Cobb-Douglas Demand Functions

Why Cobb-Douglas Demand Is Useful

A Cobb-Douglas utility function gives a particularly clear link between preferences and spending. Each good receives a fixed share of the consumer’s budget, determined by its utility exponent. Once that share is known, the Marshallian demand for a good—the quantity chosen to maximize utility at given prices and income—follows directly.

The result is useful both for calculation and interpretation. It predicts how a consumer’s bundle changes when income or a price changes, while also providing a straightforward way to check whether a proposed bundle fits the budget and satisfies the consumer’s preferences.

Definition: A two-good Cobb-Douglas utility function has the form $U(x,y)=x^\alpha y^\beta$, where $x$ and $y$ are quantities of two goods and $\alpha>0$, $\beta>0$. The consumer chooses $x,y\geq 0$ to maximize utility subject to $p_xx+p_yy\leq m$, where $p_x>0$ and $p_y>0$ are prices and $m>0$ is income. Goods are divisible, and there are no constraints beyond nonnegative quantities and the budget.

Because both exponents are positive, a bundle with positive quantities of both goods gives greater utility than a bundle that uses the entire budget on just one good. With positive prices and more utility always preferred to less, the optimal bundle is interior and the budget binds. Thus the consumer spends all income: $p_xx+p_yy=m$.

Deriving the Marshallian Demand Functions

A convenient method is to maximize the logarithm of utility. Since the natural logarithm is strictly increasing, maximizing $U(x,y)$ also maximizes $\ln U(x,y)$. For positive quantities, $\ln U(x,y)=\alpha\ln x+\beta\ln y$. This transformation gives simpler first-order conditions without changing the optimal bundle.

Write the transformed objective.
Maximize $\alpha\ln x+\beta\ln y$ subject to $p_xx+p_yy=m$. The logarithmic objective is defined for positive quantities, which is appropriate because the optimum is interior under the stated assumptions.

Set up the Lagrangian and first-order conditions.
Let $\mu$ be the multiplier on the budget constraint. The Lagrangian is $\mathcal{L}=\alpha\ln x+\beta\ln y+\mu(m-p_xx-p_yy)$. Differentiating with respect to $x$, $y$, and $\mu$ gives the conditions below.

Convert the conditions into spending amounts.
Multiplying the first two conditions by their respective quantities gives $p_xx=\alpha/\mu$ and $p_yy=\beta/\mu$. Their sum must equal income.

Solve for the multiplier and quantities.
Adding the spending amounts yields $m=(\alpha+\beta)/\mu$, so $\mu=(\alpha+\beta)/m$. Substitution gives both demands.

\begin{aligned} \mathcal{L}&=\alpha\ln x+\beta\ln y+\mu(m-p_xx-p_yy),\\ \frac{\partial\mathcal{L}}{\partial x}&=\frac{\alpha}{x}-\mu p_x=0, &\frac{\partial\mathcal{L}}{\partial y}&=\frac{\beta}{y}-\mu p_y=0,\\ p_xx&=\frac{\alpha}{\mu}, &p_yy&=\frac{\beta}{\mu},\\ m&=p_xx+p_yy=\frac{\alpha+\beta}{\mu}, &\mu&=\frac{\alpha+\beta}{m}. \end{aligned}

Theorem: Under the Cobb-Douglas preferences and budget assumptions above, the Marshallian demand functions are

x^*=\frac{\alpha}{\alpha+\beta}\frac{m}{p_x}, \qquad y^*=\frac{\beta}{\alpha+\beta}\frac{m}{p_y}.

The consumer spends the fraction $\alpha/(\alpha+\beta)$ of income on good $x$ and the fraction $\beta/(\alpha+\beta)$ on good $y$. The exponents need not add to one; it is their relative sizes that determine the budget shares.

The derivation also establishes why these are the optimal quantities rather than merely a bundle that balances the budget. The logarithmic objective is strictly concave for positive quantities, and the budget set is convex. The first-order conditions therefore identify the unique maximum. The resulting quantities are positive and use the full budget, as required.

