Questions for Review
Dr. Donna Feir
Economics 313
Problem Set 4: Solutions
General Equilibrium: Allocative efficiency
1. There are two producers of goods x and y. Firm 1’s PPF is y = 30 – x and firm 2’s PPF is y = 30 – ½x. Firm 1 is producing no units of x and 30 units of y. Firm 2 is producing 40 units of x and 10 units of y.There are two consumers of goods x and y. Consumer A’s utility function is UA = x + 2y and consumer B’s utility function is UB = xy. Consumer A has the consumption bundle consisting of 10 units of x and 25 units of y. Consumer B has the consumption bundle consisting of 30 units of x and 15 units of y.
- Is the definition of productive efficiency satisfied in this economy? Explain.
At any production choice, MRT1 = 1 & MRT2 = ½, so the MRTs are not equal. But this does not necessarily mean that the allocation of production is inefficient. For this, we need to check whether the production choice is on the aggregate PPF.
- Is the definition of distributive efficiency satisfied in this economy? Explain.
MRSA = ½ no matter what consumption bundle A has. MRSB = yB/xB. yB/xB = ½ at yB = 15 and xB = 30 so MRSA = MRSB. So we have distributive efficiency.
- Is the definition of allocative efficiency satisfied in this economy? Explain.
MRSA = MRSB = MRT at the aggregate production point, so we have allocative efficiency.
2. Suppose there is only one producer in the economy, with PPF yT = 100 – 2xT. There are two consumers, each with identical Cobb-Douglas preferences, where UA = xy and UB = xy.
a)Draw a diagram of the PPF. What is the MRT equal to in this economy?
b)Derive an expression for the MRS for each consumer in this economy.
MRSA = yA/xA & MRSB = yB/xB.
c)Use your answers to (a) and (b) to solve for allocative efficiency in this economy (you should be able to solve for an (xT, yT) production pair).
We know that we will want MRSA = MRSB = MRT, where that MRT always equals 2. Thus each consumers’ MRSs must also equal 2 in order for efficiency in all three contexts. The MRSs for the consumers are as follows: MRSA = yA/xA and MRSB = yB/xB. In order for each of these to equal 2 (the MRT), it must be the case that each consumer has twice as many units of y as of x. Thus in the economy as a while, there must be twice as many units of y as of x. Thus, the efficient mix of goods x and y in this economy must satisfy yT = 2xT. The only point on the aggregate PPF where this is true is where xT = 25 & yT = 50.
d)Use your answer to c) to draw an Edgeworth Box diagram of this economy. Show the contract curve in your diagram.
3. Now suppose there are two producers in the economy, with PPFs as given below.
Firm 1’s PPF:y = 100 – 0.01x2
Firm 2’s PPF:y = 50 – 0.02x2
a)Derive an expression for the aggregate PPF in this economy (that is, yT as a function of xT). Also derive an expression for the MRT.
We have seen in class that these PPFs give rise to the following aggregate PPF: yT = 150 – (1/150)xT2. The MRT is therefore (1/75)xT.
There are two consumers, each with identical Cobb-Douglas preferences, where UA = xy and UB = xy.
b)Derive an expression for the MRS for each consumer in this economy.
Again, MRSA = yA/xA & MRSB = yB/xB.
c)Use your answers to a) and b) to solve for allocative efficiency in this economy (again, you should be able to solve for an (xT, yT) production pair. In contrast to question 1, however, you will not get “nice” whole numbers).
The math in this one gets a little messy, but make sure you try to follow along with all the steps. Keep the economics in mind and this will be fairly straightforward.
Just like in question 2, we want MRSA = MRSB = MRT. MRSA = MRSB yA/xA = yB/xB, where yB = yT – yA & xB = xT – xA (B’s consumption of both goods is the total amount available less A’s consumption), Substituting and simplifying yields yA/xA = yT/xT, which tells us that, when we equate the MRSs for each consumer, that common MRS will equal the ratio of the total amount of good y to the total amount of good x. For allocative efficiency then, we need the MRS (which equals = yT/xT) to equal the MRT (which equals (1/75)xT). So MRSA = MRSB = MRT yT/xT = (1/75)xT yT = (1/75)xT2. From the equation for the PPF we also know that yT = 150 – (1/150)xT2, so we know (1/75)xT2 = 150 – (1/150)xT2 (3/150)xT2 = 150 xT = 150/(3)½. When xT = 150/(3)½, yT = 100. So xT = 150/(3)½, yT = 100 is the efficient production pair in this economy. Check to that the MRT will equal the equalized MRS, given this production point.
d)Use your answer to c) to draw an Edgeworth Box diagram of this economy. Show the contract curve in your diagram.
4. There are two individuals in the economy, each of whom is both a producer and a consumer. Each has the same PPF, given by y = 50 – (1/5)x2. Individual A’s preferences are given by UA = 20 ln xA + yA and individual B’s preferences are given by UB = yB.
a)Solve for the aggregate supply function for good x.
Each producer sets its MRT = price ratio for A we have (2/5)xA = px/pyxSA = (5/2)px/py. Because B’s PPF is identical, we also know that B’s supply of x is given xSB = (5/2)px/py. So aggregate supply is A’s supply plus B’s supply, which is xS = 5px/py.
b)Solve for the aggregate demand function for good x.
