Economics Problems: Production Function, Efficiency Locus, PPF

1) Suppose a production function only uses one input, labor. If production displays constant returns to scale, then it does not exhibit diminishing marginal product of labor.

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True, output is normally proportional to the amount of input due to constant returns to scale and therefore the marginal product of labor remains constant and does not diminish when labor continues to be deployed. However, there remain to be no other input to be fixed.

2) Suppose a production function utilizes both capital (K) and labor (L) as inputs. For this production function, an increase in L (holding K constant) leads to an exact proportional increase in output. That is, for instance, increasing L by a factor of two yields twice as much output. Then, this production function must display increasing returns to scale.

If K and L both rise by some proportion l, we can break the effect into two parts, the effect of L alone rising by l, and then the effect of K rising by l. By assumption, the first of these causes output to rise by l. If the marginal product of K is positive, then the rise in K causes an additional increase in output, so that the total rise in output is then more than l. But that is the definition of increasing returns to scale.

3) Suppose, as with the Workhorse model in lecture, production in a 2good (X,Y), 2factor (L, K) economy exhibits constant returns to scale.

Unlike lecture, it so happens that X and Y have exactly the same factor intensity at all factor prices. That is, there are no laborintense or capitalintense goods here.

4) What does the Efficiency Locus look like in the Edgeworth Box? Sketch it and explain its particular shape. (It will not look like the Efficiency Locus in the lecture.)

The efficiency locus is the diagonal therefore the PPF is a straight line. Due to constant returns to scale, the output of X is proportional to the distance from Ox, output of Y is proportional to the distance from Oy, and each of these distances is the length of the diagonal itself minus the other.

5) Using our Workhorse Model of Production and Trade as developed in lecture, suppose that a country is initially exporting good X. Its PPF then expands outward (we dont care about about precisely why), making it possible for it to produce more of both X and Y.

There are three possible ways the PPF could shift. One where the PPF shifts out uniformly, and X and Y increase in the same proportion. Second is a shift that is biased towards production of Y. Finally, there is a shift that is biased towards X. Assume that the country is small enough that the world market price does not change as a result of its new PPF and production levels.

4) A) Sketch a graph for each of the three possibilities listed above. Include the PPF, the budget line, and indicate possible production/consumption points. In particular, it will be helpful to follow the techniques from lecture about drawing these graphs, paying attention especially to how we identify the world price ratio and shift indifference curves.

B) Using your three graphs from part A, which (if any) of the following must be true, which might or might not be true, and which cannot be true? Refer to your graphs to justify.

a. It will produce more of good X.

b. It will produce more of good Y.

c. Its income will rise.

d. It will import more of good Y.

According to all the four statements, C is the only true statement because when the PPF shifts out, the price line tangent to it also shifts out. However, this suggests that the income rises gradually.

However there are several other ways that the PPF can shift out, as shown in the bottom two panels. On the left is a case where the shift is biased toward good Y, and as a result the output of good X falls. In that case, also, the excess supply of good X falls (and could have become negative) and the excess demand for good Y also falls, so that imports of Y fall. In the bottom right the bias is in the other direction, and the output of good Y falls.