How Income and Prices Affect Demand

The demand equations make the comparative responses visible. Holding prices fixed, increasing income raises both quantities in direct proportion. Holding income and the other price fixed, an increase in a good’s own price reduces its quantity in inverse proportion. In this specific model, changing the other good’s price leaves the quantity unchanged: the consumer keeps the same expenditure share on the first good, and its price has not changed.

\begin{aligned} \frac{\partial x^*}{\partial m}&=\frac{\alpha}{(\alpha+\beta)p_x}>0, &\frac{\partial y^*}{\partial m}&=\frac{\beta}{(\alpha+\beta)p_y}>0,\\ \frac{\partial x^*}{\partial p_x}&=-\frac{\alpha m}{(\alpha+\beta)p_x^2}<0, &\frac{\partial y^*}{\partial p_y}&=-\frac{\beta m}{(\alpha+\beta)p_y^2}<0,\\ \frac{\partial x^*}{\partial p_y}&=0, &\frac{\partial y^*}{\partial p_x}&=0. \end{aligned}

These zero cross-price responses are a special implication of Cobb-Douglas preferences, not a general rule for consumer demand. They follow from the fixed spending shares: the price of $y$ changes how much $y$ the consumer can buy with its assigned share, but does not change the share assigned to $x$. The formula applies only when the stated Cobb-Douglas model and interior-choice assumptions hold.

Worked Applications

Worked Example: Finding the Bundle from Utility and a Budget

A consumer has utility $U(x,y)=x^2y^3$, income $m=\$600$, and prices $p_x=\$4$ and $p_y=\$5$. Find the utility-maximizing bundle and verify the budget and the relative marginal-utility condition. Method: use the logarithmic first-order conditions and the implied spending shares. This avoids trying to interpret a multiplier from the untransformed utility objective as though it came from the logarithmic objective.

The exponents sum to $2+3=5$. The spending shares are $2/5$ for good $x$ and $3/5$ for good $y$. Therefore, spending on $x$ is $(2/5)(\$600)=\$240$, and spending on $y$ is $(3/5)(\$600)=\$360$.

Divide each spending amount by its price: $x^*=\$240/(\$4\text{ per unit})=60$ units, and $y^*=\$360/(\$5\text{ per unit})=72$ units.

The budget check is $p_xx^*+p_yy^*=(\$4)(60)+(\$5)(72)=\$240+\$360=\$600$, exactly the available income.

For a further check, the marginal utilities are $MU_x=2xy^3$ and $MU_y=3x^2y^2$. At the proposed bundle, $MU_x/MU_y=2y/(3x)=(2\cdot72)/(3\cdot60)=144/180=0.8$. The price ratio is $p_x/p_y=4/5=0.8$, so the interior tangency condition is satisfied. The bundle also spends all income and has positive quantities, as the model requires.

Worked Example: Separating Income and Price Changes

A consumer has $U(x,y)=xy^3$, income $m=\$960$, and prices $p_x=\$8$ and $p_y=\$4$. Find the initial bundle. Then find the bundle under each separate change: (a) income rises to $\$1{,}200$; (b) the price of $x$ falls to $\$6$; (c) the price of $y$ falls to $\$3$. In each case not named as changing, retain its initial value. Method: apply the demand equations to each scenario separately, avoiding the mistake of changing more than one input at a time.

Here $\alpha=1$, $\beta=3$, and $\alpha+\beta=4$, so the spending shares are $1/4$ and $3/4$. Initially the consumer spends $(1/4)(\$960)=\$240$ on $x$ and $(3/4)(\$960)=\$720$ on $y$. Thus $x^*=\$240/\$8=30$ and $y^*=\$720/\$4=180$. The check is $(\$8)(30)+(\$4)(180)=\$240+\$720=\$960$.

(a) Higher income. At income $\$1{,}200$, spending on $x$ is $(1/4)(\$1{,}200)=\$300$, and spending on $y$ is $(3/4)(\$1{,}200)=\$900$. At the unchanged prices, $x^*=\$300/\$8=37.5$ and $y^*=\$900/\$4=225$. The budget check is $(\$8)(37.5)+(\$4)(225)=\$300+\$900=\$1{,}200$. Each quantity is $1{,}200/960=1.25$ times its initial value.

(b) Lower own price for $x$. Return to the initial income. The expenditure shares remain $\$240$ and $\$720$. At $p_x=\$6$, $x^*=\$240/\$6=40$, while $y^*=\$720/\$4=180$. The budget check is $(\$6)(40)+(\$4)(180)=\$240+\$720=\$960$. The lower own price increases $x$, while $y$ is unchanged.

(c) Lower price of the other good. Again return to the initial income and $p_x$. At $p_y=\$3$, $y^*=\$720/\$3=240$, and $x^*=\$240/\$8=30$. The check is $(\$8)(30)+(\$3)(240)=\$240+\$720=\$960$. The fall in $p_y$ raises the quantity of $y$, but leaves the quantity of $x$ unchanged in this model.

