Academic year 1997/98
Note: This solution is incomplete.
It corresponds to the English version of the exam (Type D) which is very much like Tipo A of the Spanish version.
A consumer has the following utility function:
__ u(x,y) = \/x + y
1.1. Does the consumer has convex preferences? monotonic preferences? quasi-linear preferences? Find the demand functions. What is (are) the optimal choice(s) when px=2, py=8 and m=40 ?
The consumer has quasi-linear preferences. The linear part is y and the non linear part is the square root of x. Since the square root of x is a concave function, the preferences are convex. In addition the preferences are monotonic, as the utility function is increasing in both commodities.
To calculate the demand functions we may find when the MRS is equal to the negation of the price ratio. Thus we have:
1 Px
----- = ----
__
2\/x py
This gives:
x=[py/2px]2
Depending on the prices and income there may also be a corner solution:
x=m/px, so the demand function is:
x=min{[py/2px]2 ; m/px}
The demand for y may be computed by using the above result and the budget constraint:
In the case where x corresponds to the interior solution: m=px[py/2px]2+pyy
Which gives: y={m-px[py/2px]2}/py
Thus: y=max{{m-px[py/2px]2}/py ; 0}
when px=2, py=8 and m=40 we have: x=min{[8/2×2]2 ; 40/2} = 4 and y=max{{40-2[8/2×2]2}/8 ; 0} = 4
1.2. The government subsidizes consumption of commodity x that exceeds x=8 with a subsidy of 1 dollar per unit. Find the budget constraint and the consumer's optimum.
The budget constraint is now a union of two sets: One set corresponds to the bundles on or below the budget line: 2x+8y=40, and the other to the bundles on or below 1x+8y=32.
In order to find the consumer's optimal choice we compare the optima on the two budget lines, and choose the one who gives higher utility.
The optimum on the second budget set is: x=min{[8/2×1]2 ; 32/1} = 16 and y=max{{32-1[8/2×1]2}/8 ; 0} = 2
We have u(4,4)=u(16,2)=6 so both (4,4) and (16,2) are optima.
Sarah has the following utility function u(x1,x2)=½ln(x1)+ln(x2).
In the beginning Sarah had an income of m=100 and the prices were p1=p2=2. Now the price of commodity x1 went down to p'1=1.
2.1 Find the change in the demand of x1 that occurred due to the price change (find the initial and final bundles), and decompose it to a change due to the income effect and a change due to the substitution effect.
Explain the distinction between the two effects with the help of a diagram.
The demand functions are x1=m/3p1 and x2=2m/3p2. Thus in the initial situation x1=300/18 and in the final one x'1=600/18. So the change is an increase of 300/18.
To decompose the change we find the budget line corresponding to the pivotal change. The prices correspond to the final situation and the income is m"=1×100/6+2×200/6=500/6. The optimal quantity of x1 is x"1=500/18. Thus 200/18 units of x1 account for the substitution effect and the rest, 100/18 for the income effect.
2.2 What would you say about the income effect concerning the consumption of commodity x1 if Sarah were to have the utility function u(x1,x2)=v(x1)+x2 ? Explain your answer.
In this case the income effect would be zero, as x1 is a neutral good, and its level of consumption is independent of the income (in the relevant region of prices and income).
2.3 Comment on this statement: All inferior goods are Giffen goods.
This is incorrect. An inferior good may be non-Giffen. On the other hand a Giffen good must be inferior.
2.4 Find the change in consumers' surplus due to the price change.
The change in consumers' surplus, measured in units of x2 is
2
/
|m/p2
|----- dp1 = [100/6]ln2
|3p1
/
1
A firm has the production function y=KaLb.
3.1 For which values of a and b the production function has increasing/decreasing/constant returns to scale?
Under which circumstances it would be optimal for the firm to produce an infinite quantity? Why?
When a+b>1 the production function exhibits increasing returns to scale. When a+b=1 constant returns to scale, and when a+b<1 decreasing returns to scale.
When the firm has increasing returns to scale, or when it has constant returns to scale and the marginal costs are lower than the price of the output then the firm would like to produce an infinite amount.
3.2Now assume that a=¼ and b=½. The prices of the inputs (factors of production) are wK=wL=1. Find the short-run cost function for L=25.
We have now y=K¼25½, so the short run conditional demand for K is [y4/252] and c(y)=25+[y4/252]
3.3 Find the long-run cost function. Explain how can we find the level of production in which the long run and short run costs are equal.
We may find the long-run cost function for wK=wL=1 by equating the TRS with with the negation of the inputs price ratio. After some manipulation we get: 2K=L. We substitute that in the production function and get:
K=y4/3/22/3 and L=2y4/3/22/3 and C(y)=3y4/3/22/3.
There are two ways to find where the short run and long run cost are equal. The first is to equate the two functions. The other is to equate the conditional demand of L to its quantity in the short-run.
The firm A has the following cost function: c(y)=2+y2.
4.1 Find the marginal cost curve (MC), the average cost curve (AC), the average variable cost curve (AVC), and the average fixed cost curve (AFC). Draw them in one diagram.
Without drawing another curve, can you represent the total variable cost? Also, Explain when the different curves cross each other.
MC(y)=2y; AC(y)=y+y/2; AVC(y)=y; AFC(y)=2/y
For every level of output y0 the total variable costs are equal to the area of the rectangular: y0×AVC(y0) = y0×y0. Also it is equal to the area below the MC curve from zero to y0
The marginal cost curve crosses the AC and AVC at the quantities where the latter two obtain their minimum.
4.2 Explain the general relationship between the supply curve and the different curves related to cost. Illustrate the theory using the particular example of firm A.
The supply curve would be the part of the MC curve that lies above the AVC curve.
4.3 Firm B has the following cost curve: c(y)=2y+½[y2]+10. Find the aggregate supply curve of A and B and show it in a diagram.
We have MC(y)=2+y and AVC(y)=2+½y, so the marginal cost is always above the average variable cost. This is true also for firm A.
The Supply of firm A is yA=p/2 and of firm B it is yB=max{p-2;0}. the aggregate supply is: y=p/2+max{p-2;0}=max{(3/2)p-2;p/2}