Cobb-Douglas Consumer Choice Under A Budget

Introduction

A consumer choosing between two goods faces a basic trade-off: spending more on one good leaves less available for the other. A Cobb-Douglas utility function represents preferences in which each good contributes to utility, while prices and income limit the available choices.

The optimization problem has a precise solution. The consumer spends a fixed fraction of income on each good, and those fractions depend on the utility exponents. Prices then determine the quantities purchased. The derivation matters: the exponents give spending shares, not quantities.

The model and its assumptions

Let $x$ and $y$ be the quantities of two goods. Assume their prices are $p_x>0$ and $p_y>0$, and the consumer has income $m>0$. A feasible choice is any pair of nonnegative quantities that does not exceed the budget. Let $\alpha>0$ and $\beta>0$ be fixed exponents.

Definition: A two-good Cobb-Douglas utility function is $U(x,y)=x^\alpha y^\beta$, where $\alpha,\beta>0$. The consumer-choice problem is to maximize $U(x,y)$ subject to $p_xx+p_yy\le m$ and $x,y\ge0$.

Positive prices and finite income make the feasible set bounded: $0\le x\le m/p_x$ and $0\le y\le m/p_y$. The utility function is continuous on this set, so a maximum exists. Since the exponents are positive, utility is zero if either quantity is zero. There are feasible choices with both quantities positive and hence positive utility, so a maximizing choice must have $x>0$ and $y>0$.

At a choice with positive quantities, any unused income could be spent on a little more of either good, increasing utility. Therefore the budget must bind at the maximum. The problem reduces to maximizing utility among choices satisfying $p_xx+p_yy=m$.

Deriving the optimal expenditures

On positive quantities, maximizing $U$ is equivalent to maximizing its natural logarithm, because the logarithm is strictly increasing. The log utility is $\alpha\ln x+\beta\ln y$. Use a Lagrange multiplier $\lambda$ for the binding budget constraint.

\mathcal{L}(x,y,\lambda) =\alpha\ln x+\beta\ln y +\lambda(m-p_xx-p_yy).

The first-order conditions are

\frac{\alpha}{x}=\lambda p_x, \qquad \frac{\beta}{y}=\lambda p_y, \qquad p_xx+p_yy=m.

Multiplying the first equation by $x$ and the second by $y$ gives $\alpha=\lambda p_xx$ and $\beta=\lambda p_yy$. Adding these equations and using the budget condition yields $\alpha+\beta=\lambda m$. Thus $\lambda=(\alpha+\beta)/m$. Substituting back gives the quantities and expenditures.

Theorem: If $p_x,p_y,m,\alpha,\beta>0$, the unique utility-maximizing choice is $$ x^*=\frac{\alpha}{\alpha+\beta}\frac{m}{p_x}, \qquad y^*=\frac{\beta}{\alpha+\beta}\frac{m}{p_y}. $$ The corresponding expenditures are $$ p_xx^*=\frac{\alpha}{\alpha+\beta}m, \qquad p_yy^*=\frac{\beta}{\alpha+\beta}m. $$

The expenditure formulas show the role of the exponents directly. The consumer spends the fraction $\alpha/(\alpha+\beta)$ of income on the first good and $\beta/(\alpha+\beta)$ on the second. These fractions are positive and add to one.

Why this solution is the global maximum

The first-order conditions identify a candidate. To confirm it is the unique global maximum, consider log utility on the positive quadrant. Its Hessian matrix is

\begin{pmatrix} -\alpha/x^2 & 0\\ 0 & -\beta/y^2 \end{pmatrix}.

For any nonzero vector $(u,v)$, the quadratic form for this matrix is $-\alpha u^2/x^2-\beta v^2/y^2<0$. Therefore log utility is strictly concave. A strictly concave function on the positive part of the budget set can have at most one maximizing point, and a point satisfying the first-order conditions is its global maximum there. Choices on the boundary have zero utility, while the candidate has positive utility. Hence the candidate is also the unique maximum over the entire feasible set.

The spending-share result can also be understood by writing expenditures as $e_x=p_xx$ and $e_y=p_yy$. Then $e_x+e_y=m$, and utility becomes $e_x^\alpha e_y^\beta/(p_x^\alpha p_y^\beta)$. Prices affect the conversion from expenditure to quantities, but the optimal shares of expenditure come from the exponents.

Worked examples

Worked Example: Equal quantities from unequal prices

Suppose $U(x,y)=x^2y^3$, the prices are $p_x=4$ and $p_y=6$, and income is $m=240$. Here $\alpha+\beta=2+3=5$.

