation, Home specializes in producing good x, and Foreign specializes in producing good y.
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Brother i need these graphs to be more detailed please sir at least the
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