Production functions and the shape of cost curves

• The production function determines the shape of a firm’s cost curves.

• Diminishing marginal return to labor: If a firm keeps increasing an input, holding all other inputs and technology constant, the corresponding increase in output will become smaller eventually.

Shape of the marginal cost curve

• Shape of the marginal cost curve:– Marginal cost is the change in variable cost as

output increase by one unit:– In the short run, capital is fixed, so the only

way the firm can produce more output is to use extra labor.

– The extra labor required to produce one more output is .

– The extra labor costs the firm per unit.– As a result, the firm’s marginal cost is

VCMCq

Lq

L

VC LMCq q MP

Shape of the average cost curve

• Diminishing marginal returns to labor, by determining shape of the variable cost curve, also determine the shape of the average variable cost curve.

• The average variable cost is .• Only variable input is labor, variable

cost is , so the average variable cost is

• Because the average product of labor is q/L, average variable cost is wage divided by the average product of labor

/AVC VC q

LVC wLAVCq q

L

AVCAP

Long-Run Costs• In the long run, the fixed costs are avoidable: All inputs can be varied in the long run, no long-run fixed cost.• Input choice

– A firm wants to choose particular bundle with the lowest cost of production, which is economically efficient.• Isocost line: All the combinations of inputs that require the same total expenditure. The cost of producing a given level of output depends on the price of labor and capital. • C=wL+rK

– The firm hires L hours of labor at a wage of w per hour, so its labor cost is wL.– The firm rents K hours of machinery at a rental rate of r per hour, so its capital cost is rK.

Bundles of Labor and Capital That Cost the Firm $100

A Family of Isocost Lines

Isocost line

• By substituting , w=$5, and r=$10, we find that the $100 isocost line is K=10-1/2L1. Intersects capital axis at K==10 units and labor

axis at L=20. 2. Isocosts further from the origin have higher costs3. The slope of each isocost line is the same.

C wL rK

C wK Lr r

$100C

wK Lr

Combining cost and production

• By combining the information about costs constraint in the isocost line and production summarized by isoquant, we examine how a firm chooses lowest cost way to produce a given level of output.– Lowest-isocost rule: Pick the bundle of

inputs where the lowest isocost line touches isoquant.

– Tangency rule: Pick the bundle of inputs where the isoquant is tangent to the isocost.

Cost Minimization

Interpretation of tangency or cost minimization conditions

• At the point of tangency, the slope of the isoquant (MRTS) equals slope of the isocost– MRTS =change K/change L =-w/r

Change in Factor Price

Expansion Path and Long-Run Cost Curve

Expansion Path and Long-Run Cost Curve