The optimal expenditures are

e_x=\frac{2}{5}(240)=96, \qquad e_y=\frac{3}{5}(240)=144.

Divide each expenditure by its price to obtain the quantities:

x^*=\frac{96}{4}=24, \qquad y^*=\frac{144}{6}=24.

Check the budget: $4(24)+6(24)=96+144=240$. The utility is $24^2\cdot24^3=24^5=7{,}962{,}624$. For a marginal check, $U_x=2xy^3$ and $U_y=3x^2y^2$. At $(24,24)$, $U_x=663{,}552$ and $U_y=995{,}328$, so $U_x/p_x=165{,}888=U_y/p_y$. The marginal utility per dollar is equal across the goods, as required at this interior optimum.

Worked Example: Fractional exponents and different prices

Let $U(x,y)=x^{3/2}y^{1/2}$, with prices $p_x=8$, $p_y=6$, and income $m=240$. The sum of the exponents is $3/2+1/2=2$, so the spending shares are $3/4$ and $1/4$.

Compute the expenditures:

e_x=\frac{3}{4}(240)=180, \qquad e_y=\frac{1}{4}(240)=60.

Convert them to quantities:

x^*=\frac{180}{8}=\frac{45}{2}=22.5, \qquad y^*=\frac{60}{6}=10.

The budget check is $8(22.5)+6(10)=180+60=240$. To check the marginal condition, note that $U_x=\frac{3}{2}U/x$ and $U_y=\frac{1}{2}U/y$. Thus $U_x/p_x=(3/2)U/(p_xx)=(3/2)U/180=U/120$, while $U_y/p_y=(1/2)U/(p_yy)=(1/2)U/60=U/120$. The two marginal utilities per dollar agree.

Worked Example: A one-to-two exponent ratio

Suppose $U(x,y)=xy^2$, with $p_x=3$, $p_y=4$, and $m=180$. The exponents sum to $1+2=3$, so the consumer spends one-third of income on $x$ and two-thirds on $y$.

The expenditures are

e_x=\frac{1}{3}(180)=60, \qquad e_y=\frac{2}{3}(180)=120.

The chosen quantities are

x^*=\frac{60}{3}=20, \qquad y^*=\frac{120}{4}=30.

Check the budget: $3(20)+4(30)=60+120=180$. The utility at this choice is $20(30)^2=20(900)=18{,}000$. Also $U_x=y^2=900$ and $U_y=2xy=2(20)(30)=1{,}200$, so $U_x/p_x=900/3=300$ and $U_y/p_y=1{,}200/4=300$. Both the budget and marginal conditions are satisfied.

A common mistake: confusing shares with quantities

A frequent error is to treat the exponents themselves as fractions of income. For example, with $U=x^2y^3$ and income $m=240$, an incorrect calculation might assign expenditures $2(240)=480$ to the first good and $3(240)=720$ to the second. This spends $480+720=1{,}200$, which exceeds the available income.

The exponents must first be divided by their sum. The correct expenditures are $(2/5)(240)=96$ and $(3/5)(240)=144$, which add to $240$. Another error is to call these expenditure shares the quantities themselves. Quantities require a further division by the relevant price: $x^*=e_x/p_x$ and $y^*=e_y/p_y$.

Key takeaway: Exponents determine fractions of income; prices convert those expenditures into quantities. Always check that the expenditures add to income and that each quantity equals its expenditure divided by its price.

When the formulas apply

The formulas above require positive prices, positive income, and strictly positive exponents. If income is zero, the only feasible choice is $x=y=0$. If an exponent is zero, utility does not reward consumption of that good, so the positive-exponent result no longer applies in the same way. The stated model also assumes the consumer can choose any nonnegative quantities and has no additional constraints such as minimum purchase amounts.

For the standard positive-parameter model, the calculation follows a reliable sequence: find the exponent shares, allocate income, and divide each expenditure by its price. This separates the preference information in the exponents from the quantity information in prices.

Check Your Understanding

Use the assumptions and formulas developed above to answer these questions.

For $U(x,y)=x^4y$, what fractions of income are spent on the two goods?
If the first good receives expenditure $e_x$ and has price $p_x$, how do you obtain its chosen quantity?
For $U(x,y)=x^2y^3$, prices $p_x=5$, $p_y=10$, and income $m=300$, what are the optimal expenditures and quantities?
Why must an optimum have a binding budget when both utility exponents are positive?
What does equality of marginal utility per dollar express at an interior optimum?