Academic year 1997/98

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**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
`p _{x}=2`,

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 P_{x}----- = ---- __ 2\/x p_{y}

This gives:

`x=[p _{y}/2p_{x}]^{2}`

Depending on the prices and income there may also be a corner solution:

`x=m/p _{x}`, so the demand function is:

`x= min{[p_{y}/2p_{x}]^{2} ; m/p_{x}}`

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=p _{x}[p_{y}/2p_{x}]^{2}+p_{y}y`

Which gives:
`y={m-p _{x}[p_{y}/2p_{x}]^{2}}/p_{y}`

Thus:
`y= max{{m-p_{x}[p_{y}/2p_{x}]^{2}}/p_{y} ; 0}`

when
`p _{x}=2`,

**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

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(x _{1},x_{2})=½ln(x_{1})+ln(x_{2})`.

In the beginning Sarah had an income of `m=100` and the prices
were `p _{1}=p_{2}=2`. Now the price of commodity

**2.1** Find the change in the demand of
`x _{1}`
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 `x _{1}=m/3p_{1}` and

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
`x _{1}` is

**2.2** What would you say about the income effect
concerning the consumption of commodity `x _{1}`
if Sarah were to have the utility function

In this case the income effect would be zero, as `x _{1}` 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 `x _{2}`
is

2 / |m/p_{2}|----- dp_{1}= [100/6]ln2 |3p_{1}/ 1

A firm has the production function
`y=K ^{a}L^{b}`.

**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.

**3.2**Now assume that `a=¼` and
`b=½`. The prices of
the inputs (factors of production) are
`w _{K}=w_{L}=1`. Find the short-run cost
function for

We have now `y=K ^{¼}25^{½}`, so the
short run conditional demand for

**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
`w _{K}=w_{L}=1` by equating the TRS with
with the negation of the inputs price ratio. After some manipulation we get:

`K=y ^{4/3}/2^{2/3}` and

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+y ^{2}`.

**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 `y _{0}` the total variable costs
are equal to the area of the rectangular:

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[y ^{2}]+10`. Find the aggregate
supply curve of

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 `y _{A}=p/2` and of firm

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Nir Dagan / Contact information / Last modified: August 10, 1998.