Worked Example: A Simultaneous Change in Both Prices

A consumer with utility $U(x,y)=x^4y^6$ has income $m=\$2{,}100$, $p_x=\$7$, and $p_y=\$10$. Both prices then change: $p_x$ rises to $\$14$, while $p_y$ falls to $\$5$. Find the initial and new bundles, and state which quantity changes and by how much. Method: calculate the fixed budget shares first, then apply the new prices to those shares. This avoids assuming that a change in one good’s price reallocates spending between goods.

The exponents sum to $4+6=10$, so the expenditure shares are $4/10=0.4$ for $x$ and $6/10=0.6$ for $y$. Initially, expenditure is $0.4(\$2{,}100)=\$840$ on $x$ and $0.6(\$2{,}100)=\$1{,}260$ on $y$. The quantities are $x^*=\$840/\$7=120$ and $y^*=\$1{,}260/\$10=126$. The budget check is $(\$7)(120)+(\$10)(126)=\$840+\$1{,}260=\$2{,}100$.

After the price changes, income and the budget shares are unchanged. The new quantity of $x$ is $x^*=\$840/\$14=60$, and the new quantity of $y$ is $y^*=\$1{,}260/\$5=252$. The new budget check is $(\$14)(60)+(\$5)(252)=\$840+\$1{,}260=\$2{,}100$.

The quantity of $x$ falls from 120 to 60 because its price doubles while its assigned expenditure stays at $\$840$. The quantity of $y$ rises from 126 to 252 because its price halves while its assigned expenditure stays at $\$1{,}260$. Both changes are consistent with the demand functions.

Common Mistakes and Model Limits

A frequent multiplier error arises from applying first-order conditions to the utility function and then dividing by utility. If the original Lagrangian is $U(x,y)+\lambda(m-p_xx-p_yy)$, its first-order condition for $x$ is $MU_x=\lambda p_x$. Dividing by $U$ gives $MU_x/U=(\lambda/U)p_x$, not an equation with the same $\lambda$ on the right. The multiplier in the logarithmic-objective derivation is $\mu$; it is not automatically the multiplier $\lambda$ from the utility-objective Lagrangian. The safe approach is to use one objective and its matching first-order conditions throughout.

Another common error is to treat $\alpha$ and $\beta$ as the spending fractions even when they do not sum to one. The correct shares are $\alpha/(\alpha+\beta)$ and $\beta/(\alpha+\beta)$. For example, exponents 2 and 3 imply shares $2/5$ and $3/5$, not 2 and 3 times income.

Finally, the no-cross-price response and fixed spending shares depend on the Cobb-Douglas form. They should not be assumed for other preferences. If an exponent is zero, or a good is unavailable at a positive price, the interior logarithmic derivation does not apply as written; the assumptions used here require both exponents and both prices to be positive.

Key takeaway: For $U(x,y)=x^\alpha y^\beta$, allocate the fractions $\alpha/(\alpha+\beta)$ and $\beta/(\alpha+\beta)$ of income to the two goods, then divide each expenditure amount by its good’s price. This produces the Marshallian demands and makes their responses to income and prices transparent.

Check Your Understanding

For each problem, use the Cobb-Douglas demand functions and check the resulting spending against income.

A consumer has $U(x,y)=x^2y$, income $\$900$, and prices $p_x=\$6$, $p_y=\$3$. Find the optimal quantities and the expenditure on each good. (Answer: $x^*=100,\ y^*=100$; expenditures are $\$600$ and $\$300$.)
A consumer has $U(x,y)=x^3y^2$, income $\$1{,}000$, and prices $p_x=\$10$, $p_y=\$5$. Find the optimal bundle and verify the total expenditure. (Answer: $x^*=60,\ y^*=80$; total expenditure is $\$1{,}000$.)
A consumer has $U(x,y)=xy^2$, income $\$600$, and prices $p_x=\$5$, $p_y=\$10$. If only $p_y$ falls to $\$5$, find the new bundle and identify which quantity is unchanged. (Answer: $x^*=40,\ y^*=80$; $x^*$ is unchanged.)
A consumer has $U(x,y)=x^3y$, income $\$720$, and prices $p_x=\$9$, $p_y=\$6$. Income rises to $\$900$ with prices unchanged. Find the initial and new bundles. (Answer: initially $x^*=60,\ y^*=30$; afterward $x^*=75,\ y^*=37.5$.)
A consumer has $U(x,y)=x^2y^3$, income $\$1{,}500$, and prices $p_x=\$10$, $p_y=\$5$. Income rises to $\$1{,}800$ and $p_x$ rises to $\$12$, while $p_y$ stays fixed. Find the new bundle and verify its budget. (Answer: $x^*=60,\ y^*=216$; expenditure is $(\$12)(60)+(\$5)(216)=\$1{,}800$.)

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