Deriving demand functions given utility

Question

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$.

Derive the equation for the consumer’s demand function for clothing.

I found the first order conditions for $X$ and $Y$ and then solved for $Y$ which gave me $Y = X/P_y -10$ I then combined this with the budget constraint to get $2X - 10P_y = 100$ Please would it be possible to advise me whether whether my answers are correct as this is my first attempt at deriving demand functions. Also, is the utility quasilinear? I know that an equation of the form $U(X,Y) = f(X) + Y$ is quasilinear but I don't know if $U(X,Y) = f(x,Y) + Y$ would fit the category.

Answer 1

A demand function relates the quantity demanded of a good by a consumer with the price of the good. Thus we wish to find $Y = f(P_Y)$.

Setting up the optimization problem:

$$\max{U(X,Y)}$$

subject to: $$ I = P_x X + P_Y Y $$

where $I$ is income, $P_X$ is the price of good $X$, and $P_Y$ is the price of good $Y$.

Using the values you provided gives the optimization problem as:

$$ \max{ (XY + 10Y) } $$

subject to: $$ 100 = 1 \cdot X + P_Y Y $$

Setting this up as a Lagrange problem,

$$ L = XY + 10Y + \lambda (100 - X - P_Y Y )$$

Taking the first order conditions, we get:

$[X]:$ $\frac{ \partial U(X,Y) }{ \partial X} = Y - \lambda = 0$

$[Y]:$ $\frac{ \partial U(X,Y) }{ \partial Y} = X + 10 - \lambda P_Y = 0$

$[ \lambda ]:$ $\frac{ \partial U(X,Y) }{ \partial \lambda } = 100 - X - P_Y Y = 0$

Note, at this point you will usually take the second order conditions to ensure you have a maximum. Clearly you do have a maximum in this case since $U$ is strictly increasing in $X$ and $Y$.

Combining $[X]$ and $[Y]$ we get $X + 10 = Y P_Y$

We wish to get the demand for clothing, so we will solve for $X$ with the intention of substitution it into the budget constraint, $X = Y P_Y - 10$. Substituting into the constraint yields: $100 = 2 P_Y Y - 10$, or a final demand equation of:

$$ Y = \frac{45}{P_Y} $$

Finally, for a utility function to be quasi-linear, you must be able to express one utility as a linear function of one of the goods. Note in your case this may not be accomplished since you have an interaction between $X$ and $Y$. The reason quasi-linearity is nice is because it allows the expression of utility in terms of a numeraire good.

Answer 2

Consider a monopoly with a production function given by $q = f(x) =\sqrt x$ and a fixed cost of \$1. The input price is \$0.50. The monopolist sells her product in a market that has ten consumers.

Let $p$ denote the unit price of the good. Assume that we can represent the preferences of every consumer as follows: If consumer $i$ purchases $q$ units of the good and has $y$ dollars left to spend on all other goods, whose prices are held fixed, the consumer’s utility is $$\sqrt q+k_iy$$ where three of the consumers have $k_i = 4$, four have $k_i = 3$, and three have $k_i = 2$. Each of these ten consumers has \$1,000 to spend.

(a) Assuming that the optimal solutions q and y to the consumer’s utility maximization problem are both strictly positive for all $k_i$, find the market demand function that the monopolist faces.
(b) Find the profit-maximizing price and quantity for the monopolist.