Consumer A sets her MRS = price ratio 20/xA = px/pyxDA = 20/(px/py). Because B consumes no x (it gives him no utility), the aggregate demand for x is just A’s demand for x. So we have xD = 20/(px/py).
c)Use your answers to part a) and b) to solve for the equilibrium price ratio.
xS = xD 5px/py= 20/(px/py) px/py= 2.
d)Given your answer to c), how much of each good will A and B produce?
From the individual producer supply curves we know that if px/py= 2, then each firm will produce (5/2)(2) = 5 units of x. When x = 5, y = 45 for each producer.
e)Given your answer to c), how much of each good will A and B consume?
We know that B does not consume any good x. A’s demand for x is xDA = 20/(px/py), so A consumes 10 when px/py= 2. You can use the equation for each consumer’s BL to solve for their consumption of y. A consumes 35 units of y and B consumes 55 units of y. (Remember – the endowments in the budget lines are given by the quantities of goods x and y produced by each).
f)Why does individual B produce positive amounts good of x, given seeing that in part e) you showed that her optimal consumption of good x is zero?
Even though B does not consume x, it is still worthwhile for him to produce x, as he can then sell it to A and buy y in return. This enables him to consume more y than he would otherwise be able to, were he just producing on his PPF.
5. There are two individuals in the economy, each of whom has identical Cobb-Douglas preferences such that UA = xAyA and UB = xByB. Individual A can produce both goods according to the PPF given by y = 100 – x. Individual B cannot produce either good, but is endowed with 20 units of good x.
a)Suppose the price ratio in this economy is greater than 1. Show that there is an excess supply of good x in this case.
If px/py > 1, then A will produce all x and no y. That is, A will produce 100 units of x and 0 units of y. B is endowed with 20 units of x and no y. Aggregate supply of x is therefore 120, while aggregate supply of y equals 0. Both the consumers have Cobb-Douglas preferences, which means that each will always want to consume positive amounts of each good. We do not have any y, however, meaning that there will be excess demand for good y. If there is excess demand for y, there must be excess supply of x.
b)Suppose the price ratio in this economy is less than 1. Show that there is an excess demand for good x in this case.
If px/py < 1, then A will produce all y and no x. That is, A will produce 100 units of y and 0 units of x. B is endowed with 20 units of x and no y. Aggregate supply of x is therefore 20, while aggregate supply of y equals 100.
On the demand side, both the consumers choose the level x where their MRSs equal the price ratio. For A we have MRSA = yA/xA = px/pyyA= (px/py)xA. A’s BL is given by (px/py)wAX + wAy = (px/py)xA + yA , where we know wAX = 0, wAy =100, and – from the tangency condition – yA= (px/py)xA. Substituting these into the equation for the BL yields 100 = 2(px/py)xAxA= 50/(px/py). We know (px/py) < 1, so we know A’s demand for x > 50, while supply is only 20. There is excess demand for good x when we just look at A’s demand, so there will be even greater excess demand in aggregate (since we know B always wants to consume positive x as well).
c)Use your answers to a) and b) to deduce the equilibrium price ratio in this economy.
If px/py > 1 excess supply of x and px/py < 1 excess demand for x, the only possibility for an equilibrium price ratio is px/py = 1.
d)Given your answer to c), how much of each good will B consume?
For B we have MRSB = yB/xB = px/py = 1 yB= xB. B’s BL is given by (px/py)wBX + wBy = (px/py)xB + yB , where we know wBX = 20, wBy = 0, and yB= xB. Substituting these into the equation for the BL yields 20 = 2xBxB= 10. We know xB= yB, so yB = 10 also. So B sells 10 of his x and buys 10 y in return.
e)Given your answer to c), how much of each good will A produce and consume?
We know when px/py = 1, A is indifferent among producing at any point on her PPF. This means we cannot solve for her production of x and y just yet. But think about her consumption. We know that she will consume where MRSA = yA/xA = px/py = 1 yA= xA at her optimal consumption. Using the equation for her BL, we have (px/py)wAX + wAy = (px/py)xA + yA , where we know px/py = 1 and yA= (px/py)xA. Substituting these into the equation for the BL yields wAX + wAy = 2xA. We also know that wAX = whatever x she produces (which we’ll call xAS) and wAy = whatever y she produces (which we’ll call yAS). So we know (from the equation for the PPF) that wAy = yAS = 100 - xAS ,wAx = xAS, so that wAX + wAy = xAS + (100 - xAS ) = 100. Substituting this into the equation for the BL yields 100 = 2xA, so xA = 50. This is the x that A consumes. We know that her consumption of x = her consumption of y given the price ratio of 1, so we also know that she consumes 50 units of y.
Now we can figure out how much of each good she produces. We know that she sells 10 units of y to B, and consumes 50 herself, so she must be producing 60 units. We also know she buys 10 units of x from B and consumes 50 in total, so she must be producing 40.
f)Draw an aggregate PPF/Edgeworth Box diagram illustrating the equilibrium in this economy.
See lecture slides to get an idea of what this diagram looks like.
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