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\begin{document}
\title{Micro Homework: Problem Set 1}
\author[1]{Louis Rouanet}%
\affil[1]{George Mason University}%
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\date{\today}
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1)
a)
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b)
\(U(X ; Y) = XY = 90\)
\(Y=\frac{90}{X}\)
\(MRS_{xy} = \frac{\Delta Y}{\Delta X} = - \frac{90}{X^2}\)
With X = 6, we have:
\(MRS_{xy}=-\frac{90}{36}=-\frac{5}{2}\)
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c)
\(I = P_xX + P_yY\)
\(Y = \frac{I}{P_y} - \frac{P_x}{P_y}X\)
So the slope is equal to~\(- \frac{P_x}{P_y}\) So an increase in income
does not change the slope.
If~\(P_x = 8\) ,~\(P_y = 4\) and~\(I = 45\) then
we have:
\(Y = \frac{45}{4} - 2X\)
So the slope is equal to~\(-2\). If income increases from
45 to 60, the slope of the budget constraint does not change and the
equation is the following:
\(Y = 15 - 2X\)
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d)
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Since~\(P_x = 8\) ,~\(P_y = 4\)~ ~ , the slope of the
budget curve is equal to -2. The optimum in consumption is reached when
the marginal rate of substitution is equal to the slope of the budget
curve. We know that:
\(MRS_{xy} = \frac{\Delta Y}{\Delta X} = - \frac{90}{X^2}\)
And we want to solve the equation:
\(-2 = - \frac{90}{X^2}\)
\(X^2 = 45\)
\(X = 45^{\frac{1}{2}}\)
We substitute this result in the utility function~\(U(X ; Y) = XY = 90\):
\(\sqrt{45}Y = 90\)
\(Y = \frac{90}{\sqrt{45}}\)
\(Y = 2\sqrt{45}\)
Then we substitute the results for Y and X in the budget constraint:
\(I = 8X + 4Y\)
\(I = 8\sqrt{45} + 4*2\sqrt{45}\)
\(I = 16\sqrt{45}\)
So the income necessary so that the budget constraint is tangent to the
indifference curve for which~\(U(X ; Y) = 90\) is equal to
\(16\sqrt{45}\), i.e. for the consumer to attain 90 utils of
satisfaction at our given prices.
{[}PS{]}: To solve this question, we could use a Lagrangian so as to
``maximize income.'' We find the same results with both methods.
\par\null
2)
We have~\(U(X_1 ; X_2) = X_1X_2 ; I=60 ; P_{X_1} = 5 ; P_{X_1} = 3\)
The budget constraint is equal to:~\(60 = RX_1 + 3X_2\)
Let's set the Lagrangian:
\(\mathcal{L} = X_1X_2 - \lambda[60 - (5X_1 + 3X_2)]\)
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\(\frac{dL}{dX_1} = 0 = X_2 - \lambda 5\)
\(\lambda = \frac{X_2}{5}\)
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\(\frac{dL}{dX_2} = 0 = X_2 - \lambda 3\)
\(\lambda = \frac{X_1}{3}\)
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\(\frac{dL}{d\lambda} = 0 = 60 - (5X_1 + 3X_2)\)
\(60 = 5X_1 + 3X_2\)
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\(\frac{X_2}{5} = \lambda = \frac{X_1}{3}\)
\(3X_2 = 5X_1\)
\(X_2 = \frac{5}{3}X_1\)
We substitute this result in \(\frac{dL}{d\lambda} \) .
\(60 = 5X_1 + 3\frac{5}{3}X_1\)
\(X_1 = 6\)
\(X_2 = 10\)
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3)
We have~\(U(X ; Y) = 2XY ; I = 40 ; P_y = 4 ; P_x = 8\)
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a)
The demand curve of ~X correspond to the quantity of X demanded with
respect to the price of X. We therefore consider~\(P_x\) as
a variable.
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Let's set the Lagrangian:
\(\mathcal{L} = 2XY- \lambda[40 - 4Y - P_x X]\)
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\(\frac{dL}{dX} = 0 = 2Y - \lambda P_x\)
\(\lambda = \frac{2Y}{P_x}\)
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\(\frac{dL}{dY} = 0 = 2X - \lambda 4\)
\(\lambda = \frac{2}{4}X\)
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\(\frac{dL}{d\lambda} = 0 = 40 - 4Y - P_xX\)
\(40 = 4Y + P_xX\)
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\(\frac{2Y}{P_x} = \lambda = \frac{2}{4}X\)
\(2P_xX = 8Y\)
\(Y = \frac{1}{4}P_xX\)
Substituting in~\(\frac{dL}{d\lambda} \), we have:
\(40 = 2P_xX\)
\(X = \frac{20}{P_x}\) is the demand curve for X.
\par\null
b)
A 25\% tax on the initial price of X means that the after tax price is
of 10\$ since we know that the pre-tax price is of 8\$.
We know the demand curve for X from question (a):
\(X = \frac{20}{P_x}\)
So before the tax, since~\(P_x = 8\), the quantity X demanded is
equal to:
\(X = \frac{20}{8} = 2.5\)
The quantity demanded after the 25\% tax is equal to:
\(X = \frac{20}{10} = 2\)
The budget constraint is equal to:
\(40 = 4Y + P_xX\)
So before the tax,~\(P_x = 8\) and therefore:
\(Y = \frac{40 - 2.5*8}{4} = 5\)
After the tax, we have \(P_x = 10\):
\(Y = \frac{40 - 2*10}{4} =5\)
Since the utility function is~\(U(X ; Y) = 2XY\), the consumer's
utility is equal to 25 before the tax and 20 after the tax.
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c)
After the tax,~\(X = 2\), and since the tax yields 25\% of the
initial price of 8\$. Therefore, the tax yielded 4\$ of revenue.
SInce~\(I = 40\), the income tax has to be of 10\% so as to
yield the same amount of revenue. After tax income will be of 36\$.
From question (a), we know that:~
\(X = \frac{0.5*I}{P_x}\)
Hence:
\(X = \frac{0.5*36}{8} = 2.25\)
We also know that:
\(Y = \frac{1}{4}P_xX\)
\(Y = \frac{1}{4}*8*2.25 = 4.5\)
We can theoretically account for this differences because in (b), you
have a change in the relative price of X compared to Y and a readuction
of real income as the after tax price of X increases. In the case of the
income tax in question however, you only have an income effect. the
relative price of X and Y is not changing.
\par\null
d)
We know that the utility function is~\(U(X ; Y) = 2XY\)
With the sale tax, the individual consumes 2 of X and 5 of Y.
Therefore,~\(U(X ; Y) = 20\)
With the income tax, the individual consumes 2.25 of X and 4.5 of Y.
Therefore~\(U(X ; Y) = 20.25\)
This means that the two goods in question are complementary enough so
that an Income tax will be preferred to a tax on X.
\par\null
e)
If we have 50 consumers identical to Mike and with the same budget
constraint, then the demand of X is multiplied by 50. When the price of
X is equal to 8\$, the quantity of X demanded is of 2.5. That means that
the total market demand for X is equal to 75.
\par\null
f)
Lambda is the Lagrange multiplier and approximates the marginal impact
on the objective function caused by a change in the constant of the
constraint. In our exemple,~\(\lambda\)~approximates the impact
of a change in income on total utility, or, to put it differently,
approximates the marginal utility of money.
\par\null
4)
We have~\(X = 0.99I - 2.5P_x - 2.51P_y\)
A normal good is a good for which the quantity demanded increases when
income increases. Hence since~\(\frac{dX}{dI} = 0.99\), then X is a normal
good.
\(\frac{dX}{dP_x} = -2.5\) which means that the demand curve is negatively
sloped -i.e. an increase in~\(P_x\)~reduces the amount of X
demanded.
If X and Y are substitute, then we would expect an increase
in~\(P_y\) to increase the demand for X.
Since~\(\frac{dX}{dP_x} = -2.51\), this means that X and Y are complementary,
-i.e. the increase in the price of one good leads to a decrease in the
demand of the other